COLLAPSE I
Editorial Introduction
Robin Mackay
A new publication is obliged to justify its demands on
the attention of a reader already besieged by countless
specialist journals. This justification cannot appeal to the
orgiastic logic of the filling of market ‘niches’. With
COLLAPSE we have set out to achieve quite the opposite,
aiming rather at a breadth and an openness found
wanting.
That academic philosophy courts overspecialisation
and a certain depression of the energy of thought is news
to no-one. Whilst not wishing to denigrate the necessary
and exacting work of scholarship, we wished to create a
publication which communicated the passion for thought,
and the excitement of thinking. The work we present
here is unapologetically (and not unnecessarily)
3
COLLAPSE I, ed. R. Mackay (Oxford: Urbanomic, September 2007)
ISBN 978-0-9553087-0-4
http://www.urbanomic.com
Collapse; Philosophical Research and Development Issue #1; Numerical Materialism
Robin Mackay/Texts/Books/Editor/Collapse; Philosophical Research and Development/Collapse; Philosophical Research and Development Issue #1; Numerical Materialism.pdf
COLLAPSE I
demanding: we conceived COLLAPSE as providing a
home for conceptual work in progress, with all the rough
edges this might imply.
COLLAPSE is an experimental entity, in that it has no
fixed agenda, no institutional ties, no partisan position:
nevertheless it will be clear that neither the Editor nor the
contributors regard this experimental status as the
declaration of an open season for ludic enthusiasms. But
as much as we did not wish merely to collate flights of
philosophical whimsy, we also sought to avoid grounding
ourselves in the ‘application’ of overgeneralised
theoretical tropes to specific issues in pursuit of that most
dismal of goals, ‘relevance’ to the ‘contemporary’ reader.
What was clear from the start was that the way
forward lay in rigorously challenging philosophical
thought by confronting it with conceptual production
from elsewhere (and not in a presumptuous relation of
‘application’): there is no doubt that philosophy only
stays alive by maintaining porous boundaries with its outside—this does not detract at all from its specificity and
value as a discipline.
If part of the problem with philosophy today lies in its
professionalisation, we hope not to offend any of the
contributors to this volume by saying that we consider
every one of them to be amateurs in the true sense:
dedicated and enthusiastic lovers of abstract thought,
each engaged in adventures of ideas, each refusing to
contain these adventures within strict formal or
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Editorial Introduction
disciplinary boundaries. It should go without saying that,
even if the two are rarely found in pure form, we favour
de jure mad scientists with their bubbling
conceptual cauldrons over career academics with their
meticulously cautious conference papers.
Finally, we envisage each volume of COLLAPSE as the
intersection, in a multi-dimensional space, of diverse lines
of thought; the optimum circumstance would be if each
reader picked up COLLAPSE on the strength of only one
of the articles therein, the others being involuntarily
absorbed as a kind of side-effect that would propogate the
eccentric conjuncture by stealth, and spawn yet others.
A note on form: in several of the articles collected in
COLLAPSE I we have used the interview form, and hope
to do so in future. At its best it offers a medium in which
the play of concepts takes a natural course and order of
exposition, allowing the reader in turn to be swept up by
its movement, and to reconstruct its underlying
consistency at their leisure and with their own resources.
Such an interview should read lightly whilst its
spontaneous discursive complexity will be conducive to
repeated readings.
***
Although we decided early on that eclecticism was
neither to be scorned nor feared, a theme emerged,
unbidden, during the long process of clarifying the
journal’s aims and collecting and commisioning work for
this first volume. The working title was thus ‘numerical
5
COLLAPSE I
materialism’: an inquiry into the extent and nature of
number’s dominion over any philosophy calling itself a
materialism; but also an inquiry into the materiality of
number and numerical practices. Such a theme describes
a realm exemplary in its liminal nature, not only
connecting mathematics and philosophy but abutting
onto every theoretical discourse with any aspiration to
formal rationality.
We are privileged to have amongst our contributors
ALAIN BADIOU, widely recognised as one of the most
important philosophers alive today, and one who has
consistently pursued this line of thought, often against
the grain of philosophical orthodoxy. In our interview
with Badiou we seek to clarify the consequences of a
materialist thought which defines ‘nature’ through
mathematics, determining ‘history’ in terms of its
problematic ‘outside’. We also succeed in provoking
Badiou into addressing in detail some of the major
objections raised against his doctrine, and into
elaborating more precisely how he sees mathematics in its
relation to other sciences. He also speaks of the new
conceptual resources which his latest book Logiques des
mondes brings to his ongoing work.
Although we have made clear our aspiration for the
volume as a whole, this opening interview, together with
the contributions of G REGORY CHAITIN and MATTHEW
WATKINS particularly demonstrate the sort of conjunction we hope to effect with COLLAPSE: it would be
difficult to find any other publication in which these
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Editorial Introduction
could be found together. Here are three thinkers whose
differences are not limited to their theoretical decisions
but extend to their styles, and their respective
conceptions of the nature of their subject. Despite this
divergence, a number of common threads means that
reading them together produces a combination that is
more than the sum of its—already considerable—parts.
To say that G REGORY CHAITIN is well-known in his
field would be to tell only half the truth: in fact Chaitin
has for many years been something of a gleeful maverick.
Whilst there can be no doubt as to his credentials as a
‘serious mathematician’ Chaitin not only enjoys
communicating his mathematical discoveries to a wide an
audience as possible, but he also never fails to draw from
them (sometimes to the chagrin of fellow mathematicians) the most general speculative conclusions—just as
did Leibniz, the very figure invoked in Chaitin’s whirlwind review of ‘Epistemology as Information Theory’.
Our interview with MATTHEW WATKINS is the very
portrait of a thinker who defies categorisation.
Watkins’s singular talent at explaining complex concepts
and his fluid, resolutely non-specialist speculative
exploration of their significance makes of this interview a
kind of conceptual cinematography, and made
conducting it a genuine pleasure. As the engineer of a
virtual agency which collects and catalyses material in an
area of research which has only recently begun to
condense, Watkins exemplifies well what was intended
by the provocative subtitling of COLLAPSE as a ‘Journal
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COLLAPSE I
of Philosophical Research and Development’: thought no
longer takes place in the head—if it ever did—but involves
the development of distributions and connections, a biotechno-cognitive rhizome.
Watkins also fiercely
advocates the virtues of a marginal position which allows
him the maximum space for reflection; as well as his
substantive work we discuss the experiences formative of
this conviction.
Badiou raises the question of the experiment:
scientists continually set parameters and contexts for
their experiments. Chaitin argues—and Watkins observations seem to support the thesis—that mathematics may
also be, or may become, an experimental discipline. In
N ICK LAND’s contribution, the meta-rational orthodoxies
of experimental practice are themselves put to the test.
Can the interrogation by an exotericised qabbalistics of
the lexicographical element in which we are inscribed be
denied the status of a science on the basis of utilitarian
criteria? A delirious alphanumeric arithmetic…
The liberal media, with its indefatigable good
intentions, has continually sought wholly to separate the
ideology of terrorism from the tenets of a (poorlydefined) ‘true’ Islam. Whilst leaving no doubt as to the
contingency of the circumstances, in his article for
COLLAPSE I REZA N EGARASTANI describes the mutation,
hybridisation and militarization of certain components of
Islamic thought. The mongrel nature of this terrible
conceptual assemblage makes it all the more remarkable
that Negarastani ultimately refers it to a mathematical
8
Editorial Introduction
model; a veritable mathesis of fear.
The psychoanalyst and philosopher Lacan famously
described his project as being that of a ‘mathematisation
of the unconscious’—a research programme that ended, if
not in madness, certainly in an obscurity which endures
to this day. In his ‘Mathematics of Intensity’ THOMAS
DUZER picks up some of the threads of this project,
inflecting it with an affirmative stance which militates
against much of the psychoanalytical inheritance.
N ICK BOSTROM heads an intriguing new research
initiative where philosophical thought is put to work on
issues formerly the quarry of inconsequential media
panics and politicians platitudes; issues of truly
unimaginable magnitude. Whilst such initiatives are no
doubt to be applauded, we were interested to explore in
our interview with Bostrom not only the work of his
Future of Humanity Institute but also to ask whether
philosophical thought must make compromises in order
to break its traditional academic bonds.
K EITH TILFORD’s graphical work draws on
philosophical debates in poststructuralism: his ‘crowd’
drawings evoking especially the ongoing debates over the
nature of multiplicity and individuation. ‘I NCOGNITUM’
not only relates some intriguing details of
numerical-cultural archaology, but has also compiled for
us a selection of source materials in the shape of ABJAD
diagrams. It was always our intention that COLLAPSE
should not be purely textual, which makes these last two
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COLLAPSE I
contributions are especially welcome.
It remains only to thank the contributors for their
work and their patience during the assembling of this
volume; and the reader for supporting this new venture.
In relating so expansively our aspirations for
COLLAPSE, we do not dare to hope, nor do we mean to
claim, that this first volume fulfills them all. At least the
experiment is now underway; the Editor welcomes your
responses and contributions for future volumes.
Robin Mackay,
Oxford, August 2006.
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COLLAPSE I
Philosophy, Sciences, Mathematics
Interview with Alain Badiou
Alain Badiou’s radical equation of ontology—the traditionally
philosophical ‘account of being’—with mathematics claims to free
philosophy for its contemporary task of interrogating the relation
between the being which mathematics describes, and the events
whose very ‘impossibility’ structures that being.
In the corollary distinction between the mere ‘veridicality’ of
knowledge and the hazardous revolutionary decisions which found
new truths, and in Badiou’s account of the work of founding and
remaining faithful to truth-procedures—in which humans finally
become subjects—are rightly discerned a truly novel configuration of
political thought.
However Badiou is also a penetrating and formidably
knowledgeable philosopher of science and mathematics; and his
meticulous dedication to following these disciplines’ own ‘truth
procedures’ has informed his work from the very beginning of his
philosophical career through to 2006’s Logiques des mondes1.
Collapse asks Badiou to expand on the articulation of
philosophy, mathematics and science his ‘mathematical ontology’
assumes; and what its consequences might be for the ‘other’ sciences.
1. Alain Badiou, L’être et l’événement, tome 2: Logiques des mondes. Paris:Seuil, 2006.
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C OLLAPSE: In an interview with Peter Hallward
published in 1998, you say:
In the last instance, physics, that is to say the theory of
matter, is mathematical. It is mathematical because,
qua theory of the most objectivised strata of the presented as such, it is concerned with being qua being in
its mathematicity.”2
If mathematics as science of being qua being provides
scientific access to the presented, does that mean that
mathematics provides the paradigm of ‘scientificity’?
And are we to understand that the degrees of objectification of what is presented (‘being’) correspond to degrees
of mathematisation of scientific discourse?
ALAIN BADIOU: It is undoubtedly more complicated
than that. We must begin again from the distinction,
which I have fully developed in Logiques des Mondes,
between being and being-there, or between being and
appearing. A world is structured not only by the pure
multiples which appear in it (which ‘are’ in it), but by the
logical organisation of that world, what I call its transcendental. So that every particular figure of what, in the
1998 interview, I called the ‘presented’, is the intersection
of two formal rationalities.
I propose to call
“mathematical” ontological rationality—that rationality
which concerns being qua being, that is, the indifferent
2. ‘Politics and Philosophy: An Interview with Alain Badiou’ Angelaki: Journal of
the Theoretical Humanities, Vol.2, No.3, 1998, p. 127.
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multiple. Phenomenological rationality—that which
concerns being qua appearing in a world—I call ‘logical’.
Of course, this logic is itself mathematicised, as is
every logic today. But the distinction remains, a
distinction whose criteria I have proposed in several of
the texts included in Court traité d’ontologie transitoire3. We
can say ultimately that a kind of objectification, a
becoming-object, of pure being is inscribed within a
singular correspondence between mathematics and
logic—which ultimately means: between set-theory and
Topos-theory. It being understood that these Topos realise
a transcendental indexation of multiples to a sort of
intensity evaluator, which in the current state of science,
might be a complete Heyting algebra—for which English
logicians have found the inspired name (since it
effectively concerns the ‘site’ of being) of ‘locale’.
So we cannot simply say that objectification is proportional to mathematicity. The new concept that I propose
of what an object is (‘object’ is a key concept in Logiques
des mondes) combines a purely mathematical element and
a transcendental element which relates to the singularity
of the world within which the object figures.
C: Readers of your work might be forgiven for
suspecting that you hold physics in far higher regard than
biology, that ‘wild empiricism disguised as a science’4.
3. Paris:Seuil, 1998. Translated by Norman Madarasz as Briefings on Existence: A
Short Treatise on Transitory Ontology. NY: State University Press, 2006.
4. Mathematics and Philosophy: The Grand Style and the Little Style’ Theoretical
Writings, eds. R. Brassier and A. Toscano, London: Continuum, 2004, p. 16.
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COLLAPSE I
There follow two further series of questions from this.
Firstly, concerning the relations between your ontological ‘Platonism’ (given that the latter effects an unexpected transformation of orthodox Platonism) and epistemological ‘foundationalism’ or ‘reductionism’.
Secondly,
the
relation
between
your
Platonic/Cartesian ‘rationalism’ and the ‘naturalism’ of
contemporary cognitive science which inquires into the
genesis of intelligence; an inquiry whose ultimate
ambition is perhaps the reinscription of rationality itself
(and hence of mathematical intelligence) within the
domain of natural being (if not biology stricto sensu).
(1) If you advocate a ‘foundationalist’ view on the
unity of ‘Science’, with the set-theoretical axiomatic at the
base, then mathematical physics, chemistry, biology, etc.
embedded one after another on successive strata
according to a decreasing order of ‘mathematicisation’, do
you think that this unity requires the reducibility of these
sciences either to physics, or even to mathematics in the
last instance?
And do you consider physical phenomena to be more
‘fundamental’ than biological phenomena because of
these decreasing degrees of ‘mathematicisability’? If so,
don’t you run the risk of making the coherence of science
depend ultimately upon the reducibility of
superstructures to an infrastructure, whether physical or
mathematical? As you know, this idea of reduction
remains highly problematic even within the natural
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sciences (for example the well-known difficulties concerning the reduction of ‘emergent’ properties).
Or is it rather that this ‘foundationalist’ and
‘reductionist’ perspective only holds for you within
mathematics, that is to say for the relations between settheory (or category-theory) and other mathematical
domains?
(2) In Being and Event 5 , you say that the thesis
affirming the identity of mathematics and ontology is ‘a
thesis not about the world, but about discourse.’6 No
‘Pythagorism’, then. On one hand, it might seem that this
thesis claims to undo all the false problems engendered
by the empiricist presupposition of a representational
relation between discourse, or science, and reality; but,
on the other, can we content ourselves merely with
‘sublating’ the difference between discourse and reality
within a discourse (which, in Being and Event, remains
‘metaontological’ or philosophical rather than
‘ontological’ in the strict sense) without falling into
absolute idealism?
How can we account for the position of mathematical
and scientific ‘discourse’ given that, for you, the latter is
not fundamentally linguistic, and so is not merely an
artefact of cultural provenance; nor a natural and/or
divine faculty, as intellectual intuition would be?
Don’t the Darwinian revolution and the emergence of
5. Alain Badiou, L’être et l’événement. Paris:Seuil, 1988. Translated by Oliver
Feltham as Being and Event. London: Continuum , 2005.
6. Being and Event, Introduction.
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cognitive science as a natural science treating reason as a
material phenomena—even if the ‘materiality’ in question
is perhaps not simply neurobiological—oblige us at least
to try to take account of this scientific and mathematical
‘discursivity’?
And in this regard, could it be that the ‘sciences of
complexity’ or ‘self-organisation’, which try to apply
mathematical modelling techniques to complex phenomena—vital, social, cultural, or even (above all) cognitive—
could reduce the gap that you posit in Being and Event
between the transitivity of natural sets, that is to say the
homogeneity of nature, and the discontinuity of cultural
situations, that is to say the heterogeneity of history?
Wouldn’t this ultimately be a way of assuring a
maximum coherence between sciences and the science of
being without the need for the impracticable ideal of
reduction? Or does such a perspective seem far too
Deleuzian-vitalist to you?
AB: First of all I would like to say that my perspective is
not in the least ‘foundationalist’ or reductionist. From the
ontological dimension of mathematics I draw no
conclusion as to its superiority, its capacity to found the
other sciences, or their status as the ‘basis’ for all
scientificity. I say only that mathematics are the rational
discourse on being qua being, or on the indifferent
multiple thought as such.
Now, as you know, for me, a concept as general and
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essential for all thinking of the sciences as that of truth is
not at all reducible to ontology as such. We must even
admit that it explicitly contradicts certain of its formal
axioms, just as Lucretius’s clinamen contradicts the major
axioms of atomist and materialist ontology.
In fact, I have often compared the absolute hazard of
the event, subtracted as it is from the general determinations of being, to this clinamen, required for the
assemblage of atoms to be thought as generic truth of
worlds.
From all this, it follows simply that mathematics is a
necessary formal dimension of all scientific discourse, if
we understand by ‘science’, for the moment, the rational
theory of those phenomena in the world which do not
depend directly upon the conscious activity of man.
Those phenomena pertaining, if you like, to the ‘fossil
argument’, whose implacable rigour Quentin Meillassoux
has deployed against every form of correlationism, or of
The
the constitutive primacy of consciousness.7
mathematical exigency is formal, in so far as it supports,
as to the intelligibility of these phenomena, their most
abstract and most general strata, that which relates to
their pure being, to their multiple composition. But,
precisely, this strata cannot, in my view, represent that
which is strongest and most ‘true’ in those sciences which
are not purely mathematical. Take physics: the most
elementary axioms of modern mathematical physics, let’s
7. See Quentin Meillassoux, Après la finitude : Essai sur la nécessité de la contingence.
Paris:Seuil, 2006
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COLLAPSE I
say the principle of inertia as given by Galileo’s formula,
are not at all reducible to mathematical statements, even
if they are obliged to take that form. A crucial point
about physics is to present, to create concepts, so that
they can be mathematically expressible, all the while
retaining a relation to the world which means that they
cannot be deduced from any mathematical corpus whatsoever. This is the case with the concept of uniform
movement in the principle of inertia. Moreover, it is this
irreducible worldly dimension which opens onto the
possibility of experience, at the same time as
mathematical formalisation guarantees the universality of
experimental results, in the form of their always-possible
repetition. All this, in my view, was thought in the most
subtle and decisive fashion by Bachelard, and it is a great
shame that anglophone epistemology has done such a
thorough job of neglecting his work.
So: I neither believe that physics is ‘reducible’ to
mathematics, nor do I believe that mathematics ‘founds’
physics. Between the two, there is a rooting of concepts
in a determinate world, which the experimental method
designates and delimits, in a gesture which is of a transcendental nature.
Nor do I have any reason to think that physics
‘founds’ biology, still less that mathematics could do so.
That biology, to say nothing of the ‘human sciences’, are
still not sciences, does not result from the fact that they
are not yet in a position to propose mathematical formalisms appropriate to experimentation. This formal
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deficiency is only a secondary symptom. The root of the
problem is that their concepts are wholly insufficient, that
they fail completely to present the phenomena concerned
in the register of eternal truths.
In the case of biology, the possible founding intuitions
all go back to Darwin. Mendel’s experimental expertise
opened another path, albeit one that is limited unless it is
inscribed in the vast Darwinian innovation, which no-one
to this day has succeeded in doing, unless it be in pre-conceptual approximations, purely rhetorical combinations
of chance and necessity. The veritable conceptualisation
of all this is yet to come. Naturally, there exist within this
domain countless instances of knowledge, which are
translated into remarkable technical skills, particularly in
the medical field. But there is no science. To turn this
domain into a science, some unforeseeable events would
be required, a second post-Darwinian foundation, whose
shape we cannot anticipate. In the meantime, we can say
that there exist two sciences which cannot be hierarchised: ontology, or pure mathematics, and physics, the
science of those worlds accessible to our experience.
As for cognitive science, I would like to make three
remarks. Firstly, it is a programme rather than a theory.
Even more so than biology, it is just a mass of facts and
techniques, devoid of concepts or adequate formalisms.
The truth is that we remain totally ignorant as to the real
functioning of the brain. From what I know of the
current state of research in cognitive science, I feel
justified in concluding that, despite an impressive
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technical arsenal, it is no more advanced in its understanding of the phenomena than was Gall’s phrenology,
at the time when Hegel said the latter thought it had
proved that ‘spirit is a bone’. Today we think we have
proved that thought is a neuron…Cognitive science is
hardly further advanced than the theory of cerebral localisations in Broca, even if the active zones appear today in
bright colours, displayed directly onto screens.
Secondly, that human intelligence should be a
‘material’ phenomenon is in my eyes a mere truism.
What else could it be? I am, if this is the issue, completely monist. I do not think that any principle of being can
‘double’ indifferent multiplicities. What’s more, I always
speak of the ‘human animal’ when speaking about us,
including even our most sophisticated cognitive activities.
And this leads me to the third point. Let me repeat:
the fact that intelligence, as a faculty, is a material assemblage is self-evident and of no interest. However, that the
question of truths relates to intelligence, or to the
‘cognitive’ in general, is a philosophical (not at all a
scientific) statement, and a statement I believe to be
completely false. Qua generic localised process, a truth is
not at all reducible to anything whatsoever which takes
the form of a human capacity or property, in the sense of
the human animal.
To claim that it does is to remain within the legacy of
Aristotelianism, Kantianism and analytic philosophy,
which would locate truth in the form of judgment, with
the latter in turn being a product of cognitive
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mechanisms. Whereas, as the other tradition saw—the
Platonic, Cartesian, Hegelian tradition—a truth cannot be
encapsulated in the form of judgment at all. It is a
complex process, completely transcendent to the mere
animal capacity of cognitive judgment. In my own
conception, this process combines, ontologically, the
possibility of generic multiplicities, and phenomenologically, the intraworldly constitution of post-evental bodies
(in fact, of sheaves, in the modern algebraic sense,
sheaves which ‘rise’ from the transcendental logic of
worlds towards the objective multiplicities engaged in the
procedure).
Since every truth is in-human, we can hardly hope to
understand its genesis by poking around in the neurons
of our brains!
I would like to add that to reduce thought to nature is
in general a tautology, or a contradiction. If you understand by ‘nature’ the material state of all that is, it is a tautology: thought is certainly a part of that state. If you
understand by ‘nature’ that which, precisely, pre-exists all
thought, then it is a contradictory reduction.
C: Finally, two questions bearing principally on your Le
Nombre et les nombres 8 . The distribution of prime numbers
appears to be one of the irrecusable ‘ontological’
characteristics arising immediately from the construction
of the natural numbers, but to this day it has proven
8. Alain Badiou, Le Nombre et les nombres. Paris:Seuil, 1990.
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COLLAPSE I
resistant to our understanding.
The Riemann
Hypothesis suggests that complex numbers provide a
refractory element through which it becomes possible to
understand the prime distribution as a type of order. If
we are to speak of ontology in this context, doesn’t this
oblige us to extend ontological dignity to complex
numbers? But when you speak of number, you explicitly exclude complex numbers, along with quaternions,
from the properly ontological domain9.
Historically, the affirmation of complex numbers as
numbers, with all the consequences which followed,
would seem indeed to be an example of a mathematical
‘event’. Yet you reject it on the basis of its supposed intimate relation with geometry.
Now, even if (as is undoubtedly the case) it is precisely the geometrical and diagrammatic aspect of the
complex plane which has inspired mathematicians and
physicists, it is possible to conceive them independently
of this usage. However, must we not consider the
implication of physics, mathematics and geometry (or
more generally, the diagram) from which the incontestable power of the complex plane arises?
It seems that the prime distribution is a profound
mystery concealed within the apparently simple: the
series of natural numbers. Alain Connes, among others,
suggest that the apparent obscurity of the Riemann
Hypothesis stems from as-yet uncomprehended aspects
of the relation between
simple addition and
9. Le Nombre et les nombres, Ch. XVI, n.5
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multiplication.10
Meanwhile, recent work indicates astonishing
consonances between the theoretical models of physics
and Riemann’s zeta function11. This seems to suggest,
again, that the relation between mathematics and physics
is not simply hierarchical, and furthermore, that the most
fundamental ordering mechanisms in the universe do not
necessarily coincide with our most analytically elegant
accounts of order.
So where does the ontological begin and end? If we
apply your criteria of ordinal linearity, don’t we risk a
premature foreclosure of ontology in the interests of
elegance and philosophical taste, whilst rejecting the
structures of the real—sometimes of an intimidating
complexity, refractory to all intuition, such as Connes’
adele—but which, from a point of view which we might
call abstract-empirical, are imposed upon us by
fundamental research in the domains of theoretical
physics and mathematics?
AB: We must immediately correct a misinterpretation,
perhaps owing to ambiguities in my writing, but no less
massive for that: I have never thought that numbers were
the highest form of ontological ‘dignity’! Even less have
I underestimated the mathematical—and thus ontological—importance of complex numbers, of diagrams, of
topology! Absolutely not! I even argued with Deleuze
10,11. See ‘Prime Evolution’, in the current volume.
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COLLAPSE I
on this point, in our unpublished correspondence. He
held that in ‘reducing’ multiplicities to sets, I reduced
them to numbers. I replied to him that if numbers are all,
in their being, sets, it would be quite wrong to conclude
that all sets are numbers. The innate resources of
set-theoretical multiplicity vastly exceed the concept of
number.
As for complex numbers, I simply meant that they are
exterior to the simple concept of number, which, in the
sense I use it, includes order, comparison, and the notion
of ‘larger’ and ‘smaller’, even when numbers are
infinitely large or small. The fact that the field of
complex numbers is not be an ordered field justifies this
exclusion. Far from indicating a lack of dignity in
complex numbers, this exclusion amounts to a further
limiting of the properly numerical domain, and hence of
further separating it from the ontological domain in its
full extension. That complex numbers are sets, and
hence thinkable as such in their being, is evident: it is a
question, as we know, of the set of ordered pairs of real
numbers.
I would like to add that when I say that the concept of
number cannot be geometrical I am in no way—heaven
forbid!—speaking against geometry! I have always indicated, from Théorie du sujet12 onward, that the mathematical
dialectic is that of algebra and topology, or, if one is
Greek, arithmetic and geometry. And I know very well
12. Alain Badiou, Théorie du sujet. Paris: Seuil, 1982.
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Badiou – Philosophy, Sciences, Mathematics
that today the most profound mathematics operates at the
level of algebraic topology, a structural combination of
the two great components of historical mathematics. And
in fact this is the register frow which my new concepts
(‘transcendental’, ‘object’, ‘relation’, ‘evental field’) draw
their force. The results you mention which clarify number theory on the basis of the geometry of the
complex plane or of highly sophisticated Analysis,
perfectly illustrate this point.
But I do not speak of the mathematics of numbers in
Le Nombre…, but only of what philosophy might draw
from contemporary mathematics as to what a generic
concept of Number might be.
Meanwhile, let us note that mathematicians always
prefer, in arithmetic, an ‘elementary’ demonstration to
one that is not. And what is an ‘elementary’ demonstration? An immanent demonstration, more or less directly
derivable from the concept of whole number, and the
rules of its operational domain.
In this regard, I agree with Connes: the heart of the
question of prime numbers is that the operational
dialectic of multiplication and addition, however
perfectly clear from the axiomatic point of view, has yet
to arrive at its true concept. There is there a deficiency
of thought on the algebraic side. That this opacity might
be overcome by the diagrammatic or geometrical aspect
is quite possible.
My own approach clearly shows that what is opaque
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COLLAPSE I
in numbers is not at all their definition, which, as I
believe I have demonstrated, is very simple, even if you
take the most gigantic numerical domain. What is
obscure lies on the side of operations. Now, the concept
of ‘prime number’ is a typically operational concept. To
which we must add that it is largely negative (to not be
divisible by any number other than itself or unity). I
would be happy to propose the following heuristic
principle: in pure mathematics, what is most difficult is
always linked to a restriction (and thus to a negation) of
operational capacities.
I will conclude by giving you a very simple account of
physics’ retroaction upon mathematics. There is nothing
unusual about this: presented with the mathematical
formalisation of concepts which only make sense in
relation to a world which is not that of pure mathematics
(concepts such as: ‘movement’, ‘speed’, ‘light’, and so on),
physics requires of mathematics a sort of internal torsion,
which introduces into its world (for it is a singular world)
pure conceptual analogies with that which, initially, does
not belong in this register. The clearest contemporary
example is undoubtedly provided by quantum group
algebra, a totally abstract and very exacting discipline,
which would neither have existed, nor found its name,
without quantum physics.
Translated by Robin Mackay and Ray Brassier
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COLLAPSE I
Epistemology as Information Theory:
From Leibniz to Ω
Gregory Chaitin
The following text was originally delivered as the Alan
Turing Lecture on Computing and Philosophy at E-CAP ’05, the
European Computing and Philosophy Conference, at Mälardalen
University, Västerås, Sweden in June 2005.
I NTRODUCTION
I am happy to be here with you enjoying the delicate
Scandinavian summer; if we were a little farther north
there wouldn’t be any darkness at all. And I am especially delighted to be here delivering the Alan Turing
Lecture. Turing’s famous 1936 paper is an intellectual
milestone that seems larger and more important with
every passing year. People are not merely content to
enjoy the beautiful summers in the far north, they also
27
COLLAPSE I, ed. R. Mackay (Oxford: Urbanomic, September 2007)
ISBN 978-0-9553087-0-4
http://www.urbanomic.com
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COLLAPSE I
want and need to understand, and so they create myths.
In this part of the world those myths involve Thor and
Odin and the other Norse gods. In this talk, I’m going to
present another myth, what the French call a système du
monde, a system of the world, a speculative metaphysics
based on information and the computer.1
The previous century had logical positivism and all
that emphasis on the philosophy of language, and completely shunned speculative metaphysics, but a number of
us think that it is time to start again. There is an emerging digital philosophy and digital physics, a new metaphysics associated with names like Edward Fredkin and
Stephen Wolfram and a handful of like-minded individuals, among whom I include myself. As far as I know the
terms “digital philosophy” and “digital physics” were
actually invented by Fredkin, and he has a large website
with his papers and a draft of a book about this. Stephen
Wolfram attracted a great deal of attention to the movement and stirred up quite a bit of controversy with his
very large and idiosyncratic book on A New Kind of Science.
And I have my own book on the subject, in which I’ve
attempted to wrap everything I know and care about into
a single package. It’s a small book, and amazingly enough
it’s going to be published by a major New York publish1.One reader’s reaction (GDC): “Grand unified theories may be like myths, but
surely there is a difference between scientific theory and any other narrative?” I
would argue that a scientific narrative is more successful than the Norse myths
because it explains what it explains more precisely and without having to postulate new gods all the time, i.e., it’s a better “compression” (which will be my main
point in this lecture; that’s how you measure how successful a theory is).
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Chaitin – From Leibniz to Ω
er a few months from now. This talk will be an overview
of my book, which presents my own personal version of
“digital philosophy,” since each of us who works in this
area has a different vision of this tentative, emerging
world view. My book is called Meta Math!, which may not
seem like a serious title, but it’s actually a book intended
for my professional colleagues as well as for the general
public, the high-level, intellectual, thinking public.
“Digital philosophy” is actually a neo-Pythagorean
vision of the world, it’s just a new version of that.
According to Pythagoras, all is number — and by number
he means the positive integers, 1, 2, 3, . . . — and God is
a mathematician. “Digital philosophy” updates this as follows: Now everything is made out of 0/1 bits, everything
is digital software, and God is a computer programmer,
not a mathematician! It will be interesting to see how well
this vision of the world succeeds, and just how much of
our experience and theorizing can be included or shoehorned within this new viewpoint.2
Let me return now to Turing’s famous 1936 paper.
This paper is usually remembered for inventing the programmable digital computer via a mathematical model,
the Turing machine, and for discovering the extremely
fundamental halting problem. Actually Turing’s paper is
2.Of course, a system of the world can only work by omitting everything that
doesn’t fit within its vision. The question is how much will fail to fit, and conversely, how many things will this vision be able to help us to understand. Remember,
if one is wearing rose-colored glasses, everything seems pink. And as Picasso said,
theories are lies that help us to see the truth. No theory is perfect, and it will be
interesting to see how far this digital vision of the world will be able to go.
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COLLAPSE I
called “On computable numbers, with an application to
the Entscheidungsproblem,” and by computable numbers
Turing means “real” numbers, numbers like e or
π = 3.1415926... that are measured with infinite precision, and that can be computed with arbitrarily high precision, digit by digit without ever stopping, on a computer.
Why do I think that Turing’s paper “On computable
numbers” is so important? Well, in my opinion it’s a
paper on epistemology, because we only understand
something if we can program it, as I will explain in more
detail later. And it’s a paper on physics, because what we
can actually compute depends on the laws of physics in
our particular universe and distinguishes it from other
possible universes. And it’s a paper on ontology, because
it shows that some real numbers are uncomputable,
which I shall argue calls into question their very existence, their mathematical and physical existence.3
To show how strange uncomputable real numbers can
be, let me give a particularly illuminating example of one,
3.You might exclaim (GDC), “You can’t be saying that before Turing and the computer no one understood anything; that can’t be right!” My response to this is that
before Turing (and my theory) people could understand things, but they couldn’t
measure how well they understood them. Now you can measure that, in terms of
the degree of compression that is achieved. I will explain this later at the beginning
of the section on computer epistemology. Furthermore, programming something
forces you to understand it better, it forces you to really understand it, since you
are explaining it to a machine. That’s sort of what happens when a student or a
small child asks you what at first you take to be a stupid question, and then you
realize that this question has in fact done you the favor of forcing you to formulate your ideas more clearly and perhaps even question some of your tacit assumptions.
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Chaitin – From Leibniz to Ω
which actually preceded Turing’s 1936 paper. It’s a very
strange number that was invented in a 1927 paper by the
French mathematician Emile Borel. Borel’s number is
sort of an anticipation, a partial anticipation, of Turing’s
1936 paper, but that’s only something that one can realize in retrospect. Borel presages Turing, which does not in
any way lessen Turing’s important contribution that so
dramatically and sharply clarified all these vague ideas4.
Borel was interested in “constructive” mathematics, in
what you can actually compute we would say nowadays.
And he came up with an extremely strange non-constructive real number. You list all possible yes/no questions in
French in an immense, an infinite list of all possibilities.
This will be what mathematicians call a denumerable or
a countable infinity of questions, because it can be put
into a one-to-one correspondence with the list of positive
integers 1, 2, 3, ... In other words, there will be a first
question, a second question, a third question, and in general an Nth question.
You can imagine all the possible questions to be
ordered by size, and within questions of the same size, in
alphabetical order. More precisely, you consider all possible strings, all possible finite sequences of symbols in the
French alphabet, including the blank so that you get
words, and the period so that you have sentences. And
you imagine filtering out all the garbage and being left
only with grammatical yes/no questions in French. Later
4.I learnt of Borel’s number by reading Tasic’s Mathematics and the Roots of Postmodern
Thought, which also deals with many of the issues discussed here.
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I will tell you in more detail how to actually do this.
Anyway, for now imagine doing this, and so there will be
a first question, a second question, an Nth question.
And the Nth digit or the Nth bit after the decimal
point of Borel’s number answers the Nth question: It will
be a 0 if the answer is no, and it’ll be a 1 if the answer is
yes. So the binary expansion of Borel’s number contains
the answer to every possible yes/no question! It’s like
having an oracle, a Delphic oracle that will answer every
yes/no question!
How is this possible?! Well, according to Borel, it isn’t
really possible, this can’t be, it’s totally unbelievable. This
number is only a mathematical fantasy, it’s not for real, it
cannot claim a legitimate place in our ontology. Later I’ll
show you a modern version of Borel’s number, my halting probability Ω. And I’ll tell you why some contemporary physicists, real physicists, not mavericks, are moving
in the direction of digital physics.
[Actually, to make Borel’s number as real as possible,
you have to avoid the problem of filtering out all the
yes/no questions. And you have to use decimal digits,
you can’t use binary digits. You number all the possible
finite strings of French symbols including blanks and
periods, which is quite easy to do using a computer. Then
the Nth digit of Borel’s number is 0 if the Nth string of
characters in French is ungrammatical and not proper
French, it’s 1 if it’s grammatical, but not a yes/no question, it’s 2 if it’s a yes/no question that cannot be
answered (e.g., “Is the answer to this question ‘no’ ?”), it’s
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3 if the answer is no, and it’s 4 if the answer is yes.]
Geometrically a real number is the most straightforward
thing in the world, it’s just a point on a line. That’s quite
natural and intuitive. But arithmetically, that’s another matter. The situation is quite different. From an arithmetical
point of view reals are extremely problematical, they are
fraught with difficulties!
Before discussing my Ω number, I want to return to
the fundamental question of what does it mean to understand. How do we explain or comprehend something?
What is a theory? How can we tell whether or not it’s a
successful theory? How can we measure how successful
it is? Well, using the ideas of information and computation, that’s not difficult to do, and the central idea can
even be traced back to Leibniz’s 1686 Discours de métaphysique.
COMPUTER E PISTEMOLOGY: WHAT IS A MATHEMATICAL OR SCIENTIFIC THEORY? HOW CAN WE JUDGE
WHETHER IT WORKS OR NOT?
In Sections V and VI of his Discourse on Metaphysics,
Leibniz asserts that God simultaneously maximizes the
variety, diversity and richness of the world, and minimizes the conceptual complexity of the set of ideas that
determine the world. And he points out that for any finite
set of points there is always a mathematical equation that
goes through them, in other words, a law that determines
their positions. But if the points are chosen at random,
that equation will be extremely complex.
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This theme is taken up again in 1932 by Hermann
Weyl in his book The Open World consisting of three lectures he gave at Yale University on the metaphysics of
modern science. Weyl formulates Leibniz’s crucial idea in
5 the following extremely dramatic fashion: If one permits arbitrarily complex laws, then the concept of law
becomes vacuous, because there is always a law! Then
Weyl asks, how can we make more precise the distinction
between mathematical simplicity and mathematical complexity? It seems to be very hard to do that. How can we
measure this important parameter, without which it is
impossible to distinguish between a successful theory and
one that is completely unsuccessful?
This problem is taken up and I think satisfactorily
resolved in the new mathematical theory I call algorithmic
information theory. The epistemological model that is central to this theory is that a scientific or mathematical theory is a computer program for calculating the facts, and
the smaller the program, the better. The complexity of
your theory, of your law, is measured in bits of software:
program (bit string) → Computer → output (bit string)
theory → Computer → mathematical or scientific facts
Understanding is compression!
Now Leibniz’s crucial observation can be formulated
much more precisely. For any finite set of scientific or
mathematical facts, there is always a theory that is exactly as complicated, exactly the same size in bits, as the facts
themselves. (It just directly outputs them “as is,” without
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doing any computation.) But that doesn’t count, that
doesn’t enable us to distinguish between what can be
comprehended and what cannot, because there is always
a theory that is as complicated as what it explains. A theory, an explanation, is only successful to the extent to
which it compresses the number of bits in the facts into a
much smaller number of bits of theory. Understanding is
compression, comprehension is compression! That’s how
we can tell the difference between real theories and ad hoc
theories5.
What can we do with this idea that an explanation has
to be simpler than what it explains? Well, the most important application of these ideas that I have been able to
find is in metamathematics, it’s in discussing what mathematics can or cannot achieve. You simultaneously get an
information-theoretic, computational perspective on
Gödel’s famous 1931 incompleteness theorem, and on
Turing’s famous 1936 halting problem. How?6
Here’s how! These are my two favorite informationtheoretic incompleteness results:
• You need an N -bit theory in order to be able to
prove that a specific N-bit program is “elegant.”
• You need an N-bit theory in order to be able to determine N bits of the numerical value, of the base-two
5 By the way, Leibniz also mentions complexity in Section 7 of his Principles of
Nature and Grace, where he asks the amazing question, “Why is there something
rather than nothing? For nothing is simpler and easier than something.”
6.For an insightful treatment of Gödel as a philosopher, see Rebecca Goldstein’s
Incompleteness.
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COLLAPSE I
binary expansion, of the halting probability Ω.
Let me explain.
What is an elegant program? It’s a program with the
property that no program written in the same programming language that produces the same output is smaller
than it is. In other words, an elegant program is the most
concise, the simplest, the best theory for its output. And
there are infinitely many such programs, they can be arbitrarily big, because for any computational task there has
to be at least one elegant program. (There may be several if there are ties, if there are several programs for the
same output that have exactly the minimum possible
number of bits.)
And what is the halting probability Ω? Well, it’s
defined to be the probability that a computer program
generated at random, by choosing each of its bits using an
independent toss of a fair coin, will eventually halt.
Turing is interested in whether or not individual programs halt. I am interested in trying to prove what are the
bits, what is the numerical value, of the halting probability Ω. By the way, the value of Ω depends on your particular choice of programming language, which I don’t have
time to discuss now. Ω is also equal to the result of summing 1/2 raised to powers which are the size in bits of
every program that halts. In other words, each K-bit program that halts contributes 1/2K to Ω.
And what precisely do I mean by an N-bit mathematical theory? Well, I’m thinking of formal axiomatic theories, which are formulated using symbolic logic, not in
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any natural, human language. In such theories there are
always a finite number of axioms and there are explicit
rules for mechanically deducing consequences of the
axioms, which are called theorems. An N-bit theory is
one for which there is an N-bit program for systematically running through the tree of all possible proofs deducing all the consequences of the axioms, which are all the
theorems in your formal theory. This is slow work, but in
principle it can be done mechanically, that’s what counts.
David Hilbert believed that there had to be a single formal axiomatic theory for all of mathematics; that’s just
another way of stating that math is static and perfect and
provides absolute truth.
Not only is this impossible, not only is Hilbert’s
dream impossible to achieve, but there are in fact an infinity of irreducible mathematical truths, mathematical
truths for which essentially the only way to prove them is
to add them as new axioms. My first example of such
truths was determining elegant programs, and an even
better example is provided by the bits of Ω. The bits of Ω
are mathematical facts that are true for no reason (no reason simpler than themselves), and thus violate Leibniz’s
principle of sufficient reason, which states that if anything
is true it has to be true for a reason.
In math the reason that something is true is called its
proof. Why are the bits of Ω true for no reason, why
can’t you prove what their values are? Because, as
Leibniz himself points out in Sections 33 to 35 of The
Monadology, the essence of the notion of proof is that you
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prove a complicated assertion by analyzing it, by breaking
it down until you reduce its truth to the truth of assertions that are so simple that they no longer require any
proof (self-evident axioms). But if you cannot deduce the
truth of something from any principle simpler than itself,
then proofs become useless, because anything can be
proven from principles that are equally complicated, e.g.,
by directly adding it as a new axiom without any proof.
And this is exactly what happens with the bits of Ω.
In other words, the normal, Hilbertian view of math
is that all of mathematical truth, an infinite number of
truths, can be compressed into a finite number of axioms.
But there are an infinity of mathematical truths that cannot be compressed at all, not one bit!
This is an amazing result, and I think that it has to
have profound philosophical and practical implications.
Let me try to tell you why.
On the one hand, it suggests that pure math is more
like biology than it is like physics. In biology we deal with
very complicated organisms and mechanisms, but in
physics it is normally assumed that there has to be a theory of everything, a simple set of equations that would fit
on a T-shirt and in principle explains the world, at least
the physical world. But we have seen that the world of
mathematical ideas has infinite complexity, it cannot be
explained with any theory having a finite number of bits,
which from a sufficiently abstract point of view seems
much more like biology, the domain of the complex, than
like physics, where simple equations reign supreme.
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On the other hand, this amazing result suggests that
even though math and physics are different, they may not
be as different as most people think! I mean this in the following sense: In math you organize your computational
experience, your lab is the computer, and in physics you
organize physical experience and have real labs. But in
both cases an explanation has to be simpler than what it
explains, and in both cases there are sets of facts that cannot be explained, that are irreducible. Why? Well, in
quantum physics it is assumed that there are phenomena
that when measured are equally likely to give either of
two answers (e.g., spin up, spin down) and that are inherently unpredictable and irreducible. And in pure math we
have a similar example, which is provided by the individual bits in the binary expansion of the numerical value of
the halting probability Ω. This suggests to me a quasiempirical view of math, in which one is more willing to
add new axioms that are not at all self-evident but that
are justified pragmatically, i.e., by their fruitful consequences, just like a physicist would.
I have taken the term quasi-empirical from Lakatos.
The collection of essays New Directions in the Philosophy of
Mathematics edited by Tymoczko in my opinion pushes
strongly in the direction of a quasi-empirical view of
math, and it contains an essay by Lakatos proposing the
term “quasi-empirical,” as well as essays of my own and
by a number of other people. Many of them may disagree
with me, and I’m sure do, but I repeat, in my opinion all
of these essays justify a quasi-empirical view of math,
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what I mean by quasi-empirical, which is somewhat different from what Lakatos originally meant, but is in quite
the same spirit, I think.
In a two-volume work full of important mathematical
examples, Borwein, Bailey and Girgensohn have argued
that experimental mathematics is an extremely valuable
research paradigm that should be openly acknowledged
and indeed vigorously embraced. They do not go so far
as to suggest that one should add new axioms whenever
they are helpful, without bothering with proofs, but they
are certainly going in that direction and nod approvingly
at my attempts to provide some theoretical justification
for their entire enterprise by arguing that math and
physics are not that different.
In fact, since I began to espouse these heretical views
in the early 1970’s, largely to deaf ears, there have actually been several examples of such new pragmatically justified, non-self-evident axioms:
• the P ≠ NP hypothesis regarding the time complexity of computations,
• the axiom of projective determinacy in set theory,
and
• increasing reliance on diverse unproved versions of
the Riemann hypothesis regarding the distribution of the
primes.
So people don’t need to have theoretical justification;
they just do whatever is needed to get the job done. . .
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Chaitin – From Leibniz to Ω
The only problem with this computational and information-theoretic epistemology that I’ve just outlined to
you is that it’s based on the computer, and there are
uncomputable reals. So what do we do with contemporary physics which is full of partial differential equations
and field theories, all of which are formulated in terms of
real numbers, most of which are in fact uncomputable, as
I’ll now show. Well, it would be good to get rid of all that
and convert to a digital physics. Might this in fact be possible?! I’ll discuss that too.
COMPUTER ONTOLOGY:HOW REAL ARE REAL NUMBERS? WHAT IS THE WORLD MADE OF?
How did Turing prove that there are uncomputable
reals in 1936? He did it like this. Recall that the possible
texts in French are a countable or denumerable infinity
and can be placed in an infinite list in which there is a first
one, a second one, etc. Now let’s do the same thing with
all the possible computer programs (first you have to
choose your programming language).
So there is a first program, a second program, etc.
Every computable real can be calculated digit by digit by
some program in this list of all possible programs. Write
the numerical value of that real next to the programs that
calculate it, and cross off the list all the programs that do
not calculate an individual computable real. We have converted a list of programs into a list of computable reals,
and no computable real is missing.
Next discard the integer parts of all these computable
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reals, and just keep the decimal expansions. Then put
together a new real number by changing every digit on
the diagonal of this list (this is called Cantor’s diagonal
method; it comes from set theory). So your new number’s first digit differs from the first digit of the first computable real, its second digit differs from the second digit
of the second computable real, its third digit differs from
the third digit of the third computable real, and so forth
and so on. So it can’t be in the list of all computable reals
and it has to be uncomputable. And that’s Turing’s
uncomputable real number!7
Actually, there is a much easier way to see that there
are uncomputable reals by using ideas that go back to
Emile Borel (again!). Technically, the argument that I’ll
now present uses what mathematicians call measure theory, which deals with probabilities. So let’s just look at all
the real numbers between 0 and 1. These correspond to
points on a line, a line exactly one unit in length, whose
leftmost point is the number 0 and whose rightmost point
is the number 1. The total length of this line segment is
of course exactly one unit. But I will now show you that
all the computable reals in this line segment can be covered using intervals whose total length can be made as
small as desired. In technical terms, the computable reals
in the interval from 0 to 1 are a set of measure zero, they
have zero probability.
How do you cover all the computable reals? Well,
7. Technical Note: Because of synonyms like .345999 . . . = .346000 . . . you
should avoid having any 0 or 9 digits in Turing’s number.
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remember that list of all the computable reals that we just
diagonalized over to get Turing’s uncomputable real?
This time let’s cover the first computable real with an
interval of size ε/2, let’s cover the second computable real
with an interval of size ε/4, and in general we’ll cover the
Nth computable real with an interval of size ε/2N . The
total length of all these intervals (which can conceivably
overlap or fall partially outside the unit interval from 0 to
1), is exactly equal to ε, which can be made as small as
we wish! In other words, there are arbitrarily small coverings, and the computable reals are therefore a set of
measure zero, they have zero probability, they constitute
an infinitesimal fraction of all the reals between 0 and 1.
So if you pick a real at random between 0 and 1, with a
uniform distribution of probability, it is infinitely unlikely, though possible, that you will get a computable real!
What disturbing news! Uncomputable reals are not
the exception, they are the majority! How strange!
In fact, the situation is even worse than that. As Emile
Borel points out on page 21 of his final book, Les nombres
inaccessibles (1952), without making any reference to
Turing, most individual reals are not even uniquely specifiable, they cannot even be named or pointed out, no
matter how non-constructively, because of the limitations
of human languages, which permit only a countable infinity of possible texts. The individually accessible or nameable reals are also a set of measure zero. Most reals are
un-nameable, with probability one! I rediscovered this
result of Borel’s on my own in a slightly different context,
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in which things can be done a little more rigorously,
which is when one is dealing with a formal axiomatic theory or an artificial formal language instead of a natural
human language. That’s how I present this idea in Meta
Math!. So if most individual reals will forever escape us,
why should we believe in them?! Well, you will say,
because they have a pretty structure and are a nice theory, a nice game to play, with which I certainly agree, and
also because they have important practical applications,
they are needed in physics. Well, perhaps not! Perhaps
physics can give up infinite precision reals! How? Why
should physicists want to do that?
Because it turns out that there are actually many reasons for being skeptical about the reals, in classical
physics, in quantum physics, and particularly in more
speculative contemporary efforts to cobble together a theory of black holes and quantum gravity.
First of all, as my late colleague the physicist Rolf
Landauer used to remind me, no physical measurement
has ever achieved more than a small number of digits of
precision, not more than, say, 15 or 20 digits at most, and
such high-precision experiments are rare masterpieces of
the experimenter’s art and not at all easy to achieve.
This is only a practical limitation in classical physics.
But in quantum physics it is a consequence of the
Heisenberg uncertainty principle and wave-particle duality (de Broglie). According to quantum theory, the more
accurately you try to measure something, the smaller the
length scales you are trying to explore, the higher the
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energy you need (the formula describing this involves
Planck’s constant). That’s why it is getting more and
more expensive to build particle accelerators like the one
at CERN and at Fermilab, and governments are running
out of money to fund high-energy physics, leading to a
paucity of new experimental data to inspire theoreticians.
Hopefully new physics will eventually emerge from
astronomical observations of bizarre new astrophysical
phenomena, since we have run out of money here on
earth! In fact, currently some of the most interesting
physical speculations involve the thermodynamics of
black holes, massive concentrations of matter that seem
to be lurking at the hearts of most galaxies. Work by
Stephen Hawking and Jacob Bekenstein on the thermodynamics of black holes suggests that any physical system
can contain only a finite amount of information, a finite
number of bits whose possible maximum is determined
by what is called the Bekenstein bound. Strangely
enough, this bound on the number of bits grows as the
surface area of the physical system, not as its volume,
leading to the so-called “holographic” principle asserting
that in some sense space is actually two-dimensional even
though it appears to have three dimensions!
So perhaps continuity is an illusion, perhaps everything is really discrete. There is another argument against
the continuum if you go down to what is called the
Planck scale. At distances that extremely short our
current physics breaks down because spontaneous fluctuations in the quantum vacuum should produce mini45
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black holes that completely tear spacetime apart. And
that is not at all what we see happening around us. So
perhaps distances that small do not exist.
Inspired by ideas like this, in addition to a priori metaphysical biases in favor of discreteness, a number of contemporary physicists have proposed building the world
out of discrete information, out of bits. Some names that
come to mind in this connection are John Wheeler,
Anton Zeilinger, Gerard ’t Hooft, Lee Smolin, Seth
Lloyd, Paola Zizzi, Jarmo Mäkelä and Ted Jacobson, who
are real physicists. There is also more speculative work
by a small cadre of cellular automata and computer
enthusiasts including Edward Fredkin and Stephen
Wolfram, whom I already mentioned, as well as
Tommaso Toffoli, Norman Margolus, and others.
And there is also an increasing body of highly successful work on quantum computation and quantum information that is not at all speculative, it is just a fundamental reworking of standard 1920’s quantum mechanics.
Whether or not quantum computers ever become practical, the workers in this highly popular field have clearly
established that it is illuminating to study sub-atomic
quantum systems in terms of how they process qubits of
quantum information and how they perform computation with these qubits. These notions have shed completely new light on the behavior of quantum mechanical systems.
Furthermore, when dealing with complex systems
such as those that occur in biology, thinking about infor46
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mation processing is also crucial. As I believe Seth Lloyd
said, the most important thing in understanding a complex system is to determine how it represents information
and how it processes that information, i.e., what kinds of
computations are performed.
And how about the entire universe, can it be considered to be a computer? Yes, it certainly can, it is constantly computing its future state from its current state, it’s
constantly computing its own time-evolution! And as I
believe Tom Toffoli pointed out, actual computers like
your PC just hitch a ride on this universal computation!
So perhaps we are not doing violence to Nature by
attempting to force her into a digital, computational
framework. Perhaps she has been flirting with us, giving
us hints all along, that she is really discrete, not continuous, hints that we choose not to hear, because we are so
much in love and don’t want her to change!
For more on this kind of new physics, see the books
by Smolin and von Baeyer in the bibliography. Several
more technical papers on this subject are also included
there.
CONCLUSION
Let me now wrap this up and try to give you a present to take home, more precisely, a piece of homework.
In extremely abstract terms, I would say that the problem
is, as was emphasized by Ernst Mayr in his book This is
Biology, that the current philosophy of science deals more
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with physics and mathematics than it does with biology.
But let me try to put this in more concrete terms and connect it with the spine, with the central thread, of the ideas
in this talk. To put it bluntly, a closed, static, eternal fixed
view of math can no longer be sustained. As I try to illustrate with examples in my Meta Math! book, math actually advances by inventing new concepts, by completely
changing the viewpoint. Here I emphasized new axioms,
increased complexity, more information, but what really
counts are new ideas, new concepts, new viewpoints.
And that leads me to the crucial question, crucial for a
proper open, dynamic, time-dependent view of mathematics,
“Where do new mathematical ideas come from?”
I repeat, math does not advance by mindlessly and
mechanically grinding away deducing all the consequences of a fixed set of concepts and axioms, not at all!
It advances with new concepts, new definitions, new perspectives, through revolutionary change, paradigm shifts,
not just by hard work.
In fact, I believe that this is actually the central question in biology as well as in mathematics, it’s the mystery
of creation, of creativity: “Where do new mathematical
and biological ideas come from?” “How do they
emerge?”
Normally one equates a new biological idea with a
new species, but in fact every time a child is born, that’s
actually a new idea incarnating; it’s reinventing the
notion of “human being,” which changes constantly.
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I have no idea how to answer this extremely important question; I wish I could. Maybe you will be able to
do it. Just try! You might have to keep it cooking on a
back burner while concentrating on other things, but
don’t give up! All it takes is a new idea! Somebody has to
come up with it. Why not you?
APPENDIX: LEIBNIZ AND THE LAW
I am indebted to Professor Ugo Pagallo for explaining
to me that Leibniz, whose ideas and their elaboration
were the subject of my talk, is regarded as just as important in the field of law as he is in the fields of mathematics and philosophy.
The theme of my lecture was that if a law is arbitrarily complicated, then it is not a law; this idea was traced
via Hermann Weyl back to Leibniz. In mathematics it
leads to my Ω number and the surprising discovery of
completely lawless regions of mathematics, areas in
which there is absolutely no structure or pattern or way
to understand what is happening.
The principle that an arbitrarily complicated law is
not a law can also be interpreted with reference to the
legal system. It is not a coincidence that the words “law”
and “proof ” and “evidence” are used in jurisprudence as
well as in science and mathematics. In other words, the
rule of law is equivalent to the rule of reason, but if a law
is sufficiently complicated, then it can in fact be
completely arbitrary and incomprehensible.
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ACKNOWLEDGEMENTS
I wish to thank Gordana Dodig-Crnkovic for organizing E-CAP ‘05 and for inviting me to present the Turing
lecture at E-CAP ‘05; also for stimulating discussions
reflected in those footnotes that are marked with GDC.
The remarks on biology are the product of a week spent
in residence at Rockefeller University in Manhattan, June
2005; I thank Albert Libchaber for inviting me to give a
series of lectures there to physicists and biologists. The
appendix is the result of lectures to philosophy of law
students April 2005 at the Universities of Padua, Bologna
and Turin; I thank Ugo Pagallo for arranging this.
Thanks too to Paola Zizzi for help with the physics
references.
REFERENCES
• Edward Fredkin, http://www.digitalphilosophy.org.
• Stephen Wolfram, A New Kind of Science, Wolfram
Media, 2002.
• Gregory Chaitin, Meta Math!, Pantheon, 2005.
• G. W. Leibniz, Discourse on Metaphysics, Principles of
Nature and Grace, The Monadology, 1686, 1714, 1714.
• Hermann Weyl, The Open World, Yale University Press,
1932.
• Thomas Tymoczko, New Directions in the Philosophy of
Mathematics, Princeton University Press, 1998.
• Jonathan Borwein, David Bailey, Roland Girgensohn,
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Mathematics by Experiment, Experimentation in Mathematics,
A. K. Peters, 2003, 2004.
• Rebecca Goldstein, Incompleteness, Norton, 2005.
• B. Jack Copeland, The Essential Turing, Oxford
University Press, 2004.
• Vladimir Tasic, Mathematics and the Roots of
Postmodern Thought, Oxford University Press, 2001.
• Emile Borel, Les nombres inaccessibles, Gauthier-Villars,
1952.
• Lee Smolin, Three Roads to Quantum Gravity, Basic
Books, 2001.
• Hans Christian von Baeyer, Information, Harvard
University Press, 2004.
• Ernst Mayr, This is Biology, Harvard University Press,
1998.
• J. Wheeler, (the “It from bit” proposal), Sakharov
Memorial Lectures on Physics, vol. 2, Nova Science, 1992.
• A. Zeilinger, (the principle of quantization of information),
Found. Phys. 29:631–643 (1999).
• G. ’t Hooft, “The
http://arxiv.org/hep-th/0003004.
holographic
principle,”
•
S.
Lloyd,
“The
computational
http://arxiv.org/quant-ph/0501135.
universe,”
• P. Zizzi, “A minimal model for quantum gravity,”
http://arxiv.org/gr-qc/0409069.
• J. Mäkelä, “Accelerating observers, area and entropy,”
http://arxiv.org/gr-qc/0506087.
•T. Jacobson, “Thermodynamics
http://arxiv.org/gr-qc/9504004.
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of
spacetime,”
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The Militarization of Peace:
Absence of Terror or Terror of Absence?
Reza Negarestani
I NTRODUCTION
This essay explores the rise of a new wave of terrorism which
exploits its own dissolution, making a weapon of the doctrine of
Taqiyya or strategic (dis)simulation, dismantling the theatrical
aspect of the battlefield and selecting civilians as primary targets
and ‘molecular battlefields’. This tendency threatens not only
global civilian survival but the very horizon of survival or living (in
its most basic, abstract sense) in general. It makes survival itself a
field of exploitation for extremist terrorism.
When militarization ceases to be an exclusively
wartime process and to belong only on the battlefield,
then even peace—the temporary gap, the blank space of
unfriction between war machines and collective survival
—can be militarized. This does not mean taking
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ISBN 978-0-9553087-0-4
http://www.urbanomic.com
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advantage of peace as a temporary suspension which can
be exploited, or as a depository in preparation for the
militarization processes of future wars (who gathers the
most forces when everyone else is resting?) Rather, and
far more significantly, it means the endo-militarization of
peace itself, wherein peace is directly used as a weapon,
exploited as a new plane for invasion and insurgency, and
for offensive strikes against enemy bases and/or their
supportive lattices.
New modes of disseminating terror threaten the basic
notions of survival in general, creating a generalized state
of terror where death hangs over, regulates, every
moment that is lived. Such necrocracy is the goal of heretical Islamic agencies of Terror such as Jama’at-e Takfir1 and
its Takfiri agents – a militant Jihadi movement believing in
the absolute excommunication of infidels (Takfir originally means excommunication). These agencies have
inspired a new wave of militant religious extremists and
other obscure terrorist groups who are exploiting the
endo-militarization of peace as a new mode of warfare.
This new mode of warfare is one whose tactical lines are
not aligned with (or configured by) the plane of conflict
and visible military friction (battlefields, terrains for
guerilla warfare, street-wars, etc.); Its tactical lines do not
1. Jama’at-e Takfir (The Society of Excommunication) influenced by Qutb’s Muslim
Brotherhood emerged in Egypt as a fundamentalist group in the 1960s with
Islamic fundamentalist and militant inclinations (the former being similar to
extremist Salafism) enmeshed through decentralized and stealth operation networks. The group advocates any military course of action (whether armed battles
or not) against Jews, Christians, apostate or moderate Muslims, in order to restore
(or return to) the primal unity of the Islamic world order.
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have the localizability which is a prerequisite for direct
conflict and military formation; They are not positioned
to cut, block or replace each other depending upon their
different tendencies, transorientations and alignments;
Their operations have a wholly oblique relation to the
dynamic incompatibility which provides the basis for, and
the matrix of, militarized conflictual engagement.
A Takfiri engages as a shadow terrorist in White War –
the endo-militarization of peace, a state of hypercamouflage (best defined as complete and consequently
symmetrical overlap between two entities on a
mereotopological2 plane). In this war, the cover of
camouflage can never be penetrated or disrupted, and the
defensive employment of camouflage (best mapped as
partial overlap between two or more entities on a logical
plane) is replaced by a wholly novel, highly offensive
deployment, the space of hypercamouflage. The Takfiri’s
favoured mode of warfare is to program a new type of
tactical line which totally blends with the enemy’s lines in
such a configuration that it introduces radical instability
and eventually violent fissions into the system from
within. This happens in such a way that not only does
recovery become impossible, but in addition any corrective or restorative initiative is ineluctably turned into a
military subversion: like a chemotherapy gone awry or
an excessive scarring in which healing and the process of
2. Mereotopology is a theory of topological relationships between parts and
boundaries. See Barry Smith, “Mereotopology: a theory of parts and boundaries”,
in Data & Knowledge Engineering, Volume 20, Issue 3 (November 1996), Elsevier
Science Publishers, pp. 287 - 303.
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epithalization, in the absence of a wound, corrode the
organ in the form of fibroproliferation (a scarring process
which transforms the local injury of the wound into a
pervasive metastatic scarring), resulting in eventual lysis
and decomposition. In attempting defence, the enemy
can only necrotize and dissolve itself.
DEEP TERROR: THE DECLINE OF THE ENEMY AND THE
RISE OF OBSCURE ALLIES.
Abdu-Salam Faraj’s manifesto Jihad: The Absent
Obligation – in which malevolent political pragmatics and
tactical perversion are planted carefully in a context of
evangelistic justification and theo-tyrannical apologetics –
is a case study of this mode of warfare: White War or
the militarization of peace, comprising aggressive hypercamouflage as its primary engine.
Hypercamouflage aims to pursue to even the most
attenuated extreme, a fighting and a surviving alongside
the enemy. It invariably indicates a total withdrawal
from the perception of friends and a dissolution into the
enemy: the rebirth of a new foe.
In his book, Takfiri cultist and terrorist Faraj crafts a
fetishized form of Jihad, suggesting that the incinerating
head of Jihad must be introduced to everyone, to any
entity, regardless of their position, geographic location,
ethnicity, regardless of the relevance or otherwise of that
entity to Jihad, Islam or infidelity – Jihad as a universal
sweeping movement. The original title of the book,
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which in translation has become simplified into Jihad:
The Absent Obligation, is Jahad: Fariezato Ghaebata (or Jihad:
Fariezeh Ghaeb). Fariezeh means holy duty, but not a subjectively authoritarian duty as is demanded by Huda (Allah’s
guidance), the ‘utter submission’ (Islam) to Allah. Ghaeb
means absent, but in Islamic texts and especially Shia
books, it encompasses a huge hermeneutic potentiality
which ‘absence’ cannot hope to translate; in fact, rather
than mere absence Ghaeb indicates latent potentiality, in
the sense, for instance, of the latent period of inactivity of
a virus. This latency is to be distinguished from the
actualized and visible, which is liable to distortion and
change: Imam Mahdi (the 12th Imam and the harbinger
of Qiyamah, the Islamic Apocalypse) is absent (Ghaeb) but
affects Islam and its followers more than anything
actually present; Mahdi represents a potentiality that never
ceases to affect. In Faraj’s book, however, this definition,
which is a fundamental theological and eschatological
platform for his argument, becomes eclipsed by a
message at once more accessible and more divergent
from this original meaning: Jihad becomes a holy responsibility which is not present.
Faraj’s book adds a new twist to the tactics of heretical religious extremists such as the cult of Takfiri: the
distortion and alteration of ‘Taqiyya’ (Taghieh) from its original defensive and devout function in the dawn of Islam.
Rather than a strategic (dis)simulation – a justified
concealment of true beliefs in situations where harm or
death will definitely be encountered if the true beliefs are
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declared3 (the wider meaning of Taqiyya being “to avoid
or shun any kind of danger.”) it is reinterpreted as a silent
and fluid military infiltration, a course of action which
forms one of the elemental components of fetishized
Jihadism.
Faraj’s take on Taqiyya departs entirely from what in
the dawn of Islam originated as a defensive or protective
inclination, an evasive tactic. In The Absent Obligation,
Taqiyya is reconstituted as a type of strategic simulation or
dissimulation, in the name of a hostile politics of offence.
However, both in its traditional form and in this new
weaponized form, Taqiyya is strongly bound to the notion
of survival. In the traditional sense this is so simply
because by taking Taqiyya the believer survives in difficult
circumstances. But in militarized Taqiyya, survival is
transformed into a sort of highly-charged parasitical
endurance which inherently threatens the catalysis of all
those whose survival is afforded more easily. Survival
becomes as risky as a contagious terminal illness. Faraj
insists that Jihad cannot be separated from Taqiyya.
Whereas crusades transgress boundaries in order to
retake the holy lands, in Islamic tradition Jihad intrinsically cannot be transgressive; it must merely defend the
holy lands, and Islamic properties (which are not
necessarily associated with geopolitical agencies). But as
3. “The believers never ally themselves with the disbelievers, instead of the believers. Whoever does this is exiled from GOD. Exempted are those who are forced
to do this to avoid persecution. GOD alerts you that you shall revere Him alone.
To GOD is the ultimate destiny.” (The Quran 3:28)
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revivalist figures like Sayyid Qutb4 and Shukri Ahmad
Mustafa have twisted the entire panorama of Islamic
thought, heretical Islam’s defence of its ‘properties’ has
become a universal ‘defence’ encompassing massive waves
of acentric assault and a military subversion pervaded by
a complex tendency towards the exclusion of all beings
except for the monopolistic wasteland of the Divine (the
Desert). “The earth itself moves towards Allah by
submitting itself to the ‘exterior’ Will of Allah; or in other
words, is not Earth a part and property of Islam (utter
submission to Allah) which must be defended?”: Qutb
turns the issue inside out, all of theological thought
becoming ravaged by monopolistic dictatorship and
monomania. The Earth itself becomes a part of the
defensive politics of Jihad 5. Ahmad Mustafa, one of the
theorists of the original Takfiri cult, also suggests that “We
are returning to Islam”, and that this Grand Return
4. Sayyid Qutb (1906-1966), one of the central theorists of Islamic Revivalism and
an inspiration for later extremists such as Faraj; his Ma'alim fi-l-Tariq (Milestones)
is perhaps the first theoretical work of modern extremist Islamism, integrating
pragmatic exhortations with self-centered politico-religious doctrines. On Qutb,
see Paul Berman’s analysis of terrorism inspired by the caliph’s militarism and
heretical Islamic revivalism: Berman, P. (2004) Terror and Liberalism, W. W. Norton
& Company.
5. “The Islamic civilization can take various forms in its material and organizational structure, but the principles and values on which it is based are eternal and
unchangeable. These are: the worship of God alone, the foundation of human
relationships on the belief in the Unity of God, the supremacy of the humanity of
man over material things, the development of human values and the control of
animalistic desires, respect for the family, the assumption of the vice-regency of God on
earth according to His guidance and instruction, and in all affairs of this vice-regency, the rule
of God's law (Shari'a) and the way of life prescribed by Him ...” (Sayyid Qutb, Milestones);
see Qutb (1991) Milestones, American Trust Publications, p. 286.
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involves surrendering to the Desert of the Divine: this is
not a reactionary response to the infidels, he insists, but
merely the path to Islam, which is condemned and met
with overreaction by the rest of the world – a world
whose entire horizon, moreover, is in fact already a part
of Islam. Mustafa’s discussion takes a twisted monotheism and heretically embeds it within the foundations of
Islam – creating a retrograde movement back to a selfdeluding, romantically-imagined phantasy of the original
Islam. To this notion of Jihad which seeks to retake the
Earth as a part of Islam (Earth as a part of the universe is
on the route of utter submission – Islam – to Allah) Faraj
cunningly adds the politics of Taqiyya. As Faraj himself
confesses, this (re)taking of the Earth is not an easy task;
hence the necessity of being armed with Taqiyya and its
potential for insinuation and diffusion within the systems
and peoples of non-Islamic countries.
According to Faraj the new doctrines of weaponized
Taqiyya can be enumerated as follows:
(1) TAQIYYA AS THE DISSOLUTION OF YOURSELF AND
THE OTHER: Taqiyya becomes a politics aimed at drawing
the war out of the battlefield (In this extremist Jihad, war
must be put to work everywhere but the battlefield; war
is external to the conventional battlefield. ‘War is not a
theater, you infidels’, Faraj shouts). This is to be achieved
by introducing the Jihadi entities to civilians and all other
seemingly militarily irrelevant political economic or
cultural entities, by blending with the crowd which exists
far from the front lines. ‘Towards the real omnipresence
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of war which progressively effaces the theatrical platitude
of the battlefield’, the doctrine of Terror voluntarily
transforms itself into a sinister movement of utter selfdissolution. The use of Taqiyya as a (para)offensive
politics, however, is not the invention of Faraj, or the
Takfiri cult, or even of Wahhabi extremists. It can be
traced back to Hassan i-Sabah but it is not the invention
of Hassan either (although he improved and strictly
militarized it). The sole credit for Taqiyya as a (para)offensive polytics aimed at blending with the crowd (as opposed
to Taqiyya as a dissimulating tool for evading harm, as
devised in the early days of Islam) belongs to Abdullah
ibn Maimun or Maymun (and his Batiniyya cult, one of
the underground heretical Islamic societies and
subversive movements which he founded and which later
turned into Isma’ilie sects directed by Hassan i-Sabah):
Maimun, the Persian occultist, political saboteur and
conspiracist who undermined the reign of caliphs in
Egypt (where the Takfiri cult also originated together with
such influential figures such as Qutb, Mustafa, et al.) with
a sudden debacle, and prepared the region for his
ambiguous and mysterious allies Al Fatemids (Fatemion)
who later became the most enthusiastic enemies of the
caliphs and their conventional modes of militarism.
Faraj, following closely Ibn Maymun’s politics, suggests
that Taqiyya should not be merely a deception, a hiding
tactic; it should consist of seeking the highest degree of
participation with infidels, with their civilians: “if they
take drugs we must do the same, if they take part in every
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type of sexual activity we must drive those activities to
the point of excess”, etc. The Jihadi extremist must
become as one with the civilians of what are called
‘hardcore infidels’.
(2) TAQIYYA AS A (PARA)OFFENSIVE MILITARIZATION OF
CIVILIANS: In reference to Faraj’s politics of Taqiyya as an
inseparable element of Jihad, the French counter-terrorist
expert and President of the Paris-based World
Observatory of Terrorism, Ronald Jacquard, brilliantly
points out that a ‘Takfiri under Taqiyya’ is himself a primed
bomb, whether or not he ever sees action (is involved in
a mission). When a Takfiri becomes as one with ordinary
civilians – no longer dissimulating but moving and
behaving like a true, unfaithful civilian in every aspect of
his or her public and private life – then the weapon
begins autonomously to be activated from the other side;
the government (of a foreign non-Islamic country, for
example) itself begins to filter, purge and hunt down its
own civilians, curtailing their rights, confining them to
economic, social and political quarantine to isolate or
even purge the disease and its potential hosts at the same
time. Each individual is potentially a Takfiri cell or niche,
a site of infestation, a primary military target. So that the
most offensive, active phase of a Takfiri’s life is not when
he or she is on a high-profile mission like 9/11, but rather
when he or she becomes a mere civilian, totally unarmed
and dissociated from any line of command. A Takfiri
levels himself with everyone and consequently levels
everyone with himself; when it comes to hunting a Takfiri,
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one ineluctably ends up exterminating non-military
entities, far away from the battlefield, in the heart of one’s
own land.
(3) TAQIYYA AS A TRIGGER FOR WHITE WAR: Taqiyya
unbalances the entire conventional dynamics between
war machines, a dynamics which sees them clash with,
hunt, and consume each other. This process of unbalancing does not serve to shift the battle along the diametric
axis of ‘victory or defeat’, but rather to unbalance the
communicative links between two tactical modes: active
military lines at one pole and virally latent (un)tactical lines at
the other. The Takfiri shuts down all his military potential,
tactically ‘dies’ (not even being camouflaged anymore),
and later is resurrected again in ‘his’ true form. The
Takfiri war machines of extremist Jihad operate on
transient and divergent tactical lines. As a result, they
cannot be reached or communicated with: communication which is the prerequisite for the clash between war
machines and entropically-based military conflicts, mechanisms considered by Deleuze and Guattari as the
processes which fabricate the very machinery and space
of War.
(4) TAQIYYA AS A DESERTIFICATION TOOL: Giving
fetishized Jihad an epidemically omnipresent machinery,
Taqiyya allows Faraj to open a new era in the imagining of
mechanisms of extinction, sabotage and eradication
fueled by the pyromaniac aspects of heresy and dangerously romantic theo-tyranny. Faraj discusses the fact that
‘their’ (he rarely even names his so-called enemies: the
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US) military war machine relies heavily on the megadeath
principle or as they put it DEATH FROM ABOVE
(Overkill, Killing drones, High-tech airplanes, smart
bombs, “Shock and Awe”, MOAB (Mother of All
Bombs), invisible missiles descending from nowhere).
Faraj presents a Takfiri alternative to this megadeath
machinery6: He discusses a new doctrine of hypercamouflaged terror which he calls ‘Dieback machinery’, a term
borrowed from botany and agriculture7. What he defines
as ‘Dieback’ can be applied to an entire ‘civilization just
as well as a Tree or any arborescent mode of collectivity’:
in order to introduce a Tree to extinction, a Takfiri
terrorist never interferes with the roots, attempting to
uproot the whole tree, as this would merely remove the
taproot, leaving rootlets and other root parts in the soil
that would eventually grow and give rise to many new
trees. The terrorist or Jihadi extremist launches a dieback
6. Questions of escalation and diffusion of conflict in time and space are of massive significance for both the western military campaign and Jihadism’s terror-skirmishes. While Jihadism works with diffusion in its off-battlefield conflicts (through
its petropolitical contamination of the global politico-economic systems, its reckless
use of weaponized Taqiyya, working with strategy rather than tactics and contagious communication rather than transgression), the western techno-capitalism
maintains a escalating position in the battlefield, a position connected to the
propulsive body of techno-capitalism, its tactical precision and suprmacy.
However, both of them share a common tendency in conflict: turning human
agencies into molecular battlefields / warmachines; for Jihadism this molecularization of warriors takes an epidemically dispersive form: mainly through Taqiyya’s
para-offensive plane and for the western front, it turns into re-nomadization (in the
Deleuze-Guattarian sense of nomadic warmachines) of the State’s army and miniaturization of an entire army and its specifications on and through the body of each
soldier.
7. A disease of plants characterized by the gradual dying of the young shoots starting at the tips and progressing to the larger branches.
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disease against the tree: to be precise, he starts to
extinguish the most expendable and smallest leaves growing at the top of the tree and its branchlets, and continues
his work to the rest of leaves, without damaging the main
trunk or roots. By destroying leaves from top to bottom
and by marring branchlets, the tree will wither: excommunicated and dejected, the tree will eventually be entirely incapacitated and will start to (over)react autophagically
and allergically to the artificial dereliction effected by the
dieback disease. Taqiyya provides Takfiris with ample
opportunity to use this dieback machinery, starting from
the leaves (civilians or what they call ‘expendable
entities’) and branchlets (small organizations, etc.),
ultimately rendering the tree obsolete without ever
having launched any direct attack against its main
organs.
When a tree is infected by dieback disease, only leaves
and branches are destroyed; however, lacking leaves and
branchlets, the tree gradually becomes prone (overexposed) to environmental factors and all of its systems
become locked into malfunctioning programs, lowering
its immune system and consuming the tree from within.
Various stages in the dieback of a civilization would be:
paranoia; lack of investment; civilians as primary targets
for both fronts; dereliction. All of which result in a
reactionary response from the infected tree which, rather
than aiding recovery, is self-destructive. In a system this
self-destruction (or malfunctioning self-recovery) can be
defined as breakdown of the mechanisms responsible for
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self-tolerance, and the induction of an immune response
against components of the self. Such a cataclysm leads to
the reprogramming of the (immune) system to damage
the self.
A ‘Takfiri under Taqiyya’, then, is nothing but a civilian.
By destroying himself and civilians he can apply the
dieback mechanism to a system. Weaponized Taqiyya is
not directly connected to the dieback mechanism; but it
is a way in which a Takfiri can shift the role of Taqiyya from
mere camouflage to a powerful logistical plane on which
(para)offensive tactics and strategies can be converged
and amplified. When a Takfiri extremist goes under
Taqiyya he embeds his sabotaging mechanisms within
civilians, uses civilians as back-doors. A Takfiri under Taqiyya
is transposed from being a key operational figure in his
own army to being a civilian; at this point, Taqiyya
actually gains access not to important targets but to
ordinary civilians (the primary tactic of the dieback mechanism), allowing the Takfiri an opportunity to effectively
confound and twist all diagrams and maps which allow a
civilian to be distinguished from a terrorist. Through this
back-door, a Takfiri can both damage civilians (or expendable entities of the tree, as they are regarded) more
effectively on a massive scale, and turn their protection
systems against them by assimilating them within itself
and by being assimilated by them.
The original doctrines of the Takfiri cult originate from
the teachings of Qutb and Shukri Ahmad Mustafa. Faraj,
under the influence of the doctrine of ‘Takfir wal’Hijra’
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(Excommunication and Exodus) – imitation of the
prophet who left Mecca and the House of Allah to live in
a purified desert purged from any manifestation of idolatry
– enunciates a new vision of desert and desertification
alien to the conventional image of the Desert familiar to
socio-political dynamics, and western social dynamics in
particular. On several different levels, this desert encompasses all radical trends of Islam, from the ceaseless
exteriority of Allah to Man, to desert as the mere
functional plane of submission to this radical exteriority
(Allah who will be never disclosed), to the original desertnomadic ingredients of Jihad. A desert nomad does not
migrate, as it is minimally under the influence of climatic
factors; it burrows tunnels of its own, making its own
niches within the desert, crossing the dimensions of holey
and smooth spaces, exploiting and betraying them
equally. Scorpions are burrowers not architects, they do
not build upon compositions of solid and void, nor do
they move restlessly, they devour volumes and snatch
spaces; for them the holey space is not merely a dwelling
place, a place to reside (a niche for occupation) but more
than that, it is the Abode of War (dâr al-harb), the holey
space of unselective hunting. Mustafa hysterically
introduced the machinery and the notion of the desert
into all threads of his thought to such a degree that his
cult was mockingly called ‘The desert flogging society’. It
is rather ironic to reveal Mustafa’s real profession: he was
a very talented agronomist.
Take a Russian forest bordering the tundra, whose
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trees are emptied of life because they have been hit by
black-rot and winter dieback; In a Takfiri sense, deserted
trees are no different from a desert without trees: dieback
purifies, desertifies, the infidel organism, bringing the
Earth within the compass of the utter desolation of the
Desert of the Divine.
In the wake of militarized Taqiyya, the Takfiri is no
longer the problem; it is the original civilians of the
country, rather than immigrants, who pose a terminal
security threat. There is no more radical act of war than
fighting in molecularized and expendable battlefields
whose potentiality for conducting conflict has already
been incapacitated.
LOGICAL I NVESTIGATION OF HYPERCAMOUFLAGE VS.
NOMADIC CONFLICT.
“In the past one took a more defensive attitude,” wrote
Koch, referring to miasma theory. “We have now moved
away from this defensive point of view and have seized
the offensive ... We must be prepared, first, to detect the
infectious material easily and with certainty, and second,
to destroy it” (Koch 1903, 8, 10). For Koch, taking the
offensive meant actively seeking the parasites not only in
those obviously ill but also those “suspected” of carrying
them (die Verdächtigen) and in “the apparently healthy.”8
Every warmachine or tactical line occupies a niche
(whether in wartime or peace), a space through which it
8. Otis, L. (1999) Membranes: Metaphors of Invasion in Nineteenth-Century Literature,
Science and Politics, The Johns Hopkins University Press, pp. 34-35
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can move, feed and function; it is not only defined by the
distinctive properties of a tactical line or warmachine but
also by its enemies, the incompatible dynamism of other
tactical lines, types of predators, the exposure to environmental factors, its threshold for receiving data from the
environment, the types of data it receives and its propinquity to what it pursues or probes (there is a common
misunderstanding that attributes solid or crisp boundaries to niches; but niches are assembled wherever an
entity economizes a portion of its environment and
survives / functions in that economized space9). At a
given time t, the entity r occupies a unique10 address (or
set of addresses) as rt(x); its movement can be simplistically expressed in terms of the niches it occupies at successive intervals of time. This address is encoded and set
apart by the niche which the warmachine or tactical line
occupies.
The functions of a niche are not merely disjunctive
and exclusive (for example, directing competitions i.e.
selective movements which result in exclusion of other
portions of the environment or lines of movement) but
9. In fact, some monitoring systems basically concentrate on niches with fiat and
vague boundaries to screen and guide their occupants (tenants). Air traffic control
systems constantly analyze the volume of protected or restricted airspace – defining a circumspace or the volume enclosing a flying object – for collision avoidance,
alert systems and translocations of aircrafts. The volume of protected airspace is a
modified term for niche in traffic management, a simulation of the niche that exists
in flying or migrating birds.
10. This uniqueness is characterized by the definitive properties / qualities that the
address attributes to an entity in space-time but to have an address does not mean
to be the exclusive owner of it.
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also connective/conjunctive. In fact niches mobilize their
occupant entities with their characteristic types of
dynamism, associating them with other niches based on
the affordance necessary for following a tendency or a
plane as well as sharing it with other niches and their
inhabitants. The programming of its niche is the first
basic operation of engineering or recomposing an entity.
Therefore the significance of investigating niches or niche
types (rather than token niches or occupants) progressively increases with the development and emergence of
new dynamic lines, power formations, traffic spaces and
planes of communicative conjunctions. The State and its
grid of dominance identify the movements of an entity (r)
in a niche (whether quantitative – metron-based [measurable, scalable] – or qualitative) by the series of addresses
it authenticates and registers as it travels:
r (x1,x2,x3,...,xn)
For the State, the dynamism geared by warmachines,
the way that each warmachine perpetuates its itinerant
line, can only be traced and numerically tagged through
the logic of boundaries, the programming of dwelling/
accommodating systems and (dis)locations that the State
is able to monitor by monitoring niches and their dynamic addresses. By means of overseeing boundaries through
which entities pass, investigating the temporal effects on
(or alteration of) the forces of territoriality that moving
entities leave behind, their types of localization, and their
behaviors towards mereologic economy (the economy of
the whole), the State can fabricate a cogito (a non-human
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cognition) not only to realize but also to classify the
movement of entities and the dynamism of warmachines
whose immoderate nomadic itineracy means that they
cannot be directly apprehended or sensed by the State.
This is the cogito required for appropriation of warmachines under the State’s military protocols and forms.
Bound to (semi-)rigid segmentarity, dynamic boundaries,
affordance-based connections and static or dynamic localizations (or more accurately, in-place and outside-place
localizations), the State examines the dynamic space of
each entity and its activities – those activities corresponding to its functional, territorial and mereologic regions –
not only to read the characteristics of an entity but also to
locate it on (or according to the proximity of the entity to)
its grid of dominance. The State and all configurations of
Survival Economy track entities through the niche(s) that
they inhabit or populate. For the State’s military
Overwatch, investigating and tracing the niche is the
primary and central task; the itinerant line of an entity or
a warmachine, its communications and functional traits
are all deciphered by scanning the niche the warmachine
occupies and its type. The advanced reading-machines of
the State are even capable of extracting the quiddity of a
warmachine or an entity by analyzing the specifications
of the niche which is intrinsically bound to affordance,
dynamic forces of boundaries, and eco-logical principles.
However, as niches are connective entities (entity-asevent in a Deleuzian sense); they do not exclusively
belong to one entity or one tenant. Multiple entities can
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share one niche and niches can form nested niches (territorial forces diminish – but never disappear – in grouping
bonds) linking to each other, being connected in various
modes. Modes of connection between niches are divided
for the most part into two asymmetrical correlations:
a. Abutment (A)
b. Overlap (O)
In the Deleuze-Guattari model of the nomadic warmachine, the warmachines are external to the State’s
effective boundary, restlessly eroding it, gnawing at the
consolidated borders of the State. Logical modeling of the
interactions between the exterior nomadic warmachines
and the State is complicated mainly by the following
problems:
(a) Both the State and the warmachine retain a relative
11. Affordance is an economical network (in the sense that it is connective and
reciprocal) by which openness can be exploited as a groundwork for survival,
accommodation, dwelling and regulating communication.
The term affordance as used here diverges in certain respects from the original
term coined by James Jerome Gibson (based on the works of Ingarden, Brentano,
et al.) in his eco-cognitive studies. The regulations by which an entity can maintain
its dynamic position (in a whole, i.e. mereologic address) and survive in its environing horizon originate from a deeply meshed economic-based network of interactions, connections and regulative participations, all knitted on mutual affordability between the entity and its environment. Whole can only survive when entities
can afford each other, every type of openness on mereologic levels is demarcated
by mutual affordability ‘between’ entities. Affordance does not exclusively belong
to one pole of the economical communication but is distributed between at least
two mereologic entities. ‘I am open to you as long as I can afford you’ otherwise:
(a) you must be repulsed (b) attracted by being regulated and appropriated (c) partly filtered (d) I should appropriate myself to ‘accommodate’ you. Therefore, the
plane of being open to is intrinsically constructed on affordance or economical
affordability/communication. Through affordance, openness cannot escape
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movement to each other (each dynamic on its own plane
of tacticity) which makes the State’s militarized machines
and nomadic warmachines slippery entities with a
progressive displaceability increasing as attacks and
counter-attacks are escalated at the borders of the State.
(b) The rise of the clandestine State, that opens itself
to the nomadic warmachines to either absorb them
within its military formations by continuous contacts
with nomadic warmachines (such contacts are essentially
bound to contaminative potentials for both the State and
the nomads) or reinvents nomadic warmachines as its
mercenaries, dynamic lines for extending the State
beyond its border, a new dynamic boundary providing
the State with the opportunity of accommodating (colonizing?) or economically affording (affordance11) the
Outside instead of being cracked open by the Outside.
survivalist and economical regulations; it mainly works as the dynamic capacitator of Whole. Possibly the most elucidating (yet simplified) ‘model’ of affordance
is Aristotle’s Tetrasomia (Rotation of the Elements).
The rotational movement between elements sustains a refining dynamism for the
whole. Each phase of rotation is based on dynamic metrons (measures, scales) and
affordance (here, economical openness or mutual affordability) between elements.
Elements are open to each other either diametrically or diagonally, but they can
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Focusing on the spatiogeographic aspect of the warmachine (necessarily interconnected with its affective
aspect) – or, more precisely, investigating a mereotopological model of the nomadic warmachine and the State
through their distinctive but general mode of connection
which can be grasped as abutment or external connection
– might be the primary task for diagramming the affectspace and the lines of movement generated between concrete nomadism and the State. This modelling, both
spatiogeographic and mereotopological (and thus necessarily indicating an immanent affect space) of the
nomadic warmachine and its positioning relative to the
State’s boundary elucidates the processes at work in the
emergence of anomalous nomadic states (as in the case of
the ‘guerilla-state’ and its connection with ethnonationalism in Iran or the Bedouin nomad-tribes and their strong
but ambiguous bonds with the Saudi government in
never entirely overlap or radically communicate with each other; they need a midstate to form rotational nexuses and maintain their Wholeness. These mid-states
are valid only in a particular location of the whole rotational panorama; although
they provide the system with a propulsive polemikos or cyclic dynamism, they function locally (as a result of the elements’ affordability to each other and, at the same
time, to the whole system of Tetrasomia). For example, Earth and Water need
Menstruum (living mud) to communicate. This living mud is a communicational
entity but also a dynamic boundary which transforms/appropriates the earth and
water before opening them to each other; it can only work locally between earth
and water and not at any other location in the model of Tetrasomia. The Whole
uses these economical communications to consolidate itself and to afford Life (to
survive).
“I assume that affordances are not simply phenomenal qualities of subjective experience (tertiary qualities, dynamic and physiognomic properties, exc.). I also
assume that they are not simply the physical properties of things as now conceived
by physical science. Instead, they are ecological, in the sense that they are properties of the environment relative to an animal. These assumptions are novel, and
need to be discussed.” (J. J. Gibson)
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y
x
Abutment
x
y
Partial Overlap
x
y
Disjunction
Fig. 1
Saudi Arabia) as well as the increasing risk for nomadic
warmachines engaging clandestine states or states with an
obscure boundary.
P OSITIONING OF THE NOMADIC WARMACHINE ON A
MEREOTOPOLOGICAL PLANE.
Abutment (fig. 1) is an external connection, with
minimum trade between niches or entities (the least
contagious connection, given its tendency towards dissociation). It is demarcated by its intermediacy before partial overlapping and after disjunction, by its tangential
contact and boundary overlap.
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What was once the frontier of the State’s defence is
continuously eroded by the reckless tidal or updrift movements of nomadic warmachines. The operational significance of this mode of connection (Abutment) has been
significantly decreased with the emergence of symbiotic
and manipulative warmachines and covert militarization
processes on the one hand, and advanced reformations of
the State towards the accommodation (or colonizing) of
the Outside on the other. The latter are rooted both in
the introduction of territorial climatologic factors12 to the
dynamism of nomad-packs and in the development new
modes of survival. Now, the state knows well how to
save its foundations, even if it means assembling spaces
susceptible to the erosion of nomadic warmachines,
attracting or diverting the incoming nomadic incursions
12. On Climate and Nomadology: Following the so-called Hydraulic and
Agricultural Revolution in Iran (similar to that which Wittfogel associated with the
Chinese Empire, as well as Homer-Dixon’s more recent theories on
Hydropolitics), during the reign of Mohammad Reza Pahlavi (Shah), an hydraulic
plan – highly recommended by American consultants – was developed and proposed as a catalyst for economic development in Iran; one of multiple objectives
of this plan was to solve the problem of nomads in Iran. Apart from putting into
effect a program of hydraulic restructuring of the diverse geography of Iran (a
geography with innate potential for the forging of diversifying lines of nomadic
movement), one of the stratagems of this hydropolitical program of reform was to
originate a system for monitoring and domesticating Iranian nomads who played
key roles in resistance against the centre or induced geopolitical disintegration of
the State’s territory via their ethnonationalistic movements. The plan was neither
a method for drawing the eastern and central nomads to the governing center nor
a project for forcibly accommodating them in a sedentary sphere through the
monopolization of water-networks and direct military impositions. Rather, it suggested accompanying them, interlocking with them and replacing their dynamism
with the State’s fluxional lines of tactics, its dynamic boundary and territorial
forces. The project’s objective was to construct a soft climate (klima: zone) or a
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to specified and preprogrammed regions to protect its
critical terrains and vulnerable mechanisms, or transforming its macropolitics into a viable micropolitics which
are open at one end and grounded at the other end.
In a typified connection Cτ, abutment can be mapped
on the Euclidean plane R as:
Aτ(x,y) = df Cτ(x, y) ∧¬Οτ(x,y) (x abuts y)
(Where Οτ is a typified boundary overlap.)
Or let T = {X, cl} be a topological space, where X is
the set of points and cl is the closure operator. Let I be
any index set that includes 0. The domain, D, of a
Layered Model is a nonempty set of ordered pairs
xi=<x,i> where ∅ ≠ x ⊆ X and i ∈ Ι . (xi will be used
for <x,i>).
A(xi,yi)=:x∩y=∅&(cl(x)∩y≠∅ or x∩cl(y)≠∅) (abuts)
Since Abutment links entities on a tangential plane
(confinium), the state can effectively resist any arriving
onrush of nomadic warmachines on this mode of connection with minimum attrition damage on its critical interior (the plane of logistics and lines of command). In fact,
clandestine states seek to channel all the cumulative
damage induced by nomadic warmachines (as the
postulate of the obtrusive danger) on this mode of
connection. This is achieved by deflecting any fundamentally contagious, manipulating and undermining
threat towards distributive and recoverable eroding
processes; these latter can even be programmed to
transport the State out of its rigid segmentarity and
despotic bond with territoriality, prolonging the survival
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COLLAPSE I
of the state in a fluxional mode in a manner of an
abrasive machine of the fluvius (river) that erodes solidity
in order to transport it by the dynamic conservative vector fields of sedimentary processes – capturing fecundity
and irrigation in detrition. With warmachines tirelessly
gnawing at the State’s textum, incising and liquidating its
crisp boundaries, the State begins to leak out, but this
does not only express the collapse of the State but also the
dangerous exposure of the nomadic warmachine to the
underlying grid on which the State is assembled and
which holds its interwoven space, a network of grounding processes, mechanisms of territorial regulation and
economic repression.
The Installation of the operational cutting-edge of
nomadic warmachines on the State in the absence of any
ungrounding machineries (which incapacitate the dominant grounding, territorializing and moderating functions
of the State) is a similar case to that of the premature line
of deterritorialization which facilitates either the unconventional establishment of new immunologicallyenhanced States or a suicidal flight. Persian history, over
a long period of time (from the Achaemenians to the
Qajar dynasty (1779-1925), more than two thousand
years), narrates such a continuous conversion of nomadic
forces into State forces, before being again replaced by
another nomadic population (cyclic nomadic uprisings
against the ruling regime with a nomadic germ-cell still
active but privatized as the State’s elite, versatile military
institution). Such premature nomadic detritions of the
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State have progressively given rise to more powerful (in
terms of gravity, immunity and parasitic resistance) yet
more unstable States, causing politico-economic poverty;
inclination towards being colonized by other States; lack
of an autonomous nervous system and polarization of different populations without the possibility of positive
diversity; constant vulnerability to schisms, civil-wars,
and ethnonationalistic fault lines deleterious to an entire
country or geopolitical sphere.
When abrasion processes of nomadic warmachines
continue to hold their eroding positions – essentially
characterized by transporting dynamism of friction
(tactionis) and the process of mass-wastage – over a long
duration on the borders of the state, hyper-active territorial nexuses between the State and nomadic warmachines
emerge, increase and expand. Once such nexuses are
established (boundary overlap), the underlying ground
economy of the State (or its territorial forces), its entities
and even the State’s internal machineries directly leak out
into the space traversed by nomadic warmachines, to
such a degree that they pervade the nomadic space and
the State
the State
N
N
Fig. 2. Boundary Overlap with the State (diagrammed as a square) and Nomadic
Domestication (left), Tangential Contact (diagramed as a circle) and Nomadic
Effectivity (right)
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COLLAPSE I
turn it into a dynamic extension of the State13. In this
case, the State’s functional or territorial entities cannot be
effectively enveloped and carried away by nomadic warmachines (as in the case of tangential contact) anymore.
They cannot be cut from the State’s grid of dominance,
liberated and radically dispersed to the Outside (fig. 2).
The hazardous contact of nomadic warmachines with
the State, exposing them to the state’s regulating functional/territorial spheres, can eventually lead to the emergence of a nomadic-state on the one hand and an
ethnonationalistic nomadism (identical to the State’s
patriotic policies) on the other. Probably one of the most
significant examples of such anomalies triggered by the
over-exposure of nomadic warmachines to the State is
that found throughout Persian history.
(b) Overlap:
If, in a simplified approach, P stands for parthood and O
for overlap:
Οτ(x,y) = df ∃z(P(z,x) ∧ P(z,y))
And
Oxy=:∃z(Pzx & Pzy) (x and y overlap)
Then the following Axioms apply:
AP1 P(x,y)↔∀z(O(z,x)→Ο(z,y))
AP2 ∃x(φ(x))→∃x∀y(O(x,y)↔∃z(φ(z) ∧ O(z,y)))
Any participation (either methexis as survival-based
participation or base-participation) happens through
overlapping connections. Therefore, the majority of
13. A cache for the later movement of the State’s macropolitics towards its
micropolitical reformation
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Fig. 3. Modes of Connection
(Abutment and Overlap)
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combined connections (Tangential, Parthood, Interior,
etc.) are effectuated by different possibilities educed by
overlap between entities. Overlap draws lines of coincidence between two events or entities by specifying an
address that two entities partly or completely share in a
spatio-temporal or a functional region. Both the State’s
appropriations and counter-state insurgencies happen
through this mode of connection. Whilst it is exploitable
by the State and by affordance, this does not mean that
‘overlap’ cannot also be the main source of insurgency –
it is the connection-domain through which warmachines
leave their border-eroding externality and directly arrive
at the State’s grid, either to be specialized by the State
apparatus and turn into military formations or to be
reinvented as contagious, endo-symbiotic and parasitic
entities coinciding with the State and its machineries and
consequently
discovering a wide array of clandestine
x
y
y
{
x
z
Not overlapped:
The zone of evacuation and withdrawal from the
camouflaged position or the escape-route.
Fig. 4. Partial overlap and its interval relations in Camouflage:
(1) Between two entities; (2) Between two entities and the third entity
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and manipulative functions.
On a more technical plane, all camouflage exploitations are essentially consistent because they all involve
the use of ‘overlap’ (or, more accurately, coincidence, as
the question of overlap between entities here is the
question of overlapping niches which these entities
occupy14). Furthermore, they turn it from a mode of
connection into a politically-operational positioning that
violates both the symmetry of a niche with the address it
writes (programs) for an entity and the divisors (events,
entities, etc.) which separate and discriminate the addresses or niches of two entities in space-time coordinates.
However, this violation (that necessitates the activation of
camouflage) cannot remain durable and unchanged,
because predatory/military camouflages always employ
partial overlap, with a part constantly accessible as ‘not
camouflaged’ (either belonging to the camouflaged entity
x or the entity which it should be overlapping i.e. y15 [See
14. Two entities will be said to overlap when they share parts in common: two entities coincide when they occupy overlapping regions of space.
15. An example of the not camouflaged part (not overlapped) solely belonging to x
or y: When the ‘not camouflaged’ part merely belongs to y (in a typified
connection):
x
y
x internally overlaps y. When:
IOτ(x,y) = df∃z (IPτ(z,x) ∧ IPτ(z,y))
x is an interior part of y, and when
IPτ(x,y) = dfPτ(x,y) ∧ ¬ TPτ(x,y) and
TPτ(x,y) = dfPτ(x,y) ∧ ∃z (Aτ(z,x) ∧ Aτ(z,y)), x is a tangential part of y
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fig. 4]). This not only makes tracing and handling of the
camouflaged entity possible on a tactical level but also
provides the camouflaged entity with an escape-route or
a space for instant evacuation and withdrawal from the
camouflaged position. (An escape-route can also be
unlocked when an entity z – a third party – connects to
both x and y with different overlapping positions for each
one (See fig. 4); here, the escape is channeled through
another camouflage, a new camouflaged participation
extracted neither from x nor from y.) This ‘not camouflaged’ or ‘not overlapped’ part inhibits the camouflage
from being durable or constantly undetected, but also
makes camouflage controllable; the camouflaged entity
can move out of the camouflage at any moment.
All types of camouflage draw a disruptive function
from the overlapped part (which mainly occurs on a fragmented level) by conducting the address or niche of
another entity (for example, the prey) to the camouflaged
entity (hunter) and consequently disrupting the mereologic (part-whole) correlations at work with regard to
what should be camouflaged, making it temporally and
partly untraceable, camouflaged. Such disruptions
(which generally target a reference-point or a referencelink by which an entity is detected) can produce cognitive-glitches as well as the subversion of some specific
environing bonds that pass through both the
camouflaged entity and its object (its prey). Motion
camouflage uses a particular type of tactical dynamism
(in cases where the prey is also in motion, the movement
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of the shadower or the camouflaged predator moves on
the path of a chaotic pursuit; the movement can be modeled by projecting its pursuit curves onto the Rössler
attractor) or dynamic overlap to disrupt (i.e. shadow) its
distance and displacements from the prey (the shadowed)
by moving on a path that connects it to a fixed point
(used by the shadowed as a reference-point – a constant
unit vector) while the motion by the target is met by the
motion from the aggressor. In motion camouflage, then,
the shadower remains stationary for the target. In the
most common military camouflage – disguise by covering
objects (soldiers, vehicles, artilleries, launch pads, etc.) –
with Disruptive Pattern Materials, disruption happens
through surface modifications of a camouflaged object on
which the visual sensory organ focuses as a reference-link
between different types of surface patterns in its
surrounding space, resulting in the ignoring of the object
as a part of the safe environment. Invisibility (as a retreat
Fig. 5. If Coin stands for coincidence, O for overlap, P for part, Cov for
cover and CCoin for complete coincidence.
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from visual sensors), too, employs and modifies partial
overlap as an occlusion by obscuring surfaces, interiorizing, positioning the camouflaged (cloaked) entity where
boundaries intersect (obscuring boundaries), etc. The
primary drawback of the invisible warmachine is the
danger of being traced by semiotic regimes of the State
which are more obsessed with what is missed than with
what exists.
As the result of partial overlap, all disruptions and
subversions of mereologic bonds are subjected to eventual disclosure; and each time a camouflage is spotted, it
progressively loses its potential; any entity using such a
camouflage will be more prone to detection and
forestalling counter-measures than it usually is; this is a
symptom of the holistic connections between partial overlap and ‘localization’ which has not been functionally
incapacitated and spatially effaced yet. This is why
camouflage is rarely implemented as a primary action or
an offensive tactic but mostly as a logistic process or a
mis-ordering transitional space between different tactical
and operational lines. Transient characteristics and
stringent operational restrictions obstruct radical
weaponization of camouflage.
A Takfiri under Taqiyya (Islamic hypercamouflage)
does not occupy a niche to replace another entity, or
dwell as a hidden agent; he pushes the connection with
his environment toward a complete overlap, an unbroken
field of connection and correspondence, a complete
coincidence with its target, i.e. a complete overlapping of
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its niche with the niche of its target. He entirely overlaps
his prey and its niche and thus remains silent.
Coin(x,y)↔∃z(Cov(z,x) & Cov(z,y))
(x and y coincide if and only if there is some z that is
covered by both x and y; z standing here for niche. While
Cov is a transitive and reflexive relation, Coin is symmetrical and reflexive. The relation of coincidence is of
course broader than that of overlap, since there are pairs
of coincident objects or even processes that do not share
parts. The same question of relation arises for a Takfiri
under Taqiyya and a civilian.)
For a Takfiri under Taqiyya, occupation is neither a
military goal nor a tactic; since occupation is exclusive localization tethered to the mappings of co-localization and
parts-whole connectedness – that is to say, the despotism
of Whole – the occupier is vulnerable to environmental
forces; it can be easily distinguished, located, isolated and
finally terminated i.e. undone at the minimum attrition
cost of its environment and surroundings. Where occupation is bound to visibly militant and escalating modes of
warfare and exclusion, weaponized Taqiyya is maliciously
diffusive. In mereologics (the discourse of part-whole
modes of connectedness) we would call the positioning of
Taqiyya complete overlap: the Takfiri constitutes a sinister
survivalism whose basic function is to extinguish survival
itself. In complete overlap (see Fig. 6), every region,
function or part of the hypercamouflaged entity or
predator, the ‘Takfiri under Taqiyya’ (X) can correspond
with its identical region, function or part of the
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prey/host/civilian (Y). Therefore if every x (part or function of X) homologizes its corresponding y (part or function of Y) or more precisely, if every x corresponds with
its y ‘on all levels’ then every function of X (the tactical
movement of the Takfiri under Taqiyya, or hypercamouflaged predator) can be transferred to Y and they mutually fulfill each other.
x=y↔∀z(Oxz↔Oyz)
(Any two members of the domain that overlap the same
Fig. 6. Complete overlap and symmetric fulfillment
(Symmetry: Let S be the symbol for symmetry where n is an integer, d is the
class descriptor and compd is the complement of d:S = {n(d) ∧ compd} )
entities are identical.)
But the most horrific dimension of this arrangement
is revealed when the process is reversed: if every x fulfills
its corresponding y, then by way of the ‘exact connectingcorresponding’ space that complete overlap and complete
coincidence (CCoin(xi,yi)=:x=y)16 provides, every y (i.e.
16. Complete coincidence can be expressed in terms of covering (Cov):
Ccoin(x,y)↔Cov(x,y) & Cov(y,x)
(x and y completely coincide if and only if y covers x and x covers y)
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every function or positioning of the prey Y, which would
comprise for the most part normal survival functions and
ordinary individual or social activities) can be transferred
to its corresponding x and eventually fulfill it too. By
seizing any y, a corresponding x is triggered and covertly
unleashed; and since we are dealing with complete overlap, the very survival and communication of Y deploys,
activates and fulfills the menacing body of X, the Takfiri
under taqiyya. On the one hand, the survival of the prey/
host/civilian thoroughly agrees with the sinister enthusiasm of the terrorist; and on the other hand, peace is
generally conceived as the state of collective survival. So
that the survival of both the terrorist and the civilian
yields nothing but the (interminable?) endo-militarization
of peace, a global threat against civilians, the rise of
White War and the threat that, fuelled by the infinite
thirst of heretical Jihadism, the contagion of war might
expand unchecked, even to fill the immensely significant
horizon of survival and living in general.
Now that the survival of Y or the host/civilian
(together with its communication and modes of connection through and with its environment) fulfills the political
and military body of the ‘Takfiri under Taqiyya’, the mere
existence of the civilian is weaponized against both itself
and the immune system of the system that accommodates
and protects it to such a degree that auto-phagic overreaction looms as the only logical solution for the system.
It is the military culmination of Taqiyya to deduce irrevocable insanity from the minimum essential logic
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required for basic survival.
EXEGETICAL CONCLUSION.
“The trends explored here will obviously be decided
‘on the battlefield’ - but that increasingly means everywhere. The centrality of hypercamouflage to Jihadi strategy is already having immense consequences, inducing a
wave of ‘retro-militarization’ in State war machines,
where ‘teeth’ flow back down the ‘tail’ in a process without obvious terminus (short of the fanging-up of the
entire social body).
Saddam Hussein’s auto-disassembly of his own war
machine in the interests of a latent insurgency exemplifies
this trend from one side, whilst the moves to harden up
US logistics formations through armoring of vehicles and
combat training for all personnel complements it from the
other.
Human rights concerns about killings of civilians
could relevantly be extended from the empirical level to
that of the transcendental, where the eradication in principle of all civilian populations is taking place. The very
concept of ‘the civilian’ is becoming distinctly dated.
(Virilio’s analysis – despite betraying a somewhat antiquated perspective through terms such as ‘endocolonization’ – seems to have anticipated this trend).
The US is especially interesting because it remains a
‘peripheral’ (even ‘third world’) society in certain
respects, marked by a low domestic index of State
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monopolization of violence, thus allowing retro-militarization from the State pole to connect with an endogenous paramilitarism already rooted in the ‘civilian’ population (armed vigilantism and militia movements). As far
as militias are concerned, the world ain’t seen nothing
yet.”17
COLLAPSE VOLUME I Erratum
. p76 Note 12 should continue as follows:
…The project’s objective was to construct a soft climate (klima:
zone) or a zone of hydraulic conductivity (corresponding to the
State’s Hydropolitics) which would autonomously accommodate
nomads, making them move through itself, and thereby ease the
whole process of monitoring, domesticating and tracking nomads.
This climate being finally transformed into a tremendously
dynamic network of anti-nomadic movements, would employ its
hydraulic head to configure nomadic dynamism according to the
State’s geo-politics, economic convergence and military dynamism
(an accessible space for the State’s military entities
especially during insurgencies); the State was seeking to breed
its own territorializing (rather than territorialized)
nomadologic lines. This climate was actually to consist of
‘artificial rivers’ which were supposed to be distributed over
the country. The work of construction of these rivers was to be
undertaken over a period of years, in a country that has always
suffered from a lack of water in its central and eastern
regions. Such fluid, rich and dynamic zones or hydraulic lines
(for instance rivers and their tributaries) spread over the
country would gradually attract (on the pattern of gravity)
nomads, rendering a preferable but precisely mapped climate for
their migration; a climate which would actually be an autarkic
monitoring machine rendering its inhabitants predictable and
expanding a fluid domesticating sphere for the Iranian nomads.
17. This exegetical conclusion to the current essay was contributed by Nick Land.
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Prime Evolution
Interview with Matthew Watkins
Matthew Watkins’ Web-based Number Theory and
Physics Archive and its speculative twin Inexplicable Secrets
of Creation1 – hosted by the mathematics department of Exeter
University in the UK where Watkins is an honorary research fellow
– have grown into a unique resource. The archive brings together
work from the plurality of disciplines contributing to an as yet
unnamed field of research concerned with the startling connections
between number theory – particularly the Riemann Hypothesis on
the distribution of the prime numbers – and the physical sciences.
Watkins talks to COLLAPSE about his rôle in, and motivations for,
catalysing and disseminating the field, about the latest developments
in the search for the hypothetical ‘Riemann dynamics’, about the
nature of discovery in mathematics and its academic and cultural
status.
1. At http://www.maths.ex.ac.uk/~mwatkins/. Dr. Watkins has kindly assembled
a ‘primer’ for the mathematical concepts discussed in this interview: at
http://www.maths.ex.ac.uk/~mwatkins/zeta/collapseglossary.htm
93
COLLAPSE I, ed. R. Mackay (Oxford: Urbanomic, September 2007)
ISBN 978-0-9553087-0-4
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COLLAPSE: The primes have perennially been hailed as
‘mysterious’. In modern mathematics this mystery has
condensed around the problem which Riemann's
Hypothesis concerns. We can find primes as we count
along the number line, but we have no way of predicting
in general where and how densely they will occur. A lack
of determinable global order, then.
M ATTHEW WATKINS: But first there’s a major
question concerning what is meant by ‘order’. I’m often
asked, is there a pattern in the primes, is there an order,
but what does that mean? If you try to reformulate these
questions very precisely, you’re forced to consider what it
would it mean for there to be order, or a pattern. I mean,
there are patterns like wallpaper patterns, where you have
a block of something repeating. Well, almost by
definition the primes can’t do that. But what sort of
pattern could there be, what sort of order could there be?
The idea that there might be a pattern, the importance of
there being a pattern in the primes – these aren’t things
you can rigorously pin down.
C: Couldn’t you use an information-theoretical
definition of pattern?
MW: You could come up with a definition, one of an
endless number of possible definitions, from information
theory or some related discipline, of what a pattern is,
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and then apply it. But I think there’s still the basic fact
that when people who aren’t familiar with any of those
definitions are asking ‘is there a pattern?’ they don’t mean
anything that such a definition could capture, they mean
something else — they don’t really know quite what, but
it seems important to them that there should be; whether
or not there is a pattern in the primes they see as an
important question. And it struck me when I was thinking about this that it’s more feeling-based, it’s not a
rational question they’re posing. You can try to construct
rational questions around it. People have, and such
questions have given rise to a large part of that body of
work we call number theory.
C: But initially it’s more like the expression of an instinct
for pattern recognition?
MW: Perhaps. As Jung said – almost as the culmination
of his work on archetypes – the set of positive integers,
taken as a whole, corresponds to the archetype of order. So,
in a sense, all notions of order, of something coming
before something else, of things being in a sequence, all
of that ultimately can be linked back to our instinctive
grasping of there being a number system underlying our
experience. Now that number system turns out to have
embedded within it an enigma, a problem bordering on
the paradoxical: is there order in the way this thing’s put
together or not? We feel there should be, but we aren’t
entirely sure how to ask the question — basically, we don’t
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really know. So we start by asking whether there’s order
in the number system, and the unintended result of our
probing into this matter is that what we ultimately mean
by order in any sense gets indirectly thrown into
question.
People also frequently ask about the existence of a
formula – is there a formula for the prime numbers? Well,
again, that’s difficult because, yes there is, there’s the
Riemann-von Mangoldt explicit formula, which
effectively generates exactly the distribution of prime
numbers as ‘output’ – but you need the complete set of
Riemann zeros as input. This is an infinite set, and to
produce it you effectively need the complete set of prime
numbers, so there’s a circularity. So it’s a formula, but
not the kind of formula which people who ask this
question have in mind. There are also algorithms –
rigorous procedures – which can systematically generate
the primes. One could arguably call these ‘formulas’, but
they’re basically methods of computation, and the
computations quickly become intractably huge…so we’re
not talking about anything that can systematically spit
out primes one after another in the sense that people
might have in mind when they ask about the existence of
a formula.
C: As the years have gone on, mathematicians’ ingenuity
and the employment of new technologies have seen an
acceleration in the conquest of the critical line of
Riemann zeta zeros. But does the fact we’ve got, say, one
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billion of the zeros make it any less mysterious than when
we had a hundred? Does the apparent success of the
Riemann Hypothesis (RH) militate against the conception of the primes as mysterious?
MW: First of all, you can’t really talk about RH being
‘successful’, it’s still a hypothesis. RH doesn’t predict the
primes as such, but the theory of Riemann’s zeta
function, from which it emerges, allows us to understand
the distribution of primes much more deeply. At the
heart of this theory is the peculiar sequence of ‘zeros’
now known as ‘Riemann zeros’, ‘Riemann zeta zeros’ or
sometimes just ‘zeta zeros’ – these are what RH directly
concerns.
What’s happened really is that RH has displaced the
mystery. The primes are no longer mysterious, you
could argue — we now know that they are exactly
governed. Initially, it was found that they’re governed by
a logarithmic distribution, a sort of gradual thinning out,
in an almost statistical sense — that provides reliable but
approximate information about the primes. Riemann
later found that the logarithmic distribution is
‘modulated’ by an infinite set of waves, where each
wavelength corresponds to one of the Riemann zeros.
We’re in the realm of proven mathematical results here,
and these precisely pin down the primes, so in that sense,
all mystery is gone; but in actuality the mystery has been
pushed back, or displaced. The mystery now is, where the
hell did these Riemann zeros come from?
We can
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calculate hundreds of billions of them, we’ve got a vast,
intricate body of precise mathematical results concerning
them which ultimately brings us to a big, important,
question about whether they’ll all lie on the ‘critical line’
– that question is RH. But ultimately, what are they?
Since the seventies, this idea that they might be
vibrations of something has taken root and has now been
more-or-less universally accepted, on the basis of a lot of
computational evidence together with a mysterious,
suggestive mathematical ‘coincidence’ involving something called the Selberg Trace Formula — and that ties in
with certain unexpected connections with physics.
So if we’ve got vibrations of a mysterious ‘something’
underlying the number system, in a sense the primes are
no longer the mystery, the primes have been taken care
of, the mystery has been displaced. The primes are our
obvious way into the mystery, but ultimately it’s a
mystery about the system of positive integers, about
‘order’, and arguably even about time.
C: To return to the question of order, are the zeros any
more ordered than the primes?
MW: The set of primes and the set of Riemann zeros are
in some sense ‘dual’ structures. There’s a variant of
what’s called a ‘Fourier duality’ between them. To put it
simply, you can use the zeros to generate the set of
primes: if you have just the zeros and the explicit
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formula, you can effectively ‘put the zeros in and get the
primes out’. And it also works in the opposite direction.
So the two generate each other. In a sense the primes are
more well-behaved in that they’re all integers, they all fall
on this nice ‘grid’ of positive integers. The primes can be
explained to a schoolchild, a five-year-old is capable of
understanding the idea of prime numbers. They are
there among the familiar positive integers, the usual
counting numbers, and counting is a ubiquitous part of
our everyday experience.
They’re dual, so in some sense the two could be seen
as equally important, two sides of the same coin.
However, the Riemann zeros are very different – they’re
not integers, they’re what we call ‘transcendental’, irrational numbers; you need a degree in mathematics before
you can even begin to understand the definition of them,
and relative to the total population, only the tiniest handful of people have any real understanding of what is currently known about them. And they appear to have
absolutely nothing to do with ordinary everyday
experience.
C: We could say that the zeros are not a solution to the
problem, but the problem itself, expressed in a domain
that’s more difficult for us to access; the exact nature of
this domain then becomes the real focus of interest.
MW: Yes, the zeros are the problem, and thus the
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problem’s been displaced to somewhere we’re much less
familiar with. Counting, you know...Ancient Greeks and
earlier people could count pebbles out on the ground,
subdivide them into piles and contemplate different types
of numbers – ‘perfect numbers’, ‘triangular numbers’,
prime numbers – and they were able to develop a certain
amount of theory. But that’s just one side of the coin.
On the other side, there was no way they could have
contemplated the Riemann zeros: (a) you need a theory
of ‘functions of a complex variable’, and (b) in order to
calculate more than the first handful of them you need a
pretty powerful computer.
It reminds me of the central image in the film 2001:
It’s as if we’ve dug this monolithic thing up, it’s been
there for aeons, as a structure it’s overwhelmingly impressive, and everyone concerned is flabbergasted, asking
themselves how did that get there, you know: it comes
from somewhere else, somewhere beyond, and it induces a
sense of almost religious awe.
One suspects that if a mathematical structure
underlying or ‘explaining’ the Riemann zeros were to
emerge – that is, if in fifty or a hundred years someone
comes up with something new which ‘explains’ the zeros
in the way the zeros ‘explain’ the primes – then that new
structure is just going to open up another even deeper
mystery. Paul Erdös, who published more mathematics
papers than anyone else ever, and who was primarily a
number theorist, said that it’s going to be at least a
million years before we understand the primes, and even
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then we won’t really understand them.
C: Is it a properly transcendental problem, relating to the
limits of our thought: the more that we think, the further
the problem moves away from us?
MW: Well, again, we don’t know that yet: it may be, but
then who knows – maybe it’ll all neatly tie up somehow.
But it feels to me that the problem has a quest-like
quality. The fact that the metaphorical image of the Holy
Grail has been invoked a few times in the literature, as
well as a lot of language poetically invoking the feminine
and generally suggesting an ‘otherness’, suggests that I’m
not the only one thinking like this. I’ve had an interesting dialogue with some Jungians about this aspect of RH.
The problem of the primes isn’t just different from
other mathematical problems, it precedes them. All other
mathematical problems rely on the fact that there are
positive integers. Without the set of positive integers,
those other mathematical problems couldn’t exist. So the
problem of the primes is the problem in a sense, it’s
beyond the most basic, it’s there before all the others are
there. As soon as you’ve got counting, as soon as you’ve
got any notion of repetition, then the problem of the
primes is there waiting to be discovered.
If we don’t understand the prime numbers, we don’t
understand the positive integers. And if we don’t
understand the positive integers, then I don’t know if we
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understand anything at all, because all science is entirely
built on measurement, and you can’t measure anything
until you can count. All our rational scientific thought
relies on these very basic ideas of order and counting.
One of the most important quotations that I’ve reproduced on my website is this, from Gerald Tenenbaum
(Institut Élie Cartan):
As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to
such an extent that it can legitimately be asked
whether the subject of study of arithmetic is not the
human mind itself. From this a strange fascination
arises: how can it be that these numbers, which lie so
deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the
prime numbers is undoubtedly the most ancient and
most resistant.2
So, in probing the mystery of the prime numbers
we’re effectively on a sort of journey to the centre of the
mind, or of the collective human psyche, and ultimately
to the point where that interfaces with the physical world
which it finds itself inhabiting. That quote perhaps best
conveys some feeling as to why I’m so gripped by this
stuff.
C: The story of the modern theory of primes begins
2. G. Tenenbaum and M. Mendès France, The Prime Numbers and Their Distribution
(AMS, 2000) p.1
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with Gauss’s initial success in predicting approximately
the distribution of primes. How do we get from there to
the pioneering interdisciplinary work that your webarchive charts?
MW: Gauss – although he didn’t publish, he supposedly got there first – Gauss and Legendre noticed that there
was at least a ‘statistical’ thinning out of the primes that
you could quantify. Riemann later uncovered the zeros
of his zeta function – the Riemann zeros – and so was
able to pin it down much more rigorously. But there’s a
fifty-year gap where...actually, I don’t know what mathematicians felt during that time. Practically, they were trying to refine the approximations; Chebyshev and others
improved the approximation of how many primes you’ll
find in any given chunk of the number line. But whether
there was an expectation that eventually someone would
find a way to make this exact, or whether there was a general feeling that ‘this is the best we’ll ever do’, I don’t
know, and I can’t recall seeing anything in the literature
of that period where feelings about this matter were
expressed. Once Riemann’s work came along then noone was really interested in what people used to think.
The history of mathematical ignorance isn’t as well documented
as
the
history
of
mathematical
discovery.
There are some parallels with the situation we’re in
now, where there’s a mystery about this proposed
‘Riemann dynamics’, this hypothetical dynamical system
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underlying the Riemann zeros.
C: The complex plane is the most important mathematical support of RH itself. And here already a transformation takes place – de Sautoy3 talks about it as a sort of
magic mirror we step through — which seems to unfold
things we thought we knew, in a completely different
space — as it turns out, very fruitfully for mathematics
and the physical sciences alike.
There’s obviously something very powerful about the
complex plane itself which, at the very least, corresponds
in some way to physical reality, and so the fact that it was
also the complex plane which facilitated Riemann’s
insight into the prime distribution is itself suggestive.
MW: The complex plane appears to have a life of its
own. Complex numbers are absolutely necessary to
describe quantum-mechanical phenomena. Electricians
use the complex unit i just to work with AC electricity, so
something as ‘nuts-and-bolts’ as the National Electricity
Grid depends on the complex plane. And yet it is this
supremely mysterious thing. I mean, all those fractals
that started to circulate in the 1980’s – a lot of people
don’t realise what they’re looking at, but those are things
that naturally inhabit the complex plane. Without the
complex plane you wouldn’t be able to see such objects,
that’s their natural domain. And then the Riemann zeta
3. De Sautoy, M. The Music of the Primes. London: HarperCollins, 2004
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function, with all its strange properties; Riemann’s big
step was to take a function which Euler had looked at
and ask, what would that do if we extended it into the
complex plane? And what it was found to do then
spawned the great mystery of the Riemann zeros.
Another strange thing worth mentioning: One tends
to think of temperature as existing on a linear scale, a
one-dimensional scale. But in statistical mechanics, by
constructing a function of temperature, the ‘partition
function’, and extending it out to the complex plane, you
find that it has a set of ‘singularities’, off the familiar real
number line, in this other two-dimensional region that
doesn’t seem to have anything to do with temperature or
any other aspect of practical measurable physical reality.
Yet these singularities correspond to phase transitions of the
system. Without the complex plane you’d never have
known they were there. The same thing happens with
the zeta function, it’s got a set of singular points in the
complex plane, the Riemann zeros off the real line. From
the behaviour of the zeta function on the real line, you
would never have guessed they were there.
Various people have put forward models of twodimensional time – imaginary time certainly gets used,
complex time. Such models can be used in attempts to
explain otherwise inexplicable phenomena, but none of
this can be applied to our normal experience of reality,
you can’t really do anything with it. I would say that the
complex plane is still deeply mysterious. It’s ‘behind the
scenes’ of reality as we experience it.
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C: And historically, complex numbers had been
discovered long before there was any sense of their
ultimate utility. Only later did it became evident that
something which seemed to have been a mathematical
fiction, was hugely important to work in these fields.
MW: Absolutely, the word imaginary, you know –
you’ve got the ‘real’ numbers and the ‘imaginary’
numbers – it’s a very unfortunate name, but it’s simply
because of the history of the thing. For quite a while,
no-one thought these things had any ‘reality’ to them,
primarily because they didn’t correspond to anything
experiential in the way ‘real’ numbers were seen to.
C: It’s difficult to ignore this experimental evidence that
complex numbers relate to something in reality: we have
to take account of these things which just impress
themselves upon us. The traits of the complex plane are
obviously real, but they don’t correspond to any actual
object, any actual thing we can get hold of. They’re
distributed through reality itself.
MW: Yes, the system of complex numbers is there, I
don’t know ‘where’ it is, but it’s not just something we
invented. And, interestingly, it’s most directly evident at
the subatomic level. As I said, the theory of AC electricity relies on it, but then ultimately that’s a quantummechanical phenomenon, scaled up to the level where we
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can, say, run a toaster on it. Functions of a complex
variable get used in statistical mechanics, aerodynamics,
etc., but those are fairly indirect manifestations of
something very deep, I feel. The fact that the complex
plane relates so closely to quantum mechanics means that
in macroscopic reality, it permeates everything, as you
say, and yet nobody had a clue it was there until relatively recently. Even after it had been mathematically
brought into consciousness it was still seen as just a
fiction.
As for the primes, you can’t understand the
distribution of primes until you’ve grasped the Riemann
zeros. And the Riemann zeros live on the complex plane,
inarguably. The ‘nontrivial’ zeros, the ones RH
concerns, inhabit a narrow vertical strip in the complex
plane. The RH simply says that they all — the entire
infinite set of Riemann zeros — lie on the ‘critical line’
which runs up the middle of this ‘critical strip’.
Now, to prove RH would be an exact mathematical
task, so RH gets a lot of press – there’s the whole
fame-and-fortune thing, literally a million-dollar prize,
this idea of something like winning the ultimate intellectual gold medal, you know – but you’ve either done it or
you haven’t, it’s very clear-cut. But I’m more interested
in the less clear-cut questions – what are the Riemann
zeros, from where do they originate?
To answer this we may need something else as new
4. See http://www.claymath.org/millenium/
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and unexpected as the complex plane was when it was
first introduced, something we haven’t thought of yet, a
new mathematical ‘environment’ in which these things
will become perfectly clear. But that may well lead to
another body of questions which are even more baffling.
But “from where do the zeros originate” – what does
that mean? They’re seemingly vibrations of something,
but what? What is that thing going to be – is it going to
be a mathematical model of a dynamic system that
people may or may not be able to physically manifest? If
it is possible to physically manifest it and someone
does...what then are we confronted with?
One gets a very strong feeling that until we understand the what the zeros ‘are’, we won’t be in a position
to prove RH. These two issues are tied together. But the
former isn’t yet a precise question, whereas ‘is the RH
true’ is.
C: It is said that in mathematics a question isn’t even a
question if you can’t formulate it precisely: mathematics
is the art of formalising problems, so if you can’t do that
then in a sense it falls outside of mathematics.
MW: Yes, and so something with this kind of
quasi-mathematical character is generally regarded with a
certain suspicion; it’s neither one thing nor the other.
C: A mystery rather than a problem, then.
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MW: Yes, and I suppose I tend to be attracted to the
mysteries.
C: Practically speaking, how does the hypothetical positing of a Riemann dynamics change the nature of the
search for a proof of RH?
MW: It brings other people in, it brings the physicists in.
Before, you had analytic number theorists hammering
away at this problem. And now probability theorists,
geometers and physicists are all contenders, and they all
have pieces of the puzzle. It’s broadened the scene, if you
like, of people concerned with the problem. But it also
has given a deeper sense of what’s at stake; again, if there
is a dynamic system underlying the Riemann zeta
function, well then it underlies the number system; if it
underlies the number system then it underlies everything,
or at least everything that rational scientific thought
concerns itself with. And so, again, we’re force to ask
what is it, where does it ‘live’, what does it ‘do’? And
perhaps the most important question is, what is the time
parameter? Because a dynamical system always has a time
parameter according to which it ‘evolves’ – so what kind
of time are we talking about in this case? So it
basically opens a whole new can of philosophical worms.
It makes me think of what Hilbert said, when he was
asked about RH, he said that it isn’t just the most important problem in mathematics, it’s the most important
problem. And I think a lot of people might just think,
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yes, that’s because he was a mathematician, he was
biased...but I think he knew what he was talking about.
He and Pólya first proposed that there might be a
‘Riemann operator’, that the zeros might be a spectrum of
something. They didn’t suggest a dynamical system as
such, but they could be said to have laid the groundwork
for that. So I think Hilbert may have sensed something
very big going on there, which he was trying to express
in that pronouncement.
C: The first steps towards elaborating the nature of the
Riemann dynamics comes with Julia’s interpretation of
the zeta function as a thermodynamic partition function.
What is a partition function, and in what sense can one
speak of the primes as a numerical gas – Julia’s ‘free
Riemann gas’? Is it simply a useful metaphor taken from
thermodynamics, or is there a more substantial link?
MW: Well, firstly, Julia’s work doesn’t directly address
the issue of the Riemann dynamics, although there may
well be a deep connection there.
Your last question is difficult to answer, but it would
be hard to deny that there’s a sort of a metaphor here, in
that there’s a strong resemblance between certain aspects
of the zeta function and the theory of thermodynamic
partition functions. But it goes deeper than a superficial
resemblance. There are enough corresponding elements,
that Julia included what he called a ‘dictionary’ in the
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paper he first published about this.4 It consists of two
columns, with number theoretical structures on one side
and corresponding thermodynamic structures on the
other. And the correspondences are such that, if you’re
sufficiently familiar with number theory and statistical
mechanics, you can’t deny there’s something...there’s a
very strong link there. So you could call this a metaphor,
but I would maintain that it’s more than just a metaphor
in the familiar sense, i.e. a useful way of explaining what
something is by means of something else which isn’t
directly related to it.
Now what is a partition function, in statistical physics,
or statistical mechanics? Well, in classical mechanics, a
billiard table is often used as an example: you’ve got a
finite number of billiard balls bouncing off each other,
bouncing off the sides, they’re colliding, energy is being
transferred between them, there are various angles,
positions and momenta involved. And the idea is that
you’ve got a sufficiently simple system that you can keep
track of each individual object and what it’s doing. But a
problem arises when you’ve got something like a box of
gas: that’s effectively like a giant three-dimensional
billiard table, but there are too many components to keep
track of what each one is doing. You’re not actually going
to be able to do anything in that way, so you’re going to
have to study it in the sort of way sociologists study
society — they can’t possibly consider all the specifics of
4. Julia, B.L. ‘Statistical theory of numbers’, in M. Waldschmidt, et. al. (eds.)
Number Theory and Physics. Springer Proceedings in Physics 47 (Springer, 1989)
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each individual person, so they must look at overall
statistical trends in the population.
Suppose you had a quantity of gas particles in this
room, and they were all roaming freely. It would be very
surprising to find them all clustered up in one corner.
One expects a more uniform spread. But, there’s no real
reason they can’t do that. It’s like if I toss a coin fifty times,
I’d be very surprised if I got fifty heads or fifty tails, but
there’s no reason why that can’t happen. That would be
no more unreasonable than any other outcome of fifty
coin-tosses, it’s just that it’s extremely improbable because,
unlike any other outcome, there’s only one way of arriving at
it. Similarly, there are proportionally few possible configurations of those gas particles where they’re all squashed
in one corner, compared to the vast proportion of configurations where they’re more-or-less uniformly distributed.
Now suppose you have a box of gas, and the gas
consists of particles which can jump between different
energy levels in an effectively random way. This time
you’re concerned, rather than with the spatial
distribution, with the total energy of the system – that’s simply what you get when you add together all the
individual particle energies. You can ask about the
probability of the system having a particular total
energy, and it turns out to be rather like the situation with
the spatial distribution. That is, the system tends towards
a mid-range total energy on the whole, while the highest
and lowest ranges of possible total energy are much more
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improbable – because their occurrence requires something akin to a huge number of coin-tosses producing
almost all heads or almost all tails.
So what you’re looking at with thermodynamics is the
probability that you’ll find a box of gas or some similarly complex system in one state or another. And the
partition function takes a unit of probability and
‘partitions’ or subdivides it, so that you end up with a
curve describing in precise terms the relative probability
of finding the total energy of the system at any particular
level. So the partition function will basically return
probabilities that a system is in one of any number of
possible states. The partition functions Julia refers to are
functions of temperature — as the temperature of the
system varies, the probabilities also vary, and the
partition function is able to provide a precise probabilistic
distribution of possible total energies at any given
temperature.
Now partition functions, it turns out, are the key to
understanding statistical mechanical systems; they
‘encapsulate’ such systems. The partition function in this
context is a function of temperature, and temperature
would naturally be seen as a variable which varied on the
real line — on the positive real line, if you’re working with
absolute temperature.
Well, nineteenth century
mathematics suddenly allowed the possibility of
extending such a variable to the complex plane,
regardless of what a complex-valued temperature might
actually refer to. You take your partition function which
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is supposed to be returning probabilities of a system
being in some energy state or other based on a
real-valued temperature variable, extend it to the
complex plane, and you find there are singularities
hidden out there which tell you about the possible
existence of phase transitions in your system. These are
very important for understanding the system, but, as I
said earlier, you wouldn’t see them if you didn’t have
access to the complex plane.
Now Julia wasn’t the first — George Mackey got there
first, although it wasn’t widely noticed. Julia discovered
it independently and then Donald Spector, a couple of
years later — they all noticed that if you treat the primes
as your basic particles, and each prime p is thought of as
having as its ‘energy’ the natural logarithm of p – that
logarithm turns out to be very important, logarithms
show up everywhere in analytic number theory – then
the Riemann zeta function very naturally falls into the
rôle of being the partition function of an abstract
numerical ‘gas’ which is made of this set of particles –
what Julia calls the ‘free Riemann gas’. Imagine a
fluctuating integer, where prime factors are coming and
going all the time, joining and leaving, so the energy of
that integer is going up and down, the more prime factors
there are the higher the energy, and the less prime factors
the lower the energy. The zeta function naturally
becomes the partition function of such a system. The
‘pole’ of the zeta function – this unique singularity of the
zeta function at the point 1 in the complex plane where
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the zeta function isn’t defined, where it effectively
becomes infinite – corresponds very naturally to something in thermodynamics called a Hagedorn catastrophe, a
phenomenon involving the energy levels crowding
together so the system hits a critical state and shifts into
an altogether different mode. So the pole of the zeta
function is associated with this ‘catastrophe’, and based
on what I was just saying, the Riemann zeros also become
linked to phase transitions, in a way that no-one entirely
understands. And there’s more...those are just the basic
points, there are further subtleties which suggest that, in
some sense, thinking of the zeta function as a partition
function goes beyond mere metaphor. It’s a metaphor,
but it’s a metaphor that goes deep enough to suggest to
me that the number system has some sort of quasiphysical quality.
C: How are we to interpret this? There’s a perplexing
quality to these propositions, one is never sure whether
what’s being revealed is a progression, or simply a
restatement of the same problem in different terms.
MW: Possibly, but you see, the mathematics that’s come
out of studying things like boxes of gas, that that should be
applicable at all to studying something as fundamental as
the positive integers, to me comes across as sort of
uncanny. I think that’s a good word to capture how a lot
of people have reacted to these discoveries. It’s hard to
see how it’s simply a reformulation of the problem.
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You’d never have got there if you hadn’t studied the
boxes of gas in the first place. When you ask how we can
best interpret this, the only answer I can come up with is
I honestly don’t know. To me it points to something
fascinating which we haven’t yet entirely understood or
taken into account.
Now, interestingly, Alain Connes’ (College de France,
IHES, Vanderbilt) model involving what’s called a
C*-dynamical system – his attempt to try and describe the
Riemann dynamics, which hasn’t yet fully succeeded,
although it’s certainly opened up some new vistas – was
inspired by Julia’s paper, but Connes uses the partition
function in a somewhat different sense. The partition
functions I’ve been describing, the ones associated with
boxes of gas, etc., could be called ‘classical partition
functions’ as they belong to ‘classical statistical mechanics’. But there are also partition functions used in
quantum statistical mechanics, which take some of the
same concepts down to the quantum level.
Connes takes certain elements of quantum statistical
mechanics and applies them to the zeta function, treating
it as a partition function, and this reveals certain things
which again push the metaphor, in my mind, so far that
it can’t be regarded as just a metaphor.
C: So there is a direct link between the quantum-mechanical interpretation and the thermodynamic?
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MW: I think there must be, although it’s not yet entirely
clear what it would be.
Of the two most
extensive pages in my web-archive, one deals with the
spectral interpretation — Hilbert and Pólya’s suggestion
that the Riemann zeros might be vibrational frequencies
of something and Michael Berry’s (Bristol University)
physics-inspired work concerning what that ‘something’
might be. Berry and his colleague Jon Keating have
outlined a whole set of dynamical properties characterising this hypothetical Riemann dynamics. And the other
page deals with the thermodynamic or statistical
mechanics side of things – you’ve got Julia, Spector,
Mackey, who all put forward the idea that the zeta
function is a partition function, which would suggest that
the zeros are in fact phase transitions of something. So
these two currents of research are seemingly different
approaches, not obviously compatible. Alain Connes has
begun to bridge the gap, though. He has taken Julia’s
suggestion about zeta as a partition function, shifted it
into the realm of quantum statistical mechanics, and then
brought in p-adic and adelic number systems, and a lot of
other very deep mathematics including something called
noncommutative geometry, which is about as difficult as
current mathematics gets. He’s managed to describe a
dynamical system, or at least sketch out the beginnings of
one, which produces the Riemann zeros as vibrational
frequencies, but where the zeta function is also playing
the rôle of a partition function, so there is a link there.
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C: Connes’ adele is an infinite-dimensional space in which
each dimension is folded, so to speak, with the frequency
of each prime.
MW: Yes, that’s almost it. An adele is a generalised kind
of number which contains an infinite number of coordinates, one associated with each prime number,
effectively, and then an extra one, which corresponds to
the continuum of real numbers.
The adelic number system embraces all of the
different p-adic number systems — 2-adic, 3-adic, 5-adic, 7adic, etc. p-adics and adeles constitute yet another aspect
of number theory finding its way into physics, thereby
suggesting that things aren’t the way we thought they
were.
The Archimedean principle, the basic principle of all
measurement, is based on rational numbers, on ratios. If
you have a line segment and a longer line segment, by
taking the shorter line segment and joining it end to end
a finite number of times, you will always be able to
exceed the longer line segment. That seems obvious –
it’s the basis on which I can take a ruler and measure this
room. If I kept joining it end to end and I never got to
the end of the room, then measurement wouldn’t work
very well! So, the universe at the macroscopic scale is
Archimedean: the Archimedean principle applies. And
the number system we generally use, the continuum of
real numbers, is an Archimedean system.
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Now, the real number continuum is based on a
particular arbitrary choice of how we ‘close’ the system of
rational numbers. The rational numbers are fairly
simple, well-determined, or given, if you like – canonical.
You’ve got your integers, and then you start taking
ordinary fractions and that fills in the gaps – it doesn’t fill
in all the gaps, but it densely fills in the number line. The
‘holes’ that remain are the irrational numbers, which
can’t be expressed as ratios of integers, √2 being the one
that, it’s widely believed, was first discovered, and π
being undoubtedly the most famous. But there’s not just
a handful of exceptions, these irrational numbers are in
some sense more common than the rational numbers.
The question is, given the system of rational
numbers, how do you fill in the holes, how do you seal
the whole thing up? Well, the method we’ve ended up
adopting produces the system of real numbers, which is
a system in which the Archimedean principle applies.
And that’s based on defining the holes, the irrational
numbers, as the ‘limits’ of sequences of rational numbers.
But to define the limits, you have to have a sense of
distance; put simply, a sequence converges when its
elements get closer and closer to something, and the
notion of ‘closer’ requires some sense of distance. The
sense of distance we use to define the real numbers is the
obvious one: the distance between any two rational
numbers on the real number line is what you get when
you subtract the smaller from the larger. But that’s an
arbitrary way of defining distance. It turns out that,
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within the logical constraints which apply, there are an
infinite number of other meaningful, consistent ways you
can define what distance is, and each leads to a different
notion of ‘closure’ and hence to a different number
system. So you’re still starting with the rationals, but the
way you ‘fill in the holes’ is completely different, and you
end up with a different kind of mathematics. Now this
was discovered by Hensel in the late 1890’s, and very
quickly the possible ways of closing the rationals were
classified. It turns out that there are infinitely many of
them, and that they correspond to prime numbers:
there’s the 2-adic system, the 3-adic system, the 5-adic
system, the 7-adic system, all the way up, and then
finally there’s the ∞-adic system, which corresponds to
the usual system of real numbers, and which suggests the
existence of what’s called the ‘prime at infinity’, a deeply
mysterious thing, which an Israeli mathematician called
Shai Haran has written a whole book about5.
But the point is, in a 2-adic, 3-adic or 5-adic number
system, the distance between two rational numbers has
nothing to do with the traditional distance between two
points on a ruler anymore, rather it’s about arithmetic
relationships involving divisibility of numerators and
denominators by the prime p which characterises the
p-adic system in question. So things that would look very
close together on a ruler could be huge distances apart,
and vice versa, things that are vast distances apart in a
5. M.J. Shai Haran The Mysteries of the Real Prime. London Mathematical Society
Monographs, Oxford: OUP, 2001.
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normal Euclidean sense could be very close together in a
p-adic sense.
C: And the adelic system is built up of all these?
MW: An adele is a generalised number which has an
infinite number of co-ordinates. One’s a 2-adic number,
one’s a 3-adic number, one’s a 5-adic number: one for
each prime. They’re usually written as:
(2-adic number, 3-adic number, 5-adic number...; real number)
so you get one of each. When, at the end of the
nineteenth century, these p-adic number systems were
discovered, it was realised that we’ve been doing all our
physics on the basis that time and space are like the real
number continuum. That’s the assumption; all the
Einsteinian, Riemannian, Minkowskian manifolds, spacetime manifolds, were based on real numbers extending in
different dimensions. But why should we assume the
universe is ‘real’, in that sense? You could formulate a 17adic manifold and do space-time physics in it, or a 37-adic
manifold; but then, why pick one prime rather than
another? Hence the idea arose, why not chuck them all
in, create a system which involves all of them at once —
this is the adelic approach, described in very crude terms.
Hence p-adic and adelic physics — there are people developing models of p-adic physics where the p is just left as
an arbitrary p, where it would work for any prime,
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basically re-building physics according to these new
number systems. So you’ve got p-adic models of time,
p-adic models of probability. A lot of it really turns your
ideas of the world on their head.
Now Connes has come up with a dynamical system
on a space of adeles, which generates the spectrum of
Riemann zeros. The problem is that the system he’s
starting with has already got the prime numbers built in
to it, so some people would say, well, he’s really only
reformulated the problem. But I suspect there’s a lot
more to it than that. It’s not quite the dynamical system
that is being sought in connection with RH, but it is
widely seen as a valuable step in the right direction.
Even more interesting than Connes’ work, from my
point-of-view, is that of the lesser-known Michel Lapidus
(University of California-Riverside), another Frenchman
with a staggeringly broad view of mathematics and
physics. I recently had the privilege of proofreading his
latest book – I hope it will come out this year, it’s been a
long time in the pipeline. It’s called In Search of the Riemann
Zeros and it brings all of these ideas together. And he’s
taken Connes’ idea even further. He’s got a set of ideas
involving quantum statistical mechanics, p-adics and
adeles, dynamical systems, vibrational frequencies,
partition functions, it’s all in there, but also fractals, string
theory...
C: The adele already intuitively brings to mind string
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theory, because of the way everything seems to be bound
up with the nature of these peculiarly convoluted spaces.
MW: There’s been a lot of work done on p-adic and
adelic string theory, but that’s not quite what you mean.
Lapidus has actually come up with a fascinating
connection. He was working on something he called
‘fractal strings’, but these didn’t have anything to do with
the ‘string theory’ physicists study, it was just the name
that he had given to these particular mathematical
objects. And then he generalised them to something
called ‘fractal membranes’. But since he came up with
that, oddly enough, he’s found that aspects of string
theory relate directly and unexpectedly to the mathematics.
His model involves a dynamical system, a noncommutative flow of fractal membranes in a moduli space...
C: Which sounds wonderful!
MW: Yes, at a very naïve level, I just enjoy all the
extraordinary language. But, more seriously, I have a
certain emotional investment in Lapidus actually being
onto something, because if he’s correct, it turns out that
his ‘flow’, this very strange, highly counterintuitive, noncommutative geometrical ‘flow’ projects down into a
simpler realm, into the number system, as a flow of
‘generalised prime numbers’ on a line. This is very close
to some strange speculative ideas I made public back in
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1999 6. Lapidus contacted me a few years ago to say this,
as it had come to his attention when I first put it up on
the Web. Now, it’s not that I influenced him, it’s almost
as if I caught a glimpse of some future mathematics which
will follow from his current work. I don’t know how I can
explain what happened… it’s as if I caught a glimpse of
something which was coming, but I didn’t have the
language to describe it accurately, so I just described it as
well as I could in this rather naïve way. And so in a way
I now feel somewhat vindicated concerning my slightly
crackpot idea, because of Lapidus’ work.
*
In some ways, I think all this intellectualising and
mathematics isn’t really that good for me, and isn’t really
what I ‘should’ be doing. But part of me can’t entirely
detach myself from it. The speculation I just mentioned,
which now appears to be at least partly vindicated,
gripped me in a profound way. This event had a
precedent a few years earlier when I became convinced
there was some connection between the Gaussian
probability distribution and the prime numbers: that was
driven by a sort of compulsion that was, looking back,
was quite...not psychotic – it didn’t lead to any sort of
negative behaviour – but it did rather take over my
psyche.
I was one of those kids who, it was obvious fairly
early on, could excel at mathematics, and being a fairly
6. See http://www.maths.ex.ac.uk/~mwatkins/isoc/evolutionnotes.htm
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scrawny, unattractive young person, one latches on to
anything one is good at — it provides a sense of
importance. It was as simple as that, it wasn’t any sort of
noble motive for seeking the truth or anything. I was
living in the States as a teenager, starting to get interested
in things like radical art movements and philosophy. If
circumstances had been different I’d have probably
studied something else, but I wanted to get out of the
States, that was quite a big thing for me then, so to get
into a British university my best bet was to apply to do a
maths degree, which I did. And then I sailed through
that, and got offered a place on a PhD programme, which
seemed like a great idea – effectively being paid to
explore ideas which I found quite interesting and which I
seemed to have an aptitude for exploring. So that was all
fairly accidental, and there was no real motive behind it,
if you like. It was just the way my life unfolded. But by
the time I was doing the PhD I was starting to engage
with a lot of other non-mathematical ideas and people,
and there was a real sense that, hang on, where is this
going, is this really what I want to be doing? At that
point I was more interested in ‘seeking the truth’ – it
sounds a bit grandiose, but I wasn’t interested in a stable
career, and the idea of deriving some sort of self-esteem
from being an accomplished mathematician, that no
longer seemed to be of any importance. So I started
thinking, if I’m seeking the truth, is the truth to be found
here, is this really what I should be doing? And then the
disillusionment set in. After a year of being on a Royal
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Society European fellowship, there was a distinct sense
that modern mathematics was becoming irreparably
fragmented, and I felt like I was being made very
comfortable in an ivory tower, in a vast field of other
ivory towers, between which there was relatively little
communication. And then there were all sorts of personal factors, just the way my life was going, people I knew,
a sense of imminent global catastrophe...
This was 1995, so perhaps there was a touch of
millenarian hysteria involved! There was a sense that, as
a mathematician, I was part of the problem rather than
part of the solution. A lot of my friends were involved in
ecological activism and things like that, and I started to
formulate a worldview wherein science had become the
new, unacknowledged, religion of industrialised society,
and mathematics was the inner priesthood of science. To
put it in very simple terms, Western culture runs on
science, and science runs on maths. So I saw myself as
being trained up for this priesthood which was unconsciously steering the world to complete destruction and
meaninglessness. And so there was a sense of guilt,
almost, that I was involved in this. So I just broke out
and floated around doing all sorts of interesting things for
a few years, had a great time — I don’t regret that at all.
I never imagined that I’d get involved in mathematics
again.
But then certain ideas about prime numbers started to
percolate in my mind. I’d never really looked at number
theory in any detail, had just a very basic number theory
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course as an undergraduate. But shortly after I ‘dropped
out’, I started thinking about prime numbers and the fact
that they have a sort of ‘random’ quality…and at the
same time thinking about the Gaussian distribution, the
bell curve, and the ubiquity of that, the fact that almost
anything that you can name, count, measure, and gather
data on tends to scatter along this particular ideal
exponential curve. I remember posting a question on an
Internet newsgroup back in 1995, trying to get somebody
to explain to me why this thing shows up everywhere:
not just in the biological realm, but in much more
convoluted ‘cultural’ realms – I expect that you could
count the number of appearances of a letter of the alphabet on the front page of a newspaper over so many years
or months, and you’d find the same thing. And the
purely mathematical explanations put forward made
sense to some extent, but I still felt there was some huge
mystery lurking behind the Gaussian distribution, the
fact that it shows up everywhere. I scribbled all sorts of
half-baked ideas down, some of which seem ridiculous
now, some still of great interest. But I became convinced
– and I still don’t know where this came from – I became
utterly convinced that the distribution of prime numbers
in some sense was very deeply linked to this, to the
ubiquity of the Gaussian distribution, that they were two
sides of something.
And what’s strange is that almost seven years later, I
discovered there was something called the Erdös-Kac
theorem, which was proved in 1940, and which I’d never
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even heard mentioned before. This was the beginning of
probabilistic number theory, and it basically states that the distribution of prime factors of large integers follows a
Gaussian distribution.
Obviously, the larger the
integer, the more prime factors it is likely to have, but you
rescale in a way that takes that into account, so you’re
dealing purely with the seeming randomness in the fact
that some numbers have got lots of prime factors and
some numbers have only got one – and you end up with
a bell curve. And not just an approximate one, this is
what really struck me: if I was to measure the population
over time of sparrows in the garden out there, or the way
that those sunflower seeds fall on the ground [pointing to
bird-feeder hanging in a tree], if I had large enough
numbers I may well get very nice approximations of the
bell curve. A high-resolution computer image might even
match the ideal mathematical bell curve in every detail.
But they’re always approximate; in fact all use of
statistical inference in science is based on finite amounts
of data, which give rise to approximate bell-curves or
other distributions. With the Erdös-Kac theorem on the
other hand, the n, the number of elements in your data
set, actually tends to infinity. This is what really struck me
about all this: n can tend to infinity only when you have
an infinite amount of whatever it is you’re dealing with.
And integers are the only thing, effectively, which we
have – at least theoretically – access to an infinite amount
of.
So, I haven’t fully delved into this, but there’s a
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problem with the use of infinity in statistics and probability theory. It’s fine in some sort of abstract Platonic
sense, but when you start applying it to the world, there is
no infinity. But it does apply absolutely, precisely – and
this is the theorem which Erdös and Kac proved – that as
n tends to infinity, the distribution of prime factors tends
to this distribution. So, in some sense, that’s the only
‘true’ Gaussian distribution there really is, the ‘oldest’
one, the most primordial. As soon as you’ve got positive
integers, that’s hidden there within them. Any other
instances of the Gaussian distribution, you know, bird
populations or currency fluctuations or anything else like
that, not only are these approximate, but they require all
sorts of complicated categories and definitions. So,
anyway, I still can’t quite explain why I was so gripped by
this idea of the prime numbers and the Gaussian distribution being linked, but I was, and it’s as if I was somehow
unconsciously aware of something and couldn’t manage
to pin it down, you know. I tried endlessly to find some
way of relating these things and failed. Had this been
2005 rather than 1995 I probably would have quickly
found out about the Erdös-Kac theorem using websearches.
So as a result of this unresolved compulsion, I had a
certain amount of prime number-related activity going on
in my mind. Then, in the winter of 1998 I went back to
the States to visit my parents who were still out there, and
I had a lot of free time. I found a long thin piece of
cardboard and drew a number line, circled all the prime
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numbers, and then started drawing arcs between the
prime numbers and their multiples. So every number
was connected to all of its prime factors by arcs
emanating out from that number to the left. The
number fifteen would have two arcs emerging from it,
one going to number three, the other to five. And a
prime number would have no arcs going to the left, only
arcs going to the right. Now obviously you can never
draw the complete thing, but I drew enough of it that you
could get a sense of there being something, a connectedness, a ‘messy’ connectedness, like a nervous system, or
mycelium, or...I don’t know, I can’t quite describe it, but
I just spent a long time looking at this, I had it up on the
bedroom wall. And as a result of internalising that image,
I started to think that it was perhaps the gaps between the
primes that were most important…but I was somehow
naïve enough to think that possibly no-one else had
thought of that, whereas in fact quite a lot of work has
been done on the gaps between the primes and yes, they
are important. But I started thinking that maybe the
gaps, suitably rescaled, are the things which distribute in
a Gaussian way. I tried to run some computer models,
to calculate the gaps and analyse their distribution — but
not having access to the necessary computational power,
that wasn’t really going anywhere.
And then this image of the interconnectedness of the
primes, the whole number system as a single connected
entity, with each prime as a sort of ‘nexus’, the whole
thing exploded in my mind — it was something very
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sudden, and the initial impression I got was that the
primes themselves were imbued with a sort of ‘charge’...I
think I’d read somewhere that average gaps between
consecutive primes are logarithmic, that is the average
gap between a prime p and the next prime is log p, the
natural logarithm of p. Obviously the gaps can vary
wildly from this average, but the average is a precise
mathematical result, becoming increasingly precise as we
allow p to tend to infinity. I was suddenly gripped by this
idea that the primes themselves were imbued with a kind
of charge, something like an electrical charge, and that
that log p was the clue, that was the charge of the prime
p. At the time I was unaware of Julia’s thermodynamic
approach which associates with each prime p the energy
log p, and also that certain proposed dynamical schemes
involve ‘orbits’ with period log p associated with each
prime p.
C: So the magnitude of the gap before the prime would
be its charge?
MW: Well, for sufficiently large primes p, the gap before
and the gap after would both be approximately log p.
And I had the idea that these primes were in some sense
repelling each other and that the bigger the prime, the
greater the charge and the stronger the repulsion, hence
the bigger the gap. This all came tumbling in as a single
thought, really — the account I’m giving now is an
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attempt to reconstruct and coherently describe it. But
rapidly following this initial impression was the idea was
that, well, if there’s that kind of repulsion involved then
what I’m looking at is a frozen image of something which was
previously in motion — this is what I got a very strong innervisual sense of. I try to describe it to people like this:
imagine attaching a wire to a wall and then stretching it
away from the wall, effectively off to infinity, and then
marking out with tiny white dots equal spaces
representing the integers, and then imagine little tiny
magnetic beads, mutually repulsive particles, positioned
along the wire at positions 2,3,5,7,11, etc., that is, at the
positions of what we call the prime numbers. Now set up
a camera, and then subject the whole area to a huge
fluctuating magnetic field, causing the beads to move up
and down the wire, driven not just by the field, but by
their mutual repulsion. Film that, and then run the film
backwards. What you’d see is all these particles moving
around on the wire and repelling each other, responding
to each other, and then eventually coming to rest at the
positions we associate with the primes. That’s the image.
Now I was well aware of the obvious question: how
do we interpret the time parameter here? This is a huge
problem – we’re not talking about time in the familiar
clock sense, not in the historical sense. I certainly wasn’t
under any illusion that anything like this had ‘happened’
at any point in the past. I was suggesting that the system
had a ‘past’, but that it wasn’t part of the historical past,
rather of some other time-like dimension. And rather
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than thinking, that’s ridiculous, I won’t think that, I tried
to suspend disbelief and see where it would take me. So
the basic thought then was, okay, if what we’re looking at
is a frozen image of something which was previously in
motion, the motion must have subsided for some reason
– so what we’re looking at must be something in a state
of equilibrium. So, what kind of equilibrium? Well, I
came up with a crude notion of ‘arithmetic equilibrium’:
Why have the magnetic beads come to rest where they
are? Well, if we freeze the motion at any moment, so
you’ve got an infinite sequence of tiny beads whose
positions don’t necessarily correspond to positive integers
– they could be any real numbers – and then generate all
possible finite multiplicative combinations of those
numbers, that would produce something analogous to
the positive integers. The positive integers, recall, can be
generated as the set of all finite multiplicative combinations of the primes. But these new ‘integers’ would not
be anything like the familiar integers, they’d generally be
all over the place. They wouldn’t be nicely arranged,
equally-spaced. But if the particles ever happened to
reach the point where they collectively inhabited the
positions associated with what we now call the primes,
the ‘integers’ they’d generate would be equally spaced.
So, I thought, it’s equal-spacedness which is a key to this
‘arithmetic equilibrium’ which, according to my scheme,
has been achieved in the number system.
C: Something like an entropic sequence, heading towards
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an attractor.
MW: Something like that, I was thinking in terms of all
sorts of ideas I had partial understanding of – my understanding of physics is very piecemeal, it was even more so
then. So many ideas were feeding in. I started to think,
how would it begin? Maybe something like a big bang,
where you’ve got all the particles squeezed together at the
wall, at the end of the wire, but with something like an
infinite magnetic field produced by the wall, and then you
let go, and they all explode outwards. At any moment
you could freeze the image and generate all the finite
multiplicative combinations, the set of ‘integers’ that they
generate: I called these ‘generalised primes’ and
‘generalised integers’. Well, it turns out that Arne
Beurling, a relatively obscure Norwegian mathematician,
had come up with this idea of generalised primes and
generalised integers many decades previously. To better
understand the familiar primes he’d started looking at the
question, suppose we ‘change’ the primes, what can we
then say about the associated integers and their
asymptotic distribution?
Martin Huxley (Cardiff
University), who’s quite an eminent number theorist, got
in touch with me as a result of my original website, to say,
oh yes, there is actually a name for those, they’re called
‘Beurling generalized primes’.
C: The distinction being between the primes as we know
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them and, as it were, a generalised function of ‘priming’
by which a number system is generated.
MW: Yes, it’s a bit like that, taking the idea of the primes
not as indivisible integers, but as a set of generators. But
the idea of them flowing or moving, no-one as far as I
knew had ever put that idea forward. And so I came up
with what I decided was almost a ‘creation story’, some
sort of strange mythological mathematics – the creation
story behind the number system. Whether there was this
‘big bang’ thing at the beginning or not, I wasn’t
sure...but the idea was that, okay, these generalised
primes were somehow set in motion. Remember, there
are these generalised prime particles, and then there’s a
kind of invisible set of generalised integers that they’re
embedded in, that they’re generating, which are also in
motion. And, at any moment, the ‘heterogeneity’ of these
generalised integers, their lack of equal-spacedness, is
creating some kind of ‘tension’ which is affecting the
particles’ charges. The idea of fixed log p charges gave
way to the idea of fluctuating charges, governed by the
spacing within the generalised integers at any given
moment. So you can almost think of the distribution of
these generalised integers trying to space itself out by
‘influencing’ the generalised primes and their charges so
that their mutual repulsion eventually leads them to a
stable configuration, an attractor point – that would be
the arithmetic equilibrium. Having reached that – the
familiar configuration of primes – the generalised integers
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would be nothing but the familiar positive integers
1,2,3,... The perfect equal-spacedness of these would
result in all forces on the generalised primes dropping
away, and the number system has then ‘come into being’.
That was the ‘story’ I came up with, that all came a
bit later, trying to make sense of this image that I originally had of the primes being charged, mutually repulsive,
and in motion — or having been in motion. At the time, it
had felt like, this is profoundly important and I have to
act on it, I was being somehow compelled to act on it. It
felt like the most important...certainly the strangest idea
ever to enter my mind. And, insofar as I can grasp what
is meant by ‘numinous’, it was charged with a numinous
quality.
I was hoping to be able to actually describe the
scheme in serious mathematical terms, to reveal that
there was some mathematical integrity behind it, but that
never happened...So all I had was this nebulous idea
about an evolutionary dynamical system underlying the
primes. And it was an idea which seemed very strange,
I can’t emphasise that enough — I couldn’t really justify it
using any sort of logical or mathematical reasoning, and
yet it gripped me psychologically with such force that I
couldn’t let go of it, I was driven to try and make sense
of it. And that led me to create a website...you know, this
is what you do in 1998, you create a website, and then
you start emailing various eminent mathematicians and
physicists to try and get them to look at what you’re
doing. And as a result of that, a few people were quite
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helpful and responsive, I was sent some relevant
literature, and I started to realise that actually, there are a
lot of strange, unexplained connections between number
theory and physics. These things seemed to me to be
circumstantial evidence supporting my strange insight,
whatever it might have been, or whatever value it might
have had. They too suggested the number system had
some mysterious ‘quasi-physical’ character. This may
have been wishful thinking on my part, but the material
was undeniably fascinating in its own right, so I started
compiling it into a web-archive, intended to, at least
indirectly, back up my idea. Eventually, though, my
original idea began to become a bit of an embarrassment
to me – it seemed quite nave and ill-informed. So, as the
archiving took on a life of its own, and I became
fascinated with all this serious maths and physics that I
had become aware of, I gradually buried the original idea
inside a vast web-archive. But I never entirely removed
it, somehow still sensing, or hoping, that there was something of value there.
All my attempts to come up with a mathematical
model, a dynamical system that would correspond to that
image, had failed. I had struggled because I didn’t have
anything like the mathematical abilities that would be
required for that. And in fact, I now feel vindicated in
that it’s not that I wasn’t capable enough to do it; in order
to describe anything like a flow in this space of Beurling
prime configurations wherein what’s called the classical
prime configuration, the usual primes, constitutes some
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kind of dynamical equilibrium – in order to describe anything like that you need to do what Michel Lapidus has
done, and introduce a noncommutative flow on a moduli
space of fractal membranes. And there was no way in
1998-9 that I could have had access to those ideas. So –
and again, this isn’t a serious proposition, but the only
way I can make sense of this for myself – it was as if I’d
caught a little precognitive glimpse of some future
mathematics, sensed the importance of it, tried to get it
down, but didn’t have the language to get it down, did
the best I could, and put it out on the Web. This then led
on to me putting a lot of time and effort into what was
effectively public service web-archiving for a few years,
which has been quite fulfilling, but it was initially just a
consequence of the original ‘flash’, and the compulsion it
induced in me. Now I’m feeling somewhat vindicated
that someone appropriately qualified has shown that
there does appear to be something like this underlying
the number system.
C: Is there an analogy between what you’re describing
and what happened historically with non-Euclidean space
— could it be seen as an arithmetical version of that, with
the unknown time parameter as something as unanticipated as the curvature of space?
MW: Yeah, in the sense that you’re breaking out of what
is considered to be the only possible version of
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something, into a whole range of possible versions, and
that initially seems ‘mad’ to many onlookers.
C: At the time, the idea that space could be folded or that
space could be curved seemed insane. Nevertheless, such
new generalisations are arguably the very movement of
science itself.
MW: I think it was Gauss, Boylai and Lobachevsky who
simultaneously came up with the same basic idea of
parabolic geometry, and at least one of them was afraid
to even mention it to anyone. If I had still been involved
in serious mathematical research in 1998-9, if there had
been a career at stake, my guess is that, having had the
same experience, I may well have thought twice about
going public with these ideas. Whereas as it was, it
didn’t really matter.
C: An interesting example of how being embedded in a
discipline, having a reputation, and no doubt having
funding depending on it, would actually stop you from
saying something — there wouldn’t be any channel
through which to get it out.
MW: In a way, I was in a perfect position to just have a
go, to push it out there.
I’ve read accounts of mathematicians trying to
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describe how they made certain great conceptual leaps.
The big difference is that the leaps they made were into
something that could actually be mathematically
described, and ultimately, you know, were incorporated
into legitimate mathematics. Whereas I just had a sort of
mad flash, a glimpse of something which, as yet, is not
legitimate mathematics, it’s just a vague impression.
C: Yet the structural detail in which you described it
makes it something more than simply a vague idea.
MW: Well I’m not sure that the detail of what I’ve
described adds any validity. Had it not been for Lapidus’
work coming along, I probably would have entirely
disowned it by now. But at the time, there was a
conviction that there was something in it, but it was hard to
know what to call it. There was an awkwardness
because, it falls between the usual categories...I suppose it
could be called phenomenology or something, there’s
probably a legitimate-sounding name that someone could
come up with. But when I put it out on the Web I was
quite careful, because I was well aware of all sorts of
cranks on the Web ranting about how they’ve discovered
this or that revolutionary idea, or proved Einstein wrong,
or whatever. And I so I tried to be very understated in
how I presented it – you know, I’ve had this idea, and I
don’t know what it means, it may well be meaningless,
but I invite people to either show me why it’s
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meaningless, or else indicate what it might lead to. And
gradually it began to happen. But I don’t know if it really contributed to anything. I think Michel Lapidus
would probably have reached the same conclusions
regardless. Perhaps it did influence him, I don’t know,
but I don’t think so. So in a way, if I did catch a glimpse
of some sort of future mathematical discovery, it would
have occurred anyway, so what’s the value of what I did?
C: At least, it does lead one to think about mathematics
not in terms of the points at which people draw
everything together, make it into a formal system, but
rather these discontinuous moments when, inexplicably,
things move, things split apart and something new opens
up?
MW: A crack opens up and something doesn’t quite
make sense.
C: From what you know of the mathematical community, is it the case that the sort of research you are pursuing
is not accepted, that they’re not interested in it?
MW: There’s a small enclave of perhaps slightly more
open-minded, more unusual mathematicians, who are
prepared to discuss these sorts of things privately. The
vast majority are slightly bemused or just not interested,
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they’re too busy with their own work to stop and think
about what it all might mean. Mathematicians aren’t
generally encouraged to think about ‘meaning’. They
don’t really need to, they’ve got a very exact discipline,
they’ve got theorems to prove and things like that.
Basically, what I’m doing, I couldn’t call it mathematical
research. You’ve called it fundamental research...you
could call it that, I know what you mean. The way I see
it I’m just trying to raise certain questions and generate
discussion, and I’d say the vast majority of the mathematical community just isn’t going to engage with that, which
is okay. Because I’m not actually doing mathematics, I’m
not engaged in mathematics research in the way they are;
I’m playing a different game, asking questions about what
mathematics means, what it is, how we relate to it. But
at the same time I’m not part of the philosophy-of-mathematics community either, which is involved in something much more rigorous and disciplined than what I’m
doing.
I suppose because I’ve got more time I’m in a better
position to just stop and think: what’s the point, why are
we looking at this stuff anyway, what does it mean?
Professional mathematicians these days tend to be
extremely busy, they’ve got to theorems to prove, papers
to publish, conferences to attend. They need to keep
their careers afloat, and so they’ve got a lot less time to
think about what this stuff might mean.
But the thing about the Web – and this is quite an
important factor in what I’m doing – is that it’s possible
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for me to say what I think and to discuss it with large
numbers of other people in the academic world, without
having any formal academic status and without having to
get anything published. And I can change it as I go along
– there’s no final document, that’s the other thing. I don’t
publish articles, I can just put together vague rambling
webpages and then keep changing them as my ideas
change.
C: This is a striking aspect of your research – the
presentation of it is very open: no need to hold back until
you’ve got an completely solid hypothesis and then put it
online tentatively as a preprint. The site is continually
updated, and you’re creating this network which
connects together all these scientists who it seems are
working on related problems but don’t always know of
each other: in some cases you’re actually notifying them
of each other’s work.
MW: I’ve spent a lot of time emailing relevant
researchers and alerting them to the existence of new
articles or preprints which they may well be interested in.
And it’s difficult to quantify, but I do seem to have
stimulated a certain amount of interdisciplinary work.
I’ve created a rôle for myself which hasn’t really got a
name yet, and as far as I know, no-one’s prepared to fund
me, but I’m doing my bit to weave together these threads
of research.
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Part of what caused my disillusionment with
mathematics, which caused me to drop out in the first
place, was...well, the overriding impression was the biblical image of the Tower of Babel. It occurred to me that
if you were to put the names of all professional research
mathematicians in the world into a hat and pick out two,
the chance of there being any real overlap in their
research interests would be quite small, and this
continues to get smaller. It was as if mathematical
research was getting so fragmented that there was no
longer any effective communication possible. So in a
way, I suppose what’s needed, if one wants to try and fix
this, is people who are not specialising, but rather trying
to get an overall picture and to weave it all together by
creating lines of communication. I didn’t come into this
with that intention, but that seems to have been the rôle
I’ve created for myself. I haven’t got any answers at all.
I just feel that there are questions that are important and
which aren’t being asked – possibly because there just
isn’t the language in which to ask them coherently yet.
But at the same time, because there are no real constraints
on me, I don’t have to prove myself to anyone, publish
anything, or stay within any particular boundaries, I can
just throw out certain ideas, get people thinking about
things, suggest connections between things in such a way
as to indicate the existence of something which we can’t
yet pin down perhaps, but which will come into focus the
more we look at it.
In the mathematical community, at least the
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proportionally small number of people I’ve communicated with, I do get a sense that there’s a sense of wonder
there which is something unquantifiable, something that
you couldn’t prove a theorem about, but which is
nonetheless there. It’s something to do with these individuals’ emotional, psychological or even spiritual
orientations, I suppose. But a lot of mathematicians, I’m
afraid, do tend towards the familiar stereotype of socially
inept, almost mildly autistic people who have very little
time for the unquantifiable aspects of life. And so there
is an almost scathing disregard from some quarters. I
think – I feel – that anything that’s vague or a little bit
ephemeral, they see that as worse than useless, perhaps
because their own self-esteem and status is tied up in a
self-image of being the guardians of some sort of absolute
inarguable exactitude and truth.
C: Your guiding thread is a fascination with how
mathematics relates to reality, rather than with mathematics per se.
This seems to be related to the fundamental
problematic which appears right at the very origin, you
could say the co-origin, of mathematics, philosophy and
natural science: with the Pythagoreans, who realised that
operations carried out on numbers applied – rigorously,
but for them somewhat magically – to natural phenomena, and so put forward the idea that reality was actually
nothing but numbers, reality was structured by number.
In a sense they put forward a type of mathematical
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empiricism, i.e. the idea that you could go out and
explore the world, and what you would expect to find
was relationships between numbers, and you could
understand the natural world like that. Now this came to
a catastrophic end with the discovery of irrational numbers...
MW: Yeah, the legendary drowning at sea of Hippasus
of Metapontum – it’s fascinating stuff, a pivotal event in
human history...
C: Certain aspects of the natural world were shown to
exceed number – or number as it was conceived then.
Certain quantities which can be mathematically
described (the diagonal of a square with side length 1, the
area of a circle with radius 1, the golden ratio) cannot be
expressed as ratios of integers, they are ‘alogos’ or, as we
now say, irrational.
After a long period under the influence of Aristotle’s
instrumentalism, for which every sublunary physical phenomena was subject to an inevitable degradation, meaning that exact mathematics was applicable only to astronomy, the celestial and sublunary worlds were (blasphemously) reunified, most of all by Kepler, under a single
mathematical physics, reinvigorating the Pythagorean
dream of a mathematical natural science.
Then in the nineteenth-century mathematics seemed
to exceed its reference to the real world, to claim its own
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autonomous consistency, and any necessary link with the
natural sciences was removed, mathematics asserted its
independence from any application; its applicability to
the physical world even seemed to become a sort of mathematical ghetto.
Now, in the work you’re looking at, it seems that we
return once again to a Pythagoreanism but with a strange
twist...
MW: Yes, something’s been turned on its head. I’ve
been fascinated by Pythagoras and the Pythagoreans for
a long time. Sometimes I think, you know, in a way I’m
acting a bit like a ‘neo-Pythagorean’…but as you say,
there’s a strange twist there. I think a lot of people
forget, when Pythagoras is discussed as ‘the first
mathematician’, that he had one foot in mathematics and
another one in a sort of shamanic, mystical-type reality.
C: Whereas the Pythagoreans discovered in numbers the
semi-divine property of rigorously elucidating nature, we
have this experimentally and theoretically-vindicated
body of method and knowledge taken from natural
science, with whose aid we’re trying to illuminate what
now seems like a somewhat opaque and mysterious
numerical realm; and there are these things within
number which still don’t really make sense.
Mathematicians such as Chaitin [see article in the current
volume—ed.] have said that mathematics must now become
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a quasi-empirical practice – this is in relation to his own
work, but it might perhaps equally be applied here.
MW: Some of the quotes I have on the site agree: Martin
Gardner said something about how some problems of
number theory might be undecidable and might need a
sort of mathematical ‘Uncertainty Principle’. Timothy
Gowers wrote that the primes somehow feel like experimental data, but at the same time he’s well aware that they are
rigidly determined.
We find ourselves in a situation where Michael Berry,
studying spectra of quantum mechanical systems, can
take techniques he’s developed to classify or better understand certain types of physical systems and apply them to
the Riemann zeros, in order to produce a hypothesis that
we will get a particular ‘number variance’ in the far
reaches of the spectrum of Riemann zeros – then years
later, you know, computer power reaches the point where
zeros can be calculated at that scale, the ‘number
variance’ computed...and the graphs match up perfectly.
It’s the first time I’m aware of when a physicist was able
to tell pure mathematicians something new based
entirely on his familiarity with physical systems.
C: Does the field then become de facto an experimental
one? You have the a hypothetical physical system which
will produce the system of vibrations which the Riemann
zeros seem to correspond to. And the only way to find
out whether there’s really any system which is adequate
to that would be by experimentation – in the same way
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that the Higgs Boson hypothesised to glue together the
results of quantum physics must now be sought
experimentally – hence the construction of CERN’s
much-anticipated Large Hadron Collider. Does someone
have to build the Riemann dynamical system?
MW: Michael Berry has said he’s absolutely convinced
that, if such a thing is physically possible, someone will
make one of these things in a lab, and then the Riemann
zeros will actually come out on the instrument readings.
But at the moment there’s no-one actually conducting
any experiments which are getting anywhere near that, or
even attempting to. You do have physicists taking certain
ideas – largely mathematical models intended for physical systems – and applying them to aspects of the zeta
function. There is an experimental branch of study of
course, you’ve got people looking at the Riemann zeros
themselves, which contain a wealth of data – we’ve got, I
believe, hundreds of billions of them calculated now –
this is being done with grid computing8. The gaps
between them and all kinds of other things you can
measure when you’ve got a set of seemingly random real
numbers, are being analysed using a variety of statistical
methods, random matrix theory is being applied. So
these are, to some extent, experimental studies. Marek
Wolf (Institute of Theoretical Physics, Wroclaw)
experimentally detected a widespread physical phenomenon called ‘1/f noise’ in the distribution of prime
numbers.
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The prime numbers continue out to infinity, we’ve
known they go on forever since Euclid, but we can only
calculate them up to a point. We tend to think our
current computers are ‘powerful’ , and we think we can
find ‘big’ prime numbers – you know, now and again one
will even make it into the news. But there’s no such thing as
a ‘big’ number, this is what I always try to get across to
laypeople – because the number system goes on
forever, however far we look, proportionally it’s still an
infinitesimal step into an infinite unknown .
C: And, of course, in consequence, no matter how many
zeros are found, one never comes any closer to a proof of
RH.
MW: Yes, exactly. There’s the duality between Riemann
zeros and primes, and so the same idea applies with the
zeros. We can never calculate more than an infinitesimal
proportion of them. Sometimes I use the analogy of large
telescopes: you’re looking out into space, and the more
you can see, the more you can deduce about the nature
of the universe you live in. Analogously, we can ‘see into’
the number line a certain distance, what we think is a
‘long way’ – but again, it’s meaningless, really, to say a
‘long way’ or a ‘big number’. Of course we can see
further than we’ve ever seen before, so we can detect
certain apparent patterns which can give rise to
hypotheses that we can then attempt to prove. Similarly
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we can look further than ever up the critical line now, and
with hundreds of billions of Riemann zeros we can test
certain hypotheses and generate new ones. So there’s an
experimental element in that. But as far as the
hypothesised Riemann dynamics goes, the quest to try
and pin down something like a Riemann dynamics isn’t
really being furthered by experimental science as such,
rather the progress seems to be coming from mathematicians like Connes, Lapidus and Christopher Deninger
(University of Münster). But these people – well,
certainly Connes and Lapidus – do have a very broad
interest in large areas of both mathematics and physics,
which is what makes their work so interesting.
It would be misleading to suggest that mathematics
has become an empirical science, since exact formulations
are still possible – even in these more hazy areas – at least
we can’t rule out the possibility of exact formulations.
But an empirical approach has become potentially useful.
In connection with this, I should mention the emergence
of probabilistic number theory, which in itself raises huge
questions. Probabilistic number theory effectively started
in 1940 with the Erdös-Kac theorem which I mentioned
earlier, the discovery that the number of prime factors in
‘large’ integers has a kind of random distribution which
follows the Gaussian distribution or bell curve. That
discovery led to a whole outpouring of theorems and
conjectures which have collectively become known as
probabilistic number theory, where you apply the
methods of probability theory, and make use of the key
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idea that divisibility by a prime p and divisibility by a
different prime q are ‘statistically independent events’, one
has absolutely no bearing on the other. When you deal
with probability you deal with this idea of independent
events – well, these are arguably the most independent ‘events’
there can ever be. Physical events in any well-prepared
experiment, you might think they are independent; but
ultimately every particle of the universe is
gravitationally pulling on every other particle, everything
is linked, although the effects are generally negligible and
impossible to quantify. The only place where things are
totally independent is in the number system – the
divisibility of an integer by two different prime numbers.
So here is a place where you can apply probability
theory, where everything is entirely exact, where you can
let your n tend to infinity and that actually refers to something. Probabilistic number theory allows you to prove
things about prime numbers and about the number
system generally, using the techniques of probability theory, and that seems highly counterintuitive. The fact that
it works at all raises questions which are more like
‘mysteries’ than formal mathematical problems.
There are three separate areas worth mentioning
here: the emergence of probabilistic number theory, the
effectiveness of the analogy with statistical mechanics –
partition functions, etc. which I described earlier – and
then the rôle of random matrix theory, which was
developed for modelling subatomic phenomena, but then
was accidentally found in the 70’s to apply directly to the
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theory of the Riemann zeros. So you’ve got three
separate areas of randomness-based thinking, stochastic
disciplines if you like. They deal with large systems
which have too many components to keep track of
individually – these components must be treated almost
sociologically, as populations, and subjected to probabilistic or statistical thinking. All three areas have been
effective in furthering our understanding of the number
system. Now, again, mathematicians would tend to focus
on at most one of these things, see what could be
achieved and perhaps make a few sober remarks on what
it all might mean. But to me, the fact that you’ve got
these three areas, all of a stochastic nature, shedding light
on the primes and the Riemann zeros, points to something very strange. We’ve got primes, the most basic
things in the universe as we experience it – the sequence
of prime numbers is the most basic non-trivial information there is, it’s the one thing you can’t argue with anyone about, it’s the one thing all lifeforms in the universe
could potentially relate to. And yet in some ways they
seem to be best understood using a type of analysis more
appropriate to weather systems, roulette wheels, boxes of
gas, etc.
I’ve always thought of probability theory as a slightly
‘tainted’ branch of mathematics for three reasons: Firstly,
it’s origins are not entirely honourable – I seem to recall
that it has its roots in an historical accumulation of gambling techniques which got distilled into a formal theory
by Pascal. Secondly, it deals with ‘events’, repeatable
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‘events’, which are categories of physical phenomena,
‘occurrences’ of one type or another which can be
quantified, measured, counted, numerically analysed, etc.
whereas truly ‘pure’ mathematics doesn’t rely on
anything in physical reality in quite this way. Finally, by
its very nature, probability theory tends to deliver
imprecise information – there’s always a margin of error.
And yet this system of thought, which has been
developed in order to deal in an approximate way with
large, complicated physical systems, seems so perfectly
applicable to something which is so fundamental, which
is characterised by an absolute precision, and which
underlies everything else – the distribution of primes! It’s
as if we’ve got something back-to-front. It’s similarly
interesting that probability should have such a
fundamental rôle in quantum mechanics: an ultrasimplified account of what QM tells us is that, insofar as
it can be understood as being made of particles, the
universe can also be understood as being made of ‘fields
of probability’ . Probability theory in a casino, yes; or in
a meterology lab... But prime numbers? The fundamental level of matter? These are things we instinctively feel
should be totally deterministic and rigid. And to me, this
suggests we’re looking at something the wrong way
‘round – something’s been turned on its head. It’s as if
‘randomness’, or some essential, almost esoteric quality
associated with randomness – that quality evidenced in
our failure to really understand what we mean by ‘randomness’ – is emanating up from these fundamental
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realms. We’ve been dragging it down from the macroscopic scale, the casino scale, down to this micro-level, in
a numerical and physical sense, and finding that it helps
us understand something. But I feel something’s back-tofront there.
A mathematician called R.C. Vaughan states in one of
my archived quotations that it’s obvious that the prime
numbers are random, but we don’t know what randomness is. And there is a real problem with defining
randomness. There are several definitions, information
theorists, probability theorists, have put forward
definitions of what it means for something to be random.
The definitions overlap to a large extent, but ultimately,
when is a string of digits random? If I give you a block
of a thousand 0’s and 1’s, it might look completely
random, it might even pass numerous tests run on it for
randomness…but then I could reveal, well actually, no,
it’s a thousand digits of π starting from the two-millionth
digit. And then it’s not random anymore. So there’s the
whole question of what randomness is. This is one of the
central themes that fascinates me: where does this notion
come from, where does it lead us in our understanding of
the reality we inhabit, and why does it tie in so closely
with both the fundamentals of the number system and of
particle physics?
And then there’s the difficulty of talking about having
two of anything, that in order to have two of anything
you have to have a category which those two objects both
belong to. But the categories are always imprecise. We
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have to partition spacetime into blocks with ‘fuzzy’
boundaries, and then attempt to match aspects of these
blocks up with some ideal which exists in a sort of mental hyperspace, a Platonic realm of sorts. So we’re
projecting these categories onto the universe which actually aren’t intrinsic in the universe; we’re setting out these
boundaries, but the boundaries are blurry. Yet, despite
the possible problems this blurriness might cause, on a
practical level we’re able to then extract data which fits
remarkably well against certain probability distributions.
The most ubiquitous and I think the most important one
is the Gaussian or bell curve – and this, as we can see
from the Erdös-Kac theorem, has a mysterious and
fundamental relationship with the number system we’re
using to count members of our fuzzy-boundaried
categories in the first place.
The effectiveness of statistical inference in the hard
sciences and the social sciences – I’m sure this would be
widely disputed, but I feel there is a mystery there which
isn’t really being acknowledged, and it has to do with
how we can name and count anything, and how, when
we do name, count and measure things they seem to
collectively accord with these ideal mathematical blueprints or templates. That says more about the way our
mental hyperspace is being mapped onto the physical
universe than about anything intrinsic in the physical
universe.
C: When you look at the local you expect to find
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precision, whereas with the global you’re happy with
statistical data. Here we’re looking at these local, precise
conditions and there seems to be randomness ‘built into’
them in a way that’s not immediately comprehensible:
After all, they’re not statistical aggregates in any obvious
sense.
MW: Yes, the set of positive integers is in a category of
its own, there’s just one number system. Yet, it’s as if this entity – if we take the positive integers, the primes and the
zeta function as aspects of a single thing, different aspects
of the same entity – rather than being a carved-in-stone,
unique thing, is actually just one example of a class of
things, and we’re able to apply statistical analysis because
of that. This is why, when I started finding out about
these things, I felt my ‘prime evolution’ thing might have
something in it, this idea of the number system being a
frozen state of something which had previously inhabited
many different states. I’ve had certain quite critical, serious-minded people react to some of my more sensational
suggestions by saying, well all this number theory and
physics, there’s nothing mysterious at all — the universe
follows mathematical laws, so of course we’d expect certain aspects of number theory to show up in the physical
world. If they’d look a bit deeper into this, they’d see
what I meant: yes, it’s not surprising, given that maths
underlies all of physics, that we might get, say, particular
values of the zeta function showing up in string theory, or
the theory of integer partitions relating to Bose-Einstein
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condensates or whatever: you get these odd
little instances of number theory/physics correspondence;
I’ve catalogued a lot of these in my web-archive. But
that’s not the really interesting stuff. What’s much more
surprising is the way physics seems to be pointing the
way for understanding the zeta function, and often this is
statistical or stochastic physics, as if the zeta function –
and in some sense, then, the number system — is just one
example of a more general phenomenon. And I don’t
think anyone disputes the spectral nature of the Riemann
zeros now. But it’s not one archetypal ubiquitous spectrum we see showing up all over physics. If we saw ‘the
zeta spectrum’ – as it might be called – everywhere, then
it would somehow feel a lot less mysterious. We’d
probably feel quite comfortable with such an affirmation
of the old idea that the number system directly underlies
the structure of the physical universe. But the Riemann
zeros take the form of an almost disconcertingly
arbitrary-looking spectrum, never known of by humans
prior to the late 1850’s. In the very recent past we’ve
been confronted with the fact that it has all the fingerprints of membership in certain classes, very wide
classes, of very specific physical systems, as if it’s just one
element of a whole class, a population of things. So it’s a
bit like the way you might be able to, based on the postcode of a UK resident, predict certain things about his or
her attitudes, abilities, tastes, whatever – because you’ve
got statistical information about the population, you can
make plausible hypotheses about this specific individual.
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And it’s as if the primes-zeta entity, whatever you want to
call it, despite its seemingly fundamental, unique status, is
just one individual in a wider class of things. But the
space in which that class exists is something we haven’t
even begun to imagine might exist, or we haven’t got any
access to.
So we have this image of a frozen system, something
congealing into a state, and then…it’s as if you walked
into a concert hall and caught the last note of a
symphony, and everybody’s applauding ecstatically and
you’re wondering, what’s all the fuss about? You didn’t
witness the process that led up to that last note, and it’s
like, with the prime numbers, we’re just walking in on the
last moment, the culmination of something. As if there
was a whole ‘symphony’ that led up to that, and
humanity may be on the verge of revealing it.
*
C: All of the foregoing seems to suggest that what we
think of as simple and elegant foundations may in fact be
the eventual product of something which is rather
complex, even beyond our comprehension. So we’d have
to separate out what seems simple and elegant to us, from
what is actually fundamental in the universe, and this is
another sense in which mathematics mirrors the
condition of theoretical physics, in which, characteristically, the further we go towards the fundamental, the
stranger things become (string theory being a case in
point).
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Rather than defining the primes on the basis of the
supposedly fundamental and simple number line, in fact
it seems that, when we look through this complex theoretical-mathematical prism you have described, there’s
actually something more fundamental about the primes.
The primes themselves produce...
MW: ...the number line, yes, you can see it that way. I
came up with this naïve idea, before I really learned any
of the more serious stuff, this was after I had been thinking about the Erdös-Kac theorem, the primes and the
Gaussian distribution, but before I ‘experienced’ the
dynamical aspect of the primes. I was thinking about how
we tend to construct the primes. We’re taught to construct the number line starting with one and then using
the Peano axioms, you know, there’s an axiom that basically says, whatever number you arrive at you can always
add another one to it. And I thought, hold on, where
does this come from, this idea that you can always add
another one, and I started to question that as something
that might not be as obvious as it first seems. There’s
some hidden assumption there about order, time or something, I felt.
And I thought, well, there’s an alternate approach we
could adopt here, we could start with an infinite alphabet
of meaningless symbols, an infinite alphabet of meaningless yet distinct symbols, and then create the dictionary of
all possible words of finite length out of that alphabet.
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This alphabet of symbols would correspond to the prime
numbers. By combining the symbols in all finite possible
combinations, you generate the set of words in your infinitely-long dictionary — this corresponds to the fact that
if you combine the primes in all finite multiplicative combinations, you get the set of positive integers. Except
now there’s no sense of order: Because we’re not starting with
the positive integers, we don’t need to think of one prime
number as being ‘greater than’ another. The primes are
not embedded in the positive integers yet, they’re just
these free-floating abstract symbols. So I used to try and
conjure up this image of bubbles floating in an imaginary
space, each with an exotic glyph, a symbol from our
‘alphabet of primes’ on it. The idea is that you can then
join any number of these bubbles in any combination,
including repeats. All possible such bubble-clusters are to
be found floating somewhere in this space. Some are
larger than others in the sense that there are more bubbles in the cluster — that is, more prime factors — but
there’s no sense of a cluster coming ‘before’ or ‘after’
another cluster. It’s only when you cross the Rubicon of
deciding which alphabetic symbol is going to be your ‘2’
that you start to create some sense of order.
So I had these hints and intuitions — I couldn’t really
pin them down to anything very rigorous — that we’ve
been thinking about randomness and the fundamentals
of reality in a back-to-front way. We’ve got ourselves into
a kind of confusion where everything seems immensely
complicated when we delve down to the fundamentals of
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either the number system — which seems at least partly to
inhabit the realm of psyche — or of the physical world,
the world of matter — just open a textbook on analytic
number theory or quantum mechanics and you’ll see
what I mean. I felt this issue could be addressed if we
examined some of our ‘obvious’ assumptions. We think
we’ve taken the obvious construction — that is, you start
with one, then you add one, and then you add another
one, this idea you can always add another one. Rather,
what if we start with the primes, and build the number
system up that way? The whole ‘order’ thing then
becomes more of a ‘phenomenon’ than something
axiomatic...
C: Coincidentally, the ‘legendary’ Dr. Daniel Barker also
devised a notation system for the positive integers based
upon prime factorisation, which is very close to what
you’re talking about here.7 You have these
inseparable lexicographical units from which numbers
are composed, and they could be in any order. He was
interested in place value as a culturally-repressive numerical practice, and this was a way of doing away with place
value completely. Each number would just be like a
collection of boulders or something.
MW: The lexicographical approach, yes. I’ve tried to get
this across to some lay-people I’ve talked to. There’s the
7. See http://abstractdynamics.org/005047.html
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fundamental theorem of arithmetic — literally the most
important thing we know about the number system. And no
more than 0.1% the population have even heard of it, I’d
guess. It basically says that every integer breaks down
uniquely into prime factors. And we’ve got this strange
situation where almost nobody knows this, this simple
fact, the most important thing we know about the number system. This is straying into other territory, but to
me, humanity’s relationship with number is rather
unhealthy, because we’ve built this entire civilisation
around the mathematical sciences, and yet the ordinary
population knows nothing of the basics, and often finds
mathematics a source of fear and unpleasantness. I try
and conjure up this image of these bubbles, the fact that
the clusters can be as large as you want, you can have
huge ‘planets’ of prime factor bubbles joined together —
there’s no upper size limit. And so something like the greatest common divisor can then be explained very simply,
it’s just the intersection, literally where the two clusters
intersect. The least common multiple can be similarly
explained. Prime numbers distinguish themselves from
non-prime integers because they are individual bubbles.
The integer 1 is the absence of any bubble, the empty background space, the blank page in the “dictionary” I
mentioned earlier…
And then you imagine stringing the entire set of clusters out in a line according to this ‘order’ thing, and you
start to see that there’s a counterintuitive variation in the
sequence — you get small clusters, huge clusters and
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single bubbles all intermingling according to no sensible
scheme. And this is the sort of thing that I’d eventually
like to push further out into the public domain just to see
what sort of effect it would have, when people start looking at their supposedly familiar number system in this
new light. Because people tend to think of the number
system like a row of boxes of cereal in a supermarket, just
identical units stacked together, a sort of homogeneous
featureless thing that just goes on: each number is just the
previous one plus one, there’s nothing much there, nothing of interest. And it was Frank Sommen, a really
remarkable, imaginative Flemish mathematician who I
worked with during my PhD studies, who once said to
me, every positive integer is a different animal. I came to
see exactly what he meant: each one’s got its own ‘anatomy’, every one’s a different story, and that starts to
become apparent as soon as you realise that each integer
factors in a unique way into prime numbers.
C: This is a basic intuition that one finds in 'primitive'
numerological systems.
MW: Yes, and in children as well, with their favourite
numbers, and feelings about each of the first few positive
integers — ethnomathematics and children.
C: Something that gets beaten out of people by
mathematics: when people start learning mathematics,
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it’s as if the first task is to extirpate any idea that numbers
have quality. Mathematics is in fact often seen as constitutively opposed to any such intuition.
MW: Yes. Marie-Louise von Franz, one of my favourite
writers, who studied under Jung and wrote a lot about
number archetypes, she talked about number having
both quantitative and qualitative aspects. The quantitative is obvious, we all use numbers to count. Cultures
who revere certain numbers and have mystical beliefs
about them which we might laugh at, they still use them
to count with and to trade, they recognise that they have
a quantitative aspect. This is the aspect of number that
has given rise to economics and technology; but equally,
perhaps even more importantly, there’s the qualitative
aspect that only survives in our culture in children having favourite numbers, some adults having lucky numbers, not wanting to sleep on the thirteenth floor of an
hotel, the way they might choose lottery numbers, that
sort of thing. But, you know, in ‘serious’ society numbers
are supposed to be entirely quantitative. Von Franz wrote
about a traditional Chinese story involving eleven generals who, faced with some very difficult military situation,
took a vote as to whether they should attack or retreat.
Three voted to attack and eight voted to retreat. So what
did they do? They attacked, because three was a more
favourable number — it wasn’t a bigger number, but it was
a number associated with unanimity, or some other
favorable quality like that. And the attack was a success.
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So it’s interesting that they could build a civilisation that
was able to have a functioning economy and military and
to govern millions of people — clearly they were intensely aware that number had a quantitative aspect — but
there was also a serious engagement with the qualitative
aspect which is dismissed in our present culture as entirely superstitious. Now I’m not encouraging people to
engage in completely arbitrary numerology, I mean, I’ve
looked at a lot of that new age numerology literature, and
the problem is, nothing can be verified: someone can
write a book saying a particular number means something, and someone else can write another one saying it
means the complete opposite. It just confuses matters, as
there’s never any consensus or certainty in these interpretations. That’s why professional mathematicians would
almost unanimously just react against it and say it’s all
rubbish.
C: But is there any way to talk about it which doesn’t get
into that morass of mysticism?
MW: There are two approaches: one is the serious
attempt by Jung and his followers to catalogue all of the
ethnomathematical systems, undertaking a serious study
and survey of various cultures and their relationship with
number, trying to find common threads, and through
psychoanalytical work and dream studies, trying to find
extract essential patterns to build up a body of material
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from which we could possibly deduce something about
how number interfaces with the psyche at a fundamental
level. The other approach is to seriously study number
theory, because as far as I’m concerned, that is numerology, really — you’re looking at the properties of integers,
and if you study it to a certain depth it takes you into the
realms of what you could only call the mystical or the
uncanny, where cracks seem to open up in your normal
understanding of reality.
C: Is that perhaps what characterises number theory as
opposed to mathematics, what makes it a very different
discipline?
MW: Well, number theory is universally acknowledged
as a branch of mathematics. It can’t really be separated
from it like that. But it arguably has a unique status at
the very heart of mathematics. You’re working at the
very root of it all, dealing with the simplest objects, the
positive integers. And yet you come across these counterintuitively complicated structures and results. You can
separate mathematics into branches and disciplines but
they all ultimately overlap and interrelate. Gauss (who
himself was called the ‘prince of mathematicians’) called
mathematics ‘the queen of the sciences’, and number theory ‘the queen of mathematics’. The idea is that number
theory is generally seen as the pinnacle, in that it contains
the most difficult problems; also it’s concerned with the
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integers, and all of the rest of mathematics ultimately
relies on integers. Hence it’s not surprising that problems
of number theory do seep into other areas of mathematics, and even physics. What is surprising is that physics
is beginning to shed light on number theoretical
structures like the zeta function, as if it were just one of a
class of objects, whereas it’s meant to be this fundamental object underlying everything.
What I’m trying to describe with my clusters of bubbles isn’t intended as any sort of serious mathematical
proposition, it’s just a picturesque visualisation — trying
to look at the number system from another angle, if you
like. But there’s a hidden assumption within the Peano
axioms, I think, which needs to be addressed — although
I don’t think I’m the one to address it. It concerns the
axiom which allows you to always add one. Even in the
proof of the infinitude of primes, I sense some sort of subtle circularity there — the idea is that, if the number of
primes were finite, you could multiply them all together
and then add one. And that rapidly leads to a contradiction concerning primeness and divisibility…hence there
must be infinitely many primes. So that takes you back
to the Peano axioms, the idea that you can always add
one. But in my visualisation, multiplying them all together would correspond to building one mighty cluster using
one of each type of bubble. And in that visualisation
‘adding 1’ is a far less obvious operation. This ties in
with problems of time, the idea of time, repetition, even
basic physical questions: you know, this ‘adding 1’
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business presupposes that you’ve got a physical space,
something like the space we’re familiar with, in which
you can make a sequence of marks, or a time continuum
in which you can make a sequence of utterances or beats.
And I feel there may be subtle assumptions concerning
the homogeneity of time and space involved in this, too.
C: These questions of time and space must fall out from
the primes’ intimate connection to the relationship
between multiplication and addition.
MW: Brian Conrey, who’s President of the American
Institute of Mathematics, and Alain Connes have both
been quoted as saying that RH is ultimately concerned
with the basic intertwining of addition and multiplication.
And if we haven’t really got a clue how to prove RH —
which we don’t — we’re going to have to own up, we
don’t even understand how addition and multiplication
interrelate. A more succinct, precise way of describing
these two possible constructions of the primes that I have
outlined — the conventional ‘just add 1’ approach, and
my ‘lexicographical’ approach with its equivalent clusters-of-bubbles visualisation — is given by Grald
Tenenbaum, who certainly knows what he’s talking
about:
Addition and multiplication equip the set of positive
natural numbers with the double structure of an
Abelian semigroup. The first [addition] is associated
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with a total order relation as it is generated by the
single number one.
So if you’ve got addition and you’ve got this single
number 1, you can generate the postive integers just by
adding 1 plus 1, 1 plus 1 plus 1, etc. If you take 1 as your
‘additive generator’, the universe generated is the set of
positive integers.
The second [multiplication], reflecting the partial order
of divisibility,
This probably isn’t the time to get into the subtle
issues of ‘order’ in mathematics — you’ve got ‘total order’
and ‘partial order’: addition relates to total order, where
something definitively comes before or after something
else; and divisibility relates to partial order, a less distinctive type of order, although I won’t get into the details of
that…
[Multiplication], reflecting the partial order of divisibility, has an infinite number of generators, the prime
numbers.
So, now, rather than starting with just the number 1
and combining it with itself in every possible way using
addition, we start with this infinite set of primes and then
take all possible multiplicative combinations.
Defined since antiquity, this key concept has yet to
deliver up all of its secrets, and there are plenty of
them.8
8. Tenenbaum and France, op. cit.
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It has the quality of a square peg in a round hole, this
tension between addition and multiplication. It’s almost
like, despite the inarguable perfection of the number system, they don’t really fit together very well, and they generate what I feel is something like friction, and this
produces the sprawling mass of definitions, theorems,
lemmas and conjectures that we call analytic number theory. There’s a novel by Apostolos Doxiadis called Uncle
Petros and Goldbach’s Conjecture — it’s written as fiction, but
he gets some key ideas across through an elderly mathematician character. This is very well put, I feel:
Multiplication is unnatural in the same sense that
addition is natural. It’s a contrived second order
concept, no more really than a series of additions of
equal elements.9
So that’s the point, that 3x5, you can see that as
0+3+3+3+3+3 — you start with nothing, zero, and add
three five times. So in a sense you can build multiplication out of addition, whereas it doesn’t work the other
way around. So addition is a first order operation, and
multiplication is, as he’s saying, unnatural, in that it’s ‘second order’. The thing that struck me about it when I was
dwelling on this for a while was that it has to do with time,
it has to do with repetition. And it also relates to the very
deep issues concerning the whole idea of where number
comes from and how we define number. As I hinted earlier, I’ve spent a lot of time thinking about how you could
9. Doxiadis, A. Uncle Petros and Goldbach’s Conjecture. NY: Bloomsbury, 2000.
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ever have two of anything. You know, there are two people sitting here in this room right now, but that relies on
the definition of what a ‘person’ is. We define the category linguistically, and we think we know what a ‘person’
is, but you can imagine some sort of genetically-engineered mutant that may or may not be a ‘person’ depending on how the definition was formulated, and the definition’s made of words and each word is imprecise, is subject to interpretation. So any type of category you define
is going to have a ‘fuzzy’ boundary, so...although it works
quite well for day-to-day affairs, counting things works
fairly well, you’ve got fifteen sheep in your paddock. But
you can always contrive some convoluted situation
where, maybe it’s fourteen sheep or maybe it’s fifteen —
is that odd looking creature really a ‘sheep’ or is it something else?
So, it comes down to issues of language and definition. We consider chunks of spacetime, we recognise patterns and say, yes, that chunk of spacetime falls into suchand-such a category. As I said, I started to wonder how
you can really have two of anything. Every entity ultimately distinguishes itself from every other, these categories are not mathematically precise, there’s an arbitrary
element involved in deciding whether things get included
— “where do you draw the line?” as they say. And yet
these categories are the essence of counting, and if there’s
a problem with applying the concept ‘2’ to our experience
then there’s going to be a problem with all of the other
positive integers.
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Exceptionally, when you get down to the subatomic
level you can have two of something, because each individual electron is absolutely indistinguishable from the
others. So that’s interesting, that this concept makes
sense at the subatomic level but then ‘fuzzes out’ at
macroscopic scales.
But the thing is, when you say ‘3x7’, you’re effectively saying ‘three sevens’. So, seven pebbles in a row — you
count out seven by adding one plus one plus one, etc.
That feels quite ‘natural’. But then, to make the leap to
‘three lots of seven’…you can have three giraffes or three
potatos, the fuzzy boundaries mean that’s a difficult
enough issue as it is, but ‘three sevens’ presupposes that
a ‘seven’ is something that there can be more than one of in some
sense...
C: One would have to say that the multiplier and the
multiplicand are somehow of a different order, two different types of numbers are involved in the operation.
MW: Yes, one is operating on the other. If you add, it
doesn’t matter...I mean, it’s true to say that 3x7 is the
same as 7x3, you’ve got this basic ‘commutative’ property applying to the positive integers. But when you consider the ‘act’ of 3x7, the three is how many times you’re
doing something, whether it’s laying out a row of seven
beans or playing seven drumbeats, and the seven is some
kind of an extension in space or time. Whereas in adding
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3+7 or 7+3, both numbers play the same rôle. So there’s
something there, not easy to pin down, which we don’t
understand, and I have a very deep sense that we won’t
really understand it until we really understand time. It has
something to do with time. Our inability to understand
the primes, our inability to prove RH is a symptom of
our inability to understand the relationship between addition and multiplication, and that is related to our relationship with time.
C: On your site you quote J.J. Sylvester:
I have sometimes thought that the profound mystery
which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in
regard to time, which like space may be in essence
poly-dimensional and that this and other such sort sort
of truths would become self-evident to a being whose
mode of perception is according to superficially as
opposed to our own limitation to linearly extended
time.10
MW: I think he must have been thinking about the
relationship of multiplication and addition in terms of
time. This was 1888, so RH had been posed, but mathematicians long before RH understood that the enigma of
the prime numbers was rooted in the uneasy relationship
of addition and multiplication. So possibly he had a
10. Sylvester, J.J. ‘On certain inequalities relating to prime numbers’, Nature 38
(1888) pp259-262, reproduced in Collected Mathematical Papers, Volume 4. NY:
Chelsea,1973 p.600
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sense that the relationship had something to do with time.
But he says ‘the profound mystery which envelopes our
conceptions relative to prime numbers’ — in other words,
the puzzling interface of addition and multiplication —
‘depends upon the limitations of our faculties in regard to
time’. So if there were a higher dimensional, a two-dimensional ‘time surface’ or something like it — the word
‘superficially’ is being used by Sylvester in the original
sense meaning ‘relating to surfaces’ — our minds, normally constrained to a ‘timeline’, could perhaps ‘spread out
across it’ in some sense. It’s perhaps a bit like being able
to come up off the surface of the earth and look down
from a third dimension to get a sense of how things are
laid out, whereas when you’re stuck on the ground, certain things are not at all apparent...but these are all very
vague and intuitive ideas.
*
C: In one of the papers you link to in the archive11,
Volovich suggests a most extreme and startling explanation for the concurrence of physics and mathematics.
MW: Yes, and you may have noticed that he quotes
Pythagoras at the beginning, a slightly amusing Greek-toRussian-to-English compound translation of “all is number” — “the whole thing is a number”. I got very excited
when I first found that paper, because he’s suggesting that
number theory is the ultimate physical theory. That
11. I.V. Volovich, "Number theory as the ultimate physical theory", Preprint
CERN-TH 87 4781-4786 (1987)
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came out in 1987 as a preprint at CERN — he’s an
accomplished physicist — but it was never published in a
journal. The fact that it never got published and the fact
that he hasn’t responded to my questions about it could
suggest that he’s backed away from it somewhat. I can’t
speak for him, but I wonder if he’s slightly embarrassed
by its more grandiose claims, in the way I was suggesting
earlier that physicists and mathematicians can be.
But the thing is, he has done this vast body of work
on p-adic physics, which I referred to earlier. And the rise
of p-adic physics is a very interesting thing in itself
because, you see, even though the universe at the scale of
this room is Archimedean — I can lay my ruler end to end
and will eventually reach the end of the room — the
universe is not Archimedean at all scales. Below the
Planck scale, it’s no longer Archimedean. Below this 1035m or so — which to some people sounds too small to
worry about, but you just take a metre, then a tenth, then
a tenth, not that many times, really…It’s not that our
instruments aren’t precise enough to measure below that
scale, it’s that the whole idea of measurement as we’ve
formulated it ceases to make consistent sense. And effectively, space becomes non-Archimedean below that scale.
There’s a similar scale with time and other fundamental
quantities, below which they become non-Archimedean.
You can theoretically join some unit of measurement
end-to-end and never achieve a given, finite extension.
This has led people to think that maybe p-adic
physics, where you’re dealing with a non-Archimedean
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number system, would be more appropriate for application at the sub-Planck scale. And Volovich seems to be
suggesting that different non-Archimedean number systems could apply to different regions of space and time at
different scales. Again, I’m not entirely sure: large parts
of the paper are beyond the scope of my present understanding. I’m intrigued by his referenced to ‘fluctuating
number systems’, but I don’t know whether he means
fluctuating with time, or in some other more generalised
sense.
People are now starting to think about applying p-adic
mathematics to the physical world. Each p-adic number
system provides a different sense of ‘distance’ between
two rational numbers, and that notion of distance then
allows you to define all the other numbers which aren’t
rational via precise mathematical concepts involving
‘limits’. I mentioned this earlier. This distance or ‘metric’
is defined in terms of divisibility of primes. It has to do
with highest powers: for instance, in a 7-adic metric, finding the distance between two rationals involves basically
looking for the highest power of 7 that divides into the
numerator of their difference — that difference of course
is also a rational number — when it is expressed as a fraction in lowest terms. As a result of that, number theory
comes flooding into your p-adic physics: if you start looking at p-adic or adelic space and time, issues associated
with the prime numbers become directly relevant. Of all
of this number theory/physics material I’m archiving this
is the area I’m least familiar with.
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C: Saying that the means of measurement, that the possibility of measurement has changed is one thing, but
saying that numbers are actually the ‘atoms’ themselves,
so to speak, is something else: that means that there is no
longer some thing you’re measuring. The measurement
itself takes on a sort of substantiality.
MW: Yes, these are very difficult notions to grasp,
in so far as I understand what’s being proposed. I think,
perhaps like myself, Volovich caught a glimpse of something, got quite excited about it and wrote it down; he’s
quoted Pythagoras — it’s as if there’s some mystical quality to his insight.
C: There might be thousands of these papers hidden
everywhere that people haven’t published.
MW: I’m not sure it would be in the thousands, but who
knows…There’s a general hesitance to stick one’s neck
out. If I’m helping to encourage that sort of thing, then I
suppose that’s a useful contribution.
C: Exeter University has granted you an honorary fellowship and hosts the web- archive, but there is no funding available for your work. Apart from your own fascination with the subject, what drives you to continue this
labour of archiving and making your own speculative
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connections public?
MW: Over the years after I’d dropped out of formal academia, I spent a lot of time thinking through and honing
these ideas about mathematics being some sort of inner
priesthood of our scientistic culture that’s in the process
of destroying the ecosystem, and wondering what could
be done about it, how do we change this, you know? I felt
that campaigning to stop the destruction of this or that
rainforest isn’t going to be enough, you’ve got to go right
to the core, to the root of the problem, the fulcrum. And,
reading von Franz, with her ideas about ethnomathematics, and quantity and quality, and reading René Guenon,
who — although I don’t embrace his traditionalist fundamentalism — wrote a fascinating book called The Reign of
Quantity and the Signs of the Times, I started having this idea
that only when Western Culture re-evaluates its relationship with number can there be any real change in the way
we relate to the world, because we’ve got stuck in a
‘quantocentric’ view of the world. And so I have felt at
times that what I was trying to bring forth — whether it
was in my strange 1998 ‘evolutionary’ notion or just in
my networking of various people’s work via my webarchive — was an acceleration towards an imminent transformation in our relationship with the number system. I
was quite driven for a while, but I’ve become considerably more cautious and sober in my approach to this
since. I saw what I perceived to be clues...felt that it had
to be coming, and only through that sort of
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transformation will the Western project ever be able to
steer itself in a less destructive direction. At times I’ve felt
that I had an important rôle to play — not that I was
‘chosen’ to do it or anything, but that my work was cut
out for me, and it was an important mission. Other
times, I’ve been much less certain, and wondered, you
know, why am I sitting in front of this computer editing
HTML, when I could be spending the same time and
effort campaigning for, say, the rights of an indigenous
tribe having its land ravaged by a multinational corporation. I had to justify this to myself when people I knew
were involved in things like that, by telling myself, well
actually they’re just dealing with the symptoms, whereas
I’m trying to deal with the root of the problem. So it
verged on an idealism, almost an activism.
C: The point being that rather than lamenting the
destructive rôle of number and of science, one tries to
recognise that there’s something else within number, and
as you said, to re-evaluate our relationship to it, which is
not to say to reject it, but to become more numerate...
MW: Yeah, which is what I saw around me, people being
very suspicious towards mathematics, hating it, seeing it
as controlling and evil, and I thought, no, we need to get
inside it, try to understand where it comes from and how
it works.
But then I started to question whether I was just
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creating a whole set of complex and noble motivations
for myself when in fact it was just my ego or desire to be
acknowledged for what I’d achieved, or, you know, just
wanting some sort of recognition or status. I was continually wondering what it was that was motivating me, and
trying to rein myself in and consider the worst possible
motivations as well as the best.
I had a kind of motivational collapse in early 2005,
when I was struck by a very deep sense of there being
insufficient time; you know, I had this grandiose hope of
helping to effect some sort of long term change in culture
and the way in which we deal with the number system. I
started to think, maybe what I’m contributing to would
have that effect if there were a few more centuries left of
relatively leisurely culture and well-funded academia to
take these ideas on and develop them, but, you know,
we’re facing multiple global crises, and this sort of thing
is never really going to have time to take root.
I’ve since drifted in and out of this activity periodically, found what I think is a healthy level of interest in these
matters. But I don’t strongly believe that I’m part of
some current of cultural change anymore, I’m just...I suppose you just can’t know what effect you’re having, particularly with the Web, when you’re pushing ideas out.
You don’t know who’s reading them and what they’re
going to do with them — a bright teenager who reads my
website might be inspired to study mathematics and,
influenced by some of the hints, clues, suggestions, etc.
I’ve assembled, go on to make amazing discoveries...who
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knows?
There’s also the whole relationship between psyche
and matter which seems to have been at the centre of all
my interests over the years. I got involved in parapsychology for a while, online psychokinesis research in 1996,
wondering whether there really was something in that,
and what it would imply concerning the psyche-matter
interface. There’s also a very exciting interdisciplinary
field of ‘consciousness studies’ emerging, and which I’ve
been following, people trying to understand the physics
of consciousness, looking at microtubules in brain cells
and how quantum mechanical phenomena at that scale
might help to explain the origins of consciousness —
physicists, neurologists, philosophers, psychologists,
anthropologists, psychopharmacologists, etc. are all contributing to this field. Then there’s all the Jungian theory concerning myth, archetype, synchronicity and the
‘psychoid’ level of reality — a kind of psycho-physical
interface. The simple fact that mathematics is able to
describe the world at all, that’s a mystery involving mental constructs being mapped mapping onto material reality. There’s the ‘mind-brain problem’ which philosophers
debate. And then dreaming, shamanism, schizophrenia,
quantum-mechanical paradoxes, these are all things I’ve
spent a lot of time thinking about, reading about — generally wondering how it all fits together. And it had
occurred to me that these topic cluster around the central
mystery of how matter and psyche interface. But I’d been
thinking about prime numbers, etc. for a few years before
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it occurred to me that this is very much part of the same
picture. I’d been exploring the interface of physics —
which concerns matter, obviously — and number theory,
which, as that Tenenbaum quote suggested, is really an
exploration of ‘the mind itself’. And the research I’ve
been interested in archiving displays a two-way traffic:
Number theorists have been providing concepts and
structures which physicists have used to better understand the world of matter. Physicists have been able to,
using their understanding of matter, shed light on the
internal workings of the number system. Even number
theory without the physics is implicated: although number
is widely considered as a mental construct, at the same
time it manifests directly in the world of matter: when
you consider a quartz crystal or a five-petalled wildflower,
it’s hard to deny there’s an essential ‘sixness’ or ‘fiveness’
there. So, number itself is a bridge of sorts between
psyche and matter.
This last idea, that number is a bridge between psyche and matter, comes quite close to something Jung was
exploring in his later career. He left a lot of incomplete
work when he died, and I believe he left von Franz to
look at number archetypes. He’d looked at individual
integers, the first few integers and their various associations. But later, more importantly, he’d come up with the
idea that, not individual numbers with their associations,
but the set of positive integers as a single entity is in itself an
archetype, the archetype of order.
Now what has distinguished Western culture from the
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rest of humanity, what characterises the Sumerian-toBabylonian-to-Greek-to-Roman-to-Western-European cultural current that dominates the planet with its measurement and science and so on, is the way we’ve dealt with
this archetype which normally inhabits the collective
unconscious. I picture it as a sort of mysterious sea creature — we’ve hooked it and we’ve hauled it out from the
dark depths into the daylight of consciousness. We’ve
taken something that was primarily unconscious, and
which would naturally manifest primarily via the number
archetypes and number associations in other cultures.
We’ve dragged this thing out of the sea and onto the land,
cut it up and studied it, studied its anatomy in great detail
in order to obtain a new kind of magic, if you like, and
that, I came to believe, was the root of all the world’s
problems.
But then we have this emergence into consciousness
of the set of the prime numbers buried within the set of
positive integers, a hidden archetype within an archetype,
a kind of chaos within order, the black dot in the yin half
of the yin-yang symbol; the emergence of that archetype
— the prime numbers, the zeta function and everything
they entail — into mass consciousness, is just starting now,
really. The first four ‘popular’ books on RH have all
come out in the last couple of years...it’s strange that this
should all be happening so suddenly. Thinking along the
quasi-Jungian lines I’ve sketched out, the integration of
these ideas into consciousness, the idea of the Riemann
zeros having their origins in some ‘older’ or ‘deeper’
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numerical reality, something more ‘primordial’, etc. may
turn out to be of profound historical significance.
According to the insanely optimistic wishful thinking
which I’ve since distanced myself from, this could be the
event that would start to alleviate the effects of rampant
‘quantocentrism’ and put things back into balance.
C: I wonder whether the growth of ‘popular science’
could play a rôle here — thinking in particular of the
many books which have been published on RH.
MW: The fact that you’ve got four books on RH out
suddenly — why is this, why hadn’t this happened
before? I’m sure a few years ago most people involved
would have said that it’s impossible to explain RH to laypeople. But four authors have done their best, with varying degrees of success. The books have all been wellreceived, have sold fairly well. So why is this happening?
The mystically inclined might invoke an unseen force
that’s trying to bring these ideas into consciousness.
Jungians might talk about ‘compensation’ and the collective unconscious. But more simplistically, more materialistically, it’s market forces, it’s capitalism, and it’s because
people are looking for meaning. Many are turning to
New Age cultism, some are turning to born-again
Christianity, Scientology, fundamentalist Islam, whatever.
But there are a lot of people who are aware that the real
‘guardians of truth’ these days are not priests and monks,
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but scientists and mathematicians, and yet, they find
themselves in a position where they don’t know anything
about the essential subject matter. So they want someone
to explain, say, the mysteries of quantum physics to them.
I get this all the time, people really wanting me to explain
quantum physics, fractals, relativity, the golden mean,
chaos theory, p; there’s a handful of things that people get
really excited and obsessed about, you know. And of
course the market system rises to meet a demand, a growing demand for meaning. The problem is that capitalism
doesn’t care whether a book is accurate or well-written, it
just cares about sales figures. So as a result you get gross
oversimplifications hitting the market and sometimes selling quite well. Because the market has expanded, there
is more competition, and ideally, if you believe in the
effectiveness of capitalism, then the ‘best’ stuff will float
to the top — but ‘best’ in this sense doesn’t necessarily
correlate with truthfulness or accuracy, rather with how
successfully the book quenches readers’ thirst for meaning. There does seem to have been a certain amount of
progress, though. I don’t really watch much TV, but it
does now appear that with the computer graphics available, it’s possible to make some things a lot more visually accessible, so viewers can at least get a flavour of the
problem, or of what’s at stake.
But the really deep stuff, the major philosophical
problems underlying maths and physics…it’s hard to
imagine that there really is a shortcut to years and years
of disciplined study. I mean, you might be able to get the
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basics of something across to a few, a small section of the
population who are already interested and whose minds
are structured in a certain way — it’s not to do with levels
of intelligence, just a certain kind of intelligence. You’ve
got committees for the popular understanding of science
and things of that nature, but they’re very marginal.
Unless there were a major cultural shift, unless you had
major government funding, and the top layer of mathematicians and scientists committing themselves full-time
to bringing this stuff through into popular culture...but
there’s no motivation for that to happen — governments
aren’t interested in educating their populations except in
ways which will further economic growth. They want a
certain proportion of young people to be trained up to be
economists, accountants, engineers, etc. ‘Truth’ doesn’t
really come into it. So I doubt it…but, again, you never
know, some major cultural shift could occur where the
demand for this sort of knowledge reaches the point
where the best people would feel obliged to provide it.
Or, possibly, there could be some sci-fi type breakthrough
involving direct brain-to-brain knowledge transfers, you
know, you can’t rule these things out, but I’m not
holding my breath!
You’ve probably noticed, part of my website is very
formal-academic, the web-archive aspect; and part of it is
just about getting fundamental ideas across to people who
are open to them and just want to understand their reality a bit better. I have felt in the past, with my ‘activist’
hat on, that it’s important to bring some of these issues to
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widespread public attention — the basic issues of the number system. At this stage I don’t know if it is ‘important’
or not, but I’d be very interested to know what the
overall effect of that kind of exposure would be. Again, I
suppose I am still gripped by the idea that, if we transform humanity’s relation with number, that could have a
positive transformative effect. I suspect I’m still partially
motivated by that belief at an almost subconscious level.
The only thing I can really say with any confidence at
all is that I think we’re on the verge — and again, the
timescale is very indefinite here — but Western
Civilisation is on the verge of collectively realising that
the number system is something very different from what
it had previously thought it to be. I haven’t got a
particular theory about what it is, I just know it isn’t what
we think it is.
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Introduction to ABJAD
‘Incognitum’
ABJAD (
) is an acronym derived from the first
four consonants of the Hebrew/Arabic/Persian alphabets;
Alif, Ba, Jeem, Dal. It is a simply-constructed but
functionally-complex alphanumeric system condensing
different belief-dynamics from Near- and Middle-Eastern
cultures, which became particularly prominent after the
rise of the Shi’a religion in Iran. Arabic ABJAD originates
from the Semitic family of scripts.
In working with Arabic ABJAD usually three
numeric values (the ABJAD value of the letter, the order
of the letter in the ABJAD table and the alphabetic order
of the letter in the Arabic alphabet) are used as numerating values; the most interesting and complex products of
ABJAD are those operating with more than one numeric
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value (i.e. ABJAD value, ABJAD order and Alphabetic
order simultaneously) but later works by ABJAD scholars from Iran, India, Afghanistan and Arabic countries
show an inclination towards a unified ABJAD value
called ABJAD-e Kabir (Major ABJAD).
Fig. 1. ABJAD Table (ABJAD-e kabir: major abjad)
The arabic ABJAD table (ABJAD-e Kabir) is arranged
in powers of 9:
Level 1. Alif (= 1) to Toin (= 9)
Level 2. Ya (=10) to Saad (= 90)
Level 3. Ghaf (= 100) to Zoin (= 900)
Level 4. Ghain (= 1000)
The ABJAD values of all levels are based on the
connections between the ABJAD order and the ABJAD
value of each letter at the first level (1-9), where the
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ABJAD value of each letter is equal to its ABJAD order.
LEVEL 1: from 1 to 9. Take the letter Haa as an
example: its ABJAD value is based purely on its ABJAD
order i.e. 5=5.
LEVEL 2: from 10 to 18 (ABJAD order). Since the
level changes, there is a phase transition from one- to twonumeral values (using one ‘0’ as a place holder). The
ABJAD order of the letter Noon (under the letter Haa) is
14, which can then be numerically simplified as 1+4 = 5.
The single numeral x is converted to xx to indicate the
letter Noon’s location on the second level. Consequently
we have 50 instead of 5: 50 is the ABJAD alpha-numeric
value for the letter Noon.
LEVEL 3: from 19 to 27 (ABJAD order); on the third
level xx changes to xxx; the letter Sa is under the letter
Noon; its ABJAD order is 23: 2+3=5=500 (xxx)
LEVEL 4: at this level ABJAD diverges from the power
of nine. This might be considered the apogee of the
ABJAD alpha-numeric progression.
The numerical arrangement of ABJAD into four
alpha-numeric layers, three of which are built upon the
power of 9 (9-based) is strongly in accordance with the
unique politics of Islamic apocalypticism, whose
allegiance is not to sectarian ideologies but to the explicit
text of the Quran, immutable according to Islamic
scholars. We have three levels (1-9, 10-90, 100-900)
whose structures are numerically never concluded (they
never reduce to One, remaining imperfectible and
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inconclusive). Only the letter Ghayn numerically stands
on its own plane as 1000 (Kabalistically reducible to 1);
however, the One here is external to the 9-based arrangement of ABJAD: In Islam, the divine constantly remains
on the outside; it is characterized by its radical externality to its multiplicative creation, and conceived not as
conclusion or telos but as externality. This only serves to
highlight the overall imperfectability and inconclusiveness of ABJAD/Creation. The Divine loiters on the
exterior (beyond the threshold of ontological possibility)
even on the Day of Apocalypse (as in Islamic Apocalypse,
Qiyamah). God never reveals itself (Apocalyptio), shifting
the radicality of its exteriority and the inconclusiveness of
its creation to another plane.
While God was the exclusive source of the revelation
to Muhammad, God himself is not the content of the
revelation. Revelation in Islamic theology does not
mean God disclosing himself. It is revelation from
God, not revelation of God. God is remote. He is
inscrutable and utterly inaccessible to human knowledge [...] Even though we are his creatures whose
every breath is dependent upon him, it is not in interpersonal relationship with him that we receive guidance from him.1
The ambiguous monotheistic structure of ABJAD
(enshrouding obscure religio-political inclinations), and
its empathy for imperfectability, multiplicity and
1. Edmund Perry, The Gospel in Dispute: The Relation of the Christian Faith to other
Missionary Religions (New York: Doubleday, 1958), p155.
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inconclusiveness, unillumined by a rabid exteriority
ungraspable for Man, has made ABJAD the alphanumeric system most favorably-disposed to heresies and
obscure apostasies. It is no exaggeration to say that the
history of Islam has been perpetually accompanied by
ABJAD exploration and alphanumeric distortion of all
forms of official and established religious institutions and
texts (the latter being untransgressable, and consequently
prone to the generation of profound heresies).
“Everywhere that ABJAD can be found, a heresy has
already emerged,” remarks Abidulah ibn-Maymun, the
founder of the Ba’teni – later to become Ismailie – cult.
Given the fact that the syncretic configuration of
ABJAD scriptures as well as ABJAD diagrams has always
been complemented by the sheer syncretism of its
redactors and exegetes, their usage in mass culture and
belief systems of Muslim populations has sprawled over
a vast array of everyday affairs, surpassing mere occult
instrumentality and elitism. In the timeframe between
the rise of Horoufi sect (from which the most prominent
ABJAD theorists rose – 9th Century) to the Qajar dynasty
(1781-1925) ABJAD diagrams were composed for
purposes including education in elementary schools to
depict the interactions between alphabet, numbers and
religious matters in an efficient way (commonly being
used for memorizing religious stories, names or even
basic mathematical or linguistic lessons), interpreting the
Quran and other Islamic or sectarian scriptures, healing
diseases, invoking love or hostility, conjuring deities,
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operating as catalyzer-spells in alchemical experiments,
etc.
The multi-functioning (heretical) nature of ABJAD
also allowed it to be extensively employed as a language
of communication between minorities (or within minorities)2, and as an instrument for supporting the political
belief-systems of Islam’s non-Apocalypse through
mathematics, astronomy and geometry and through the
cross-fertilization of these fields with linguistics, cipherology and occultism, producing a vast field of cryptosciences or heretical knowledges. ABJAD has become an
example of a numerical system which, far from being a
nomadic numerical machine, intrinsically operates within
the State in order to initiate anomalous reciprocations (in
the case of ABJAD, the state and numbers go hand in
hand).
ABJAD Diagrams are perhaps among the more wellknown productions of the heretical knowledge and
absolute syncretism one finds in ABJAD systems, yet
they remain unexplored. They critically condense and
compile wide varieties of monotheistic and non-monotheistic elements, although they constitute elements specifying their connection with the minority belief-dynamics of
either Sunni or Shi’a. At the top they usually include the
number 786 which alphanumerically equates to Besmellahe Rahman-e Rahim (In the name of Allah, the passionate, the merciful), the opening verse of each chapter (Sura) in the
2. In fact, all sectarian ciphers in Islam and Islam-inspired religions are
constructed and can be only deciphered by ABJAD.
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Quran. The diagrams are mostly constituted of
humanoid figures with bodies in positions evidently
inspired by the Zurvanite, Sumero-Babelian and
Zoroastrian cartographies of blasphemy, or demographic
configurations. Archaeological explorations of Near and
Middle Eastern cults (especially from the time of the
Mesopotamian civilizations to the end of antiquity) reveal
that forms of demonism (‘demons’ being understood as
avatars of the outside) are mainly characterized by their
anomalous cartographies – diagrams which present the
bodies and positions of the demons, and the arrangement
of their appendages (faciality is the least significant aspect
of Eastern demonolatry. Islam also forbids the ‘facialization’ of religious figures and martyrs):
• The right hand upward and the left hand downward
is a cartography of demonism of the greatest archaeological provenance, originally emerging in the Assyrian
period, where demons of pestilence and contagious
epidemics are presented as seen in the bronze Tablet of
Disease (in the collection of M. De Clercq). A significant
number of ABJAD humanoid diagrams maintain this
cartography in the most explicit way (see page 202) while
presenting the ABJAD numbers connected to the legion
of the damned and blasphemous people mixed with holy
names and numbers on their bodies.
• From stretched hands, one pointing to east and one
pointing to west, we can identify solar demons (the
Romans ironically borrowed this same diagrammatic
position from the Babylonians in their crucifixions, the
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most prominent of which are the iconographic portraits
of the crucified Jesus). (See page 200).
In ABJAD diagrams, when it comes to the divination
of a religious figure, facialization is achieved through
composing the face, head and body with the exclusively
use of numbers and letters, with no depiction or explicit
portrayal (see page 201). In consequence, ABJAD
pushes Islam’s holy ban on facialization into a blasphemous demonic complexity which with the same
reprobate enthusiasm tears itself away from the
expressionist hegemony of facialization.
In these diagrams each part of the body (including the
head) has its own agenda of ciphers and exclusive
numbers. Usually most of the body is enveloped and
takes the form of a repetition of the letter Haa (ABJAD
value = 5) representing the five individuals of Shia:
Mohammad, Ali, Fatemeh, Hassan, Hussein (see page
201).
In western occultism, diagrams and magic squares are
usually surrounded by magic circles and other geometric
shapes which are always closed and symmetric. Their
task is to converge the power of the spell upon a certain
objective. In ABJAD diagrams these circles are replaced
by open geometric shapes such as triangles and curves
conducting open-ended and divergent experiments in
syncretism. In ABJAD diagrams these curves and shapes
are commonly known as ABJAD shields. Most ABJAD
books, although published by Shi’a authors and
containing the names of Shi’a Imams, correspond to the
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cipherology of the traditional Arabic ABJAD in which
diagrams or figures are guarded by ABJAD shields,
covered by either the letter Meem or the letter Dal (the first
and the last letters of the name Mohammad); the design
of these traditional ABJAD shields is of two intersecting
lines forming an acute or obtuse angle representing the
letter Dal (ABJAD value=4). Unlike in Sunni ABJAD (see
page 199), in Shi’a ABJAD these shields are not pointed
and, rather than the letter Dal are in the form of the letter Ha – expressing the Shi’a politics of Taqiyya versus the
Sunni politics of conflict in Jihad – but with curved lines
diagramming the calligraphic elements of the letter Haa.
These curved lines usually become overrun by the
repetition of the letter Haa (ABJAD value = 5) on their
outer surface, standing for ‘Panj Tan-e Aal-e Abba’ i.e.
Mohammad, Ali, Fatemeh, Hassan, Hussein who are the
pillars of Shia (See page 210).
Page 199: A spiteful spell for calling upon a disease or
replacing a fatal disease with a less dangerous illness. The
word Allah constitutes the outer open circle, the targetingarrow of the letter Dal (empowered by the outer circle
and representing the power of the prophet Mohammad)
can operate from the opening and directing the power of
the ABJAD square (composed by the numerated name of
a disease) in the guise of a sting towards a person.
Page 201: The Arabic and Abjad equivalents of the
words ‘No’ or ‘Closure’ compose the face of this ABJAD
figure, making her mute, blind and deaf or, symbolically,
easy to be controlled. At the center of this ABJAD
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diagram is an ABJAD square which has replaced the
heart. The symmetry of numbers and certain words (the
word speed and the name Ali) make this spell oscillate
between negative and positive intentions. This spell
should be engraved on four pieces of alloy (fusion of the
senses), buried in soil, put in fire, exposed to wind and
thrown in water to make a person possessed by an
involuntary love.
Pages 204-205: An Abjad diagram for memorizing the
names (unique characteristics) of Allah in an interacting
form.
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Existential Risk
Interview with Nick Bostrom
The Future of Humanity Institute forms part of the ambitious
research program of the 21st Century School established by
benefactor James Martin at Oxford University in the UK. If it is
surprising that the UK’s most traditionalist university by reputation
should host an interdisciplinary research institute dedicated to
evaluating the long-term prospects of the human race, it is perhaps
even more remarkable that the director of the Institute Nick Bostrom
began his academic life in analytic philosophy. He talks to
COLLAPSE about the wide-ranging research program of the FHI
which seeks to address ‘existential risks’ for mankind ranging from
natural catastrophes to biotechnical modification and the emergence
of nonhuman intelligences.
C OLLAPSE: The establishment of the Future of
Humanity Institute seems to mark one of those
‘crossover’ moments where strands of research that once
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seemed purely academic, or were at best categorised as
interesting ‘thought experiments’, are becoming pragmatically pertinent: amongst the issues you deal with are
bioscience, cryogenics, catastrophes of an ecological and
cosmological nature. Is the function of the Institute to
liaise between the academic research environment and
various public policy, governmental, or security bodies
who are looking for ways to gain traction on these issues?
N ICK BOSTROM: Yes, but its primary mission is a
research mission – to try to improve our understanding
of three broad areas. One is human enhancement,
ethical issues especially related to that, but also practical
issues, policy issues. The second is global catastrophic
risk; and the third area is methodological problems and
issues that crop up when we’re thinking about the big
picture for humanity, both in relation to the first two
areas and more generally when we’re thinking about
anticipated future technologies, and so on. So the
primary focus is to improve understanding through
research in those areas, but it has a component – or will
do, in the future – of liaising with policy-makers and
trying to create some public awareness and stuff like that.
So the interdisciplinarity is not an end in itself. It’s
just that when you try to address some of these problems
and issues, it just so happens that more than one
discipline is needed.
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C: Given the magnitude of the problems you’re proposing to treat, is it surprising that there hasn’t been such a
unified initiative to tackle them until now?
NB: I don’t know, it’s difficult to do within a traditional
disciplinary setting, that’s one reason. Because none of
these problems fit neatly into a discipline. So it might not
be so surprising that it hasn’t been done, because it
requires probably some framework that goes outside
traditional disciplinary boundaries. In one sense it is
surprising because the issues are important enough that
one would have thought that humanity would have a few
people somewhere thinking about them: extinction risks,
for example. So, from that, from the observer from Mars
perspective, I think they would be surprised.
I should also say that although these three areas are
quite a broad range, in reality what it means is that one
would have to pick specific, tractable questions within
those areas, and look for opportunities where analytic
methods and our current knowledge actually enable us to
say something.
C: You were trained as a philosopher, and we were struck
by the fact that the Institute is attached to the philosophy
faculty. How did the Institute originally come about?
NB: I was in the philosophy faculty, I was a postdoctoral fellow. And I’ve always been interested in, been
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pursuing, these kind of questions. And together with
Julian Savulescu, who is professor of practical ethics here,
we had a couple of ideas for research projects, or
programmes that we wanted to pursue or get funding for.
One of them was human enhancement ethics, another
global catastrophic risk, and there was a third one. So we
went to the development office, these are the people
responsible for fundraising in the university, to see if they
had any ideas. And they said that they knew someone
they thought would be interested in this, and had
actually mentioned me to him before. It turns out this is
James Martin, who is the benefactor for the James Martin
21st Century school, of which the Institute is now a part.
And James Martin had interviewed me earlier, about a
year before, for some television documentary he was
doing, and appeared to be quite enthused about what I
was trying to do. So by sort of coincidence – because I
didn’t know that he was a donor, or a funder, a rich man,
at that point, he was just somebody doing an interview –
this development guy managed to put one and one
together, and establish some contact, and then there were
lots of drafts back and forth, and lots of different
proposals, and discussion, and this is what resulted in the
end.
C: So, with the involvement of Martin, is it actually a
genuinely philanthropic exercise?
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NB: Yes – we have some co-funders as well, but that’s
where the bulk came from. And Julian Savulescu, he’s
the director of the Uehiro Centre for Practical Ethics,
downstairs in this same building, but he also has a
programme on Ethics and New Biosciences, which is also
part of the James Martin 21st Century School.
The whole James Martin school is really James
Martin’s attempt to do a couple of things: to encourage
interdisciplinary work and to bring Oxford into the 21st
century, but more specifically to try to get Oxford to
focus on some major problems for humanity in this
century and see what practical solutions can be found.
The whole school to some extent is shaped by that vision.
C: How closely would the Institute’s research have to
track or monitor things like ongoing developments in, for
example, AI research, and even A-Life research?
Different research programs underlie both of those, with
internal technical debates that would have ramifications
for broader questions about the nature of other
intelligences and the characteristics they would possess.
NB: There will be one level of monitoring which is just
keeping some general idea of what’s going on. But then
depending on which specific projects or which specific
paper you’re working on at the moment, you would
obviously then have to dig down very deeply in that area.
So if you were writing a paper on artificial intelligence
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let’s say, then you would have to get up to date on the
latest details of that. But generally you can’t be up to
date on the details of everything that is possibly relevant,
it’s just not humanly possible. So you have to have some
general idea of roughly what’s happening, and then you
get to focus, like I say, when you’re producing something
specific.
C: Are there practical limits? If the Institute expanded
its staff would you be able to have these subcategories…?
NB: Yeah, you still wouldn’t be able to keep track of
everything, but you could have people specialising more,
or covering a greater range of topics. We’re still in the
bootstrapping phase now.
C: What seems remarkable is that you’re trained in
analytic philosophy, and you’re applying the methods of
analytic philosophy to problems which up until now have
only been thematised in so-called continental philosophy,
i.e. the problem of death, the problem of what it is to be
human, etc. And yet within your programme you seem
to be pursuing these problems in a way which continental philosophy has been unable to do. For example
despite its obsession with the concept of death
continental philosophy doesn’t seem to have tackled the
basic fact that when the sun expires we will all die – and
what the consequences of that are for thought.
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NB: I think there is a traditional view that the
continental philosophers have all the interesting questions
whereas the analytic philosophers have had the right
answers but the boring questions…I see philosophy as on
a continuum with science really, I don’t make a sharp
distinction. It’s the more general end of the spectrum,
things that haven’t yet been crystallised out, specialised
into different particular sciences. And so whatever seems
relevant material to thinking about the particular
problem questions, I think that’s what one should use.
And also in some cases even though the questions might
seem the same they might actually be different. So
although continental philosophers might have thought a
lot about death, there are lots of different questions you
can ask about death: about its meaning for the artist, or
for the human being on one hand, or you could ask more
concrete questions such as whether current research
priorities make sense, is the allocation of research money
to biogerontology versus cancer research in the right proportion, for example.
C: How do other philosophers – not only at Oxford but
also elsewhere – regard your work?
NB: Some of the work I’ve done has a sort of dual-use
function, which is the way I have had to work up more
or less until now, since I haven’t been able to have a post
full-time for doing this. So some of the papers like Infinite
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Ethics or Are we Living in a Computer Simulation?1 can be read
by people who are really interested in the FHI from a
purely philosophical point of view, and there is enough in
it for them that even if you’re not interested in the
practical dimensions of it you’ll find some worthwhile
philosophy there. But from my point of view the reason
why I’ve written about them is that I think it relates to
this broader vision, to the practical concerns that I have.
So one could find some of these stepping stones where
one could get both philosophy and the relation to
humanity’s future down on the same paper, but it
requires a little bit of picking and choosing and figuring
out clever ways of combining the two. It’s obviously
much more desirable if one can do work solely on the
basis of the criteria that it should be relevant to the big
picture for humanity.
C: You propose to tackle analytically problems which are
of such cosmic scale that they induce what you call in
your own work ‘infinitarian paralysis’: that is to say,
given the magnitude of these questions, how is it possible
to do select one path or another, to choose between
options? One of the methods you seem to be using is the
quantification of risk, a risk-management or risk-assessment approach. This quantitative treatment of risk is
something which is of great importance in public policy,
in corporate policy, in finance, but how is it applicable to
these massive existential dilemmas?
1. Both at http://www.nickbostrom.com
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NB: Well that’s a relevant question, whether it is or not.
Specifically, in the case where you have literally infinite
stakes – which is the case this paper, Infinite Ethics, discusses – there is a set of ethical theories that run into trouble
when you take seriously the possibility that either the
universe might be infinite and have infinitely many
people in it, or even that our actions might have infinite
consequences. Even if you think the possibility of that is
small, you’re still going to have what I call this ‘infinitarian paralysis’ problem. It seems as if it’s impossible that
we could make a difference, in terms of changing the
expected value of the world. Now, that might be more a
problem for certain ethical theories rather than for
humanity, because if certain ethical theories have that
implication then we might just take the lesson that we
should have other ethical theories instead that avoid it.
So I think it’s a significant but technical point. There is a
broader sense in which it’s possible to think about some
of these things. It’s easy to get a sense of vertigo sometimes, because there are such extremes in terms of the
best outcomes and the worst outcomes, just the sheer
magnitude of them: the problems and the stakes are so
much larger than things that people spend so much more
time thinking about. But my view is that one should try
not to look away, one should just try to keep a steady
gaze, and use the same methods, as much as one can, use
the same analytical rigour, sober analysis, and step-bystep investigation to make progress on these.
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C: Is the long term proposition of the methodological
approach that you will actually be able to answer ethical
questions relating to these ‘existential’ issues on the basis
of quantitative consequentialist ethics? Myron Scholes,
for example, created a formula for quantifying risk in
financial transactions: do you propose that such a thing
is possible in ethics, or do such methods only give
guidance for further thinking about the problem?
NB: Yes, I think it’s one ingredient, I think there is more
to ethics than risk analysis. In terms of a meta-ethical
view or a foundational view, I’m not firmly committed,
I’m still trying to figure out which meta-ethical framework is correct, I don’t know. Certainly I think taking
consequences into account must be at least one thing that
any reasonable theory would do, but there might be
other things in addition, and how you should take them
into account. So the answer would be no, I don’t assume
that would be the be-all or end-all. I think it would be
one useful piece of information to have, if you could have
that: information about the risks and the expected utility
of different actions, I certainly think that would be useful.
Really, the value proposition would be to try to illuminate some aspect of the big picture for humanity, pretty
much those aspects where it’s possible to illuminate them.
Some might be very interesting, but would need some
new insights that nobody else had, to figure out how we
could say anything intelligent about them – maybe it’s
just not possible given current knowledge. But then there
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are some aspects where, with hard work or ingenuity, or
using techniques of various kinds, we can say something
that is not obvious and yet is relevant. So it's shooting for
those targets of opportunity...
C: That’s what you mean by ‘opportunistic research’?
Choosing the problems you think you can gain some
traction on.
NB: Exactly.
C: Prior to the methodological question, it seems, are
certain fundamental underlying assumptions which are
driving the research, amongst which are the ideas that
longevity is a good thing, that survival of the human race
is a good thing: and the optimism of the ‘transhumanist
values’2 that you personally espouse go beyond this.
NB: In terms of the first one, longevity, my view is that
increased longevity and great health, or people having
the option, would be a good thing. So that’s my view, it’s
not necessarily a premise of all the work that the Institute
will be doing. But that’s the sort of question the ethics of
which would be a subject of enquiry. It might turn out
also not to be a yes-or-no thing, it might under certain
circumstances be a good thing, under other
2. See ‘Transhumanist Values’ at http://www.nickbostrom.com
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circumstances it would be problematic. If other
researchers looked at it maybe it would not be…and so
whereas I am personally quite enthusiastic about the
prospect of healthspan extension that’s not a premise for
the Institute. Now as for the survival of humanity, could
that be seen as a premise? Well it’s not an unquestionable premise, you could have a philosophical seminar
where you would discuss that, and indeed we have had
such a seminar, papers have been written discussing
whether that would be a bad thing or not. So it’s certainly discussable. Now if the work moved more into liaising
with policymakers then I guess it would be hard not to
make some assumptions like that. I mean, it’s just as
when you have someone researching cancer they just
assume that it’s good if we can find a way to detect breast
cancer early, you don’t have to have a huge philosophical
conversation about that.
C: But precisely from a philosophical point of view this
becomes a problem: It’s not that one needs to denounce
the fact that philosophy liases with the world, tries to
engage with practical problems. But isn’t it problematic
that the ‘transhumanist values’ seem to have a basis in a
particular political (and implicitly, philosophical) position
already – e.g. when you talk about the ‘entrepreneurial
spirit’ being part of the set of values?
NB: But you’ve got to keep distinct transhumanism and
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the FHI. There are other organisations – the World
Transhumanist Association and so forth. That one in
particular is not so much a specific philosophy but a
conglomeration of different people and thoughts and
values, some of which I disagree with; but obviously I am
a transhumanist in a certain sense. But the FHI is not
an institute for transhumanism or anything like that.
What it does though is to focus on problems that would
be of interest to transhumanists and many other people
as well, I mean obviously global catastrophic risk would
be of interest to a transhumanist, but so would it be to
many other people. And similarly with the implications
and ethics of new technologies. So there again, you can
have different perspectives and viewpoints and
arguments, which might or might not agree with certain
transhumanist positions.
*
C: In several of your papers you talk about the possibility of transcending ‘observer selection effects’.
NB: Yes, observer selection effect is a kind of distortion
of our thinking that occurs when the evidence we get is
filtered in a certain way. The easiest way to explain it is
a selection effect that we find in many contexts: there is
the fish example where you catch a hundred fish from a
given pond, and they’re all larger than six inches. So you
draw the inference that probably the smallest fish in the
pond is not less than six inches. Then you look at the
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thing you caught them with, this net, that can’t actually
catch smaller fish. So you realise that although you have
one hundred data points, they’re all filtered through the
selection effect that the net introduces. So, once you
realise that you have to correct for it, if you want to
estimate the population of fish in the pond you have to
take that into account. So there are different methods for
doing that in standard statistics. Now, an observation
selection effect is similar, except it’s not introduced by
limitations in our measurement apparatus – the fish net –
but by the fact that all observations require there to be a
suitably-positioned observer to make the observation or
to build the measurement instruments in the first place.
In many cases these are not really relevant, but there are
certain specific questions and problems where
observation selection effects become crucial: in cosmology for example, they are of critical relevance when one is
trying to figure out what observational predictions one
can derive from current inflationary cosmological
models. Also, if you ask certain questions about the
evolution of intelligent life, they become relevant again.
The probability of us discovering extra-terrestrials: they
turn out to be relevant to answering that question. And
then they are relevant to others, like the probability of
catastrophe. And to a number of philosophical thought
experiments as well, like the ‘Sleeping Beauty’ Problem.
C: So our analysis of the possibility of certain events
depends on a prior ‘transcendental analysis’ of the
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human cognitive condition?
NB: I wouldn’t go so far as that, no. But let me give you
a particular example. There has been some concern
amongst people who are doing these high energy particle
accelerator experiments that – it seems unlikely – but
theoretically maybe they could cause a breakdown of the
metastable vacuum state, which would mean that there
would be a rip in space-time that would expand at the
speed of light in all directions, and bring utter disaster not
just to earth but eventually to the whole observable
universe. And so the question is, what reason can we
have for thinking that these things won’t happen?
The director of the Brookhaven particle accelerator
commissioned this report a few years ago, the
Brookhaven Report, to study this potential risk. And the
authors of the Brookhaven Report made a good point,
that the energies that would be attained in the reactor
when particles collide, are attained all the time in the
atmosphere: there are cosmic rays, particles from space
that hit molecules in the atmosphere. And they occur
much more frequently there. So, assuming that the
reactor events are equivalent to the naturally-occuring
ones, we can then calculate an upper bound on how
probable these reactor disasters could be, and it’s very
reassuring if you do that.
Now one problem here is, how do we know that the
natural disasters are not very common? Suppose it were
the case that on average, taking an average cosmic region,
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that it was usually sterilised after one day, so you would
have this furious destruction of cosmic regions, and then
nothing else would happen. Now how do we know that
that’s not the case? Even if that was the case then we
would still obviously expect to be alive – you don’t
observe these destructions, the destruction hits you at the
speed of light, so no-one can see these things. So the only
thing that would be seen would be parts of the universe
that survived. And if the universe is big enough there
would always be such parts. And so they had overlooked
that. But it turns out that you can get around that, and
this is a recent paper in Nature 3 I wrote with Max
Tegmark, who is a physicist, where we point out that you
can use data from planet formation rates to place an
upper bound on this. The basic idea is that if these
sterilisation events had been very common, then we
should have expected to have been on a planet that had
been formed much earlier. So there you can, using these
data, and awareness of these observer selection effects,
find a way round it. In this case it supports the original
conclusion.
*
C: As is the nature of the research undertaken under the
aegis of the Institute, issues about the possibility of some
posthuman intelligence, non-carbon-based life – there’s a
high degree of philosophical speculation, or abstract
speculation, involved in these. On what basis would you
discriminate between relevant and irrelevant degrees of
speculation? For instance, after the ‘singularity’, this
3. ‘How Unlikely is a Doomsday Catastrophe,’
at http://arxiv.org/abs/astro-ph/0512204
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unimaginable amplification of cognitive capacity, there’s
an issue about whether there would be any continuity
between intelligence as we know it, as familiar to human
beings who are organic entities with a set of organic interests, and some sort of massively distributed non-organically-individuated intelligence. Would it have the same
kinds of interest, would it want the same things that
carbon-based lifeforms want? These are massive issues,
obviously. But what’s the cut-off point where you foreclose speculation?
NB: Well I guess there are a couple of ways in which it
could satisfy the relevance criteria. One is that the
question might have independent philosophical
significance, philosophers often use thought experiments
to test different philosophical positions – the ‘brain in a
vat’ experiment, it’s irrelevant how likely it is that scientists can actually make a brain in a vat if you want to
think about internalism or externalism about mental
states, for example. So obviously anything that is
relevant apart from its empirical possibility would fall
into that category. But then there are issues that are
important because they might have practical relevance to
what we should do now, or to one of the main questions
that the Institute will study, but where it’s just not
possible to get anything better than a speculative answer.
So then again, my view is that we’ll have to make, and
use, whatever tenuous guesses we can make, but then
simultaneously recognise them as being tenuous guesses.
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Because there’s not going to be a vacuum here, and if one
doesn’t try to do something carefully, with qualification
and with as much precision as one can, then people are
going to do something there anyway, which is going to be
less good. So that’s one way in which it can be relevant.
And then there are some theories that would be very
speculative, and where you might not be able to say that’s
what’s going to happen, but you might be able to say if
this thing happened or that thing happened then you can
construct a quite good argument for why a third thing
would happen, so there might be these conditional results
as well. So given certain assumptions and premises you
can make a strong case that there would be these other
consequences. So that’s the third way. To some extent
one knows it when one sees it, if it seems to – even if it
doesn’t prove the point, if it sort of seems like an
intelligent, carefully thought out, worthwhile consideration, then that’s a kind of useful study.
C: Might some of the questions concerning superintelligence or a transhuman intelligence not be beyond our
analytic capabilities? Why would a higher form of
intelligence be something that would be recognisable to
us any more than human intelligence is recognisable as
such to an insect?
NB: If I could answer this in a personal capacity, what I
currently think about it is that a superintelligence may be
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best thought of as a very powerful optimisation process,
which means that whatever its goals are, the defining
characteristics of a superintelligence would be that it is
very good at achieving those goals. So that means that
what it would actually do, how it would manifest itself,
would depend critically on what its goals were. So the
effects might be anything from being totally invisible to
tiling the world with paperclip factories or smiley faces or
whatever. You might imagine some unwise AI programmer in the future, when they are just about to create this
first superintelligence, who thinks it would be very good
to give it a motivation system based on reward learning.
So they train it to get a reward whenever it sees a happy
face, a smiling human face, then if the humans seem
happy it gets rewarded for that, so it gets a motive to
perform those actions. And then once it becomes more
intelligent then it realises that it can produce this outcome
that it has been taught to find desirable much more
efficiently just by creating lots of smiley faces. And if
that’s the only goal it has then it might work out a very
clever way of achieving a maximum of smiley faces. So
I think with something that would be radically superintelligent – not a human that is slightly smarter than us, or
AIs in the near future, but a really radical superintelligence – I think it could be very alien, and one shouldn’t
think of it as a sort of human genius.
C: This indicates another fundamental issue, the question
of whether intelligence is necessarily connected with
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goals or aims. One could say that human intelligence
precisely has this excessive problem-solving capacity
without having a particular aim, or at least the former
exceeds the latter to a comical (or tragic) extent. Is
intelligence necessarily goal-directed? And if a superintelligence emerges, will it emerge from somebody explicitly
setting out to build it? Might it not emerge instead as
something which has not been planned at all? If you
think of the Internet, or something like Google, an
intelligent system that has in a sense developed a model
of meaning, of how words are connected together,
couldn’t something intelligent emerge from that without
it being planned?
NB: To some extent on the first question, one might
think that if a superintelligence turns out not to have any
particular goal, then it might not do or amount to very
much, and then its just a matter of waiting until sooner or
later some superintelligence arises that has these goals
and then you would get one of these scenarios where the
outcome might be very good or bad, it would all depend
on what the goals were. Another kind of answer to that
question is that it just depends on how you define goals.
So if you just have some thinker just sitting there and
looking at the world and pondering it you might think
that they have some sort of internal goal, of having their
thoughts more highly organised, or thinking true
thoughts or something like that.
And as for the second question, it’s difficult to say.
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There are three cases, there is one where somebody
deliberately sets out to build a superintelligence, and then
there is the other case where it just would arise
spontaneously from the Net or something. And then
there could be intermediate cases where people set out to
build something but then it goes farther than they
imagined it would. If some sort of self-improving system
is created, they might not really have clearly envisaged
what the outcome would be, but nevertheless you could
get a superintelligence out of that. Now which of those is
most likely is difficult to tell. Some people think that
deliberately setting out to do it is likely to get there much
sooner. You might do a comparison with other things,
like you don’t accidentally build a car or an aeroplane.
C: Are there particular technologies which you think the
emergence of other forms of intelligence is tethered to?
NB: Better brain-scanning technology. Well, we can
create thin slices of the brain, but then scanning those
slices more effectively than we can today, in an automated fashion, in a scalable way. I think that’s a real enabling
technology. Of course computing power, generally, that’s
another.
C: But isn’t that again to assume that any intelligence will
be modeled on human intelligence? Or is that the only
assumption that can be made?
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NB: No, but it’s one way we can imagine getting there:
we know one intelligent system, the human brain, it’s a
physical system, with neurons and stuff like that. So in
principle if we could figure enough out about this system
then we could see how it does it and do the same thing
on a computer. Now it might not be the thing that would
get us there most quickly, it might be better and easier
and more effective to do it based on other approaches
that don’t really follow the human mind closely. And for
those other approaches I think computing power would
be one thing that helps. But since right now the
theoretical problem of how to program any sort of system
like that is very difficult, and I think progress in
neuroscience and/or uploading is a potential technology
that could help.
*
C: The Institute seems to belong to this notable category
in the post-industrial world where research programs that
seem to be completely unconnected to any pragmatic
problems suddenly become crucial. Prime number
research, which related to a pure mathematical problem,
suddenly became the locus of massive financial
investment when cryptography became an crucial commercial factor. The last few years have obviously been
boom years for certain Islamologists, who suddenly find
themselves in great demand by media and intelligence
agencies alike, and so on.
Given that philosophy has such a huge wealth of
material that is apparently ‘useless’, one would have
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thought that its time was overdue…Are there corporate
or other interests outside the university who are interested in your research, who see that this extremely longview research, with a necessarily philosophical component, has important consequences for them?
NB: My own background is that I’m a philosopher but I
also do other things apart from philosophy, so it’s sometimes hard to see which of these is in play, especially in
the catastrophic risk area: I’ve done some consulting for
some intelligence agencies, some stuff like that.
Corporate, I haven’t really tried to reach out to that yet,
it might happen in the future. But the things that are
directly relevant to corporations tend to happen anyway,
because they tend to have money and they make them
happen. The things that don’t get done are things that
might be of great concern for humanity as a whole, that
our species would do well to invest some effort in thinking about. But any particular corporation, it’s not their
responsibility or their benefit. Like human extinction
risks, which corporations are unlikely to fund! But
nevertheless it seems like a good thing. So I’m trying to
do those things which should be done but aren’t being
done.
*
C: The mass media plays a huge role in transforming the
perception of risk; is there any space in what you’re doing
to think about the role that that mediated reflection of the
risk plays in transforming culture, and the consequence
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that fear of risk can be just as significant – if not more so
– than risk itself? Obviously if you think about the
supposed War on Terror and everything that’s involved
there, these factors are as important as the risk itself.
NB: I agree with that. There is one additional way in
which thinking about perceptions of risk might be
involved, in addition to the way the perceptions themselves make things change in the world: there is a
genuine objective risk, and then there is people’s perception of that risk. And if you can study the process
between these, the sort of distortions that occur, the fact
that there are certain types of risks that people might
overestimate because of they are vivid, and are played out
in the media; and others they might tend to underestimate because they happen bit by bit on an everyday scale
and are more abstract in some way. Now, if you could
figure out exactly what those distortions are, for the cases
when you can both know the objective risk and people’s
perception of it, then you might look at another case
where the objective risk is unknown but you can still
study people’s intuitions about it, and maybe you can
extrapolate back to get a better estimate of what the real
risk is, by a sort of counterbalancing for these known
biases. Especially with regard to certain existential risks,
I think that’s a useful thing to study. And there is quite a
lot known about people’s risk perceptions, and the biases
and heuristics we use to assess probabilities of things. It’s
also known that our intuitions about those things are
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wide of the mark, that there are systematic distortions.
C: Aren’t humans constitutively non-rational agents? It’s
impossible for us to right our perception of risk completely, isn’t it?.
NB: It’s a question of more or less closely approaching.
Take weather forecasters, they’re pretty good, they’re
well-calibrated. Calibration is when you say, if you think
there is a ninety percent probability of something happening, then ninety percent of the time when you say that,
that thing’s happened, that’s what calibration means. So
it doesn’t mean that you’re right all the time when you
guess, but your confidence
matches your real
knowledge. So weather forecasters are trained on this,
you go through lots and lots of examples when you’re
trained, so they’re quite well-calibrated. Pretty much
everyone else is ill-calibrated, maybe there are other
specific groups that are calibrated, insurance brokers or
stock market speculators. So it’s possible, given enough
feedback and enough incentive, to become a better
calibrated risk-assesssor. But that doesn’t seem to happen
spontaneously or naturally.
C: Bayesian analysis could be applied here, since it
includes a measure of confidence in probability calculations.
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NB: Certainly. With a lot of the biggest risks, you have
to rely a lot on subjective opinion. There’s just no way
around it, you can’t quantify a lot of these disasters that
have never happened in the world. The probability that
humanity will create a big war with nanotechnology
some time later this century: you can’t run a frequentist
approach and count the number of times this has
happened in the past. You don’t have a solid scientific
model like in quantum mechanics, where you can
calculate and get probabilities out, so what do you do?
Well, you have to think about it, what would the capabilities be that advanced nanotechnology would make available; how would that play out in a strategic context in the
world with competing powers. You think about these
things, you discuss with other people, you critique
arguments back and forth, but in the end you’re probably
going to have a situation where different people who are
intelligent, who have now read all of this, will assign
different probabilities. And I think that’s unavoidable.
C: So you do have to work with a synthesis of the facts
that you can represent statistically and the knowledge of
people’s reaction to these statistics.
NB: Yeah, so when we have solid statistical data then
obviously we use that. But most of the questions that are
of most interest to humanity you can’t solve by exclusively relying on that. And that’s not just when you think
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about the big picture of humanity, but when you think
about policy, political decision, or indeed your own life
decisions: if you decide you’re going to marry this person
you’ve been together with, there is no formula or
deductive proof that this is the right thing to do, you just
have to follow your gut feeling, after having taken into
account the obvious objective knowables, then you just
have to make an assessment. But that doesn’t mean there
aren’t better and worse ways of doing it. You can certainly be foolish or wise, and different people might do these
things differently well. So just because there is a
subjective component doesn’t mean that there are no
constraints.
C: Is it possible for a human being to calibrate themselves
in respect of risks on a cosmological scale? If we were
truly able to take into account the futility of being a
human being on earth, we wouldn’t ever do anything –
wouldn’t that be the truly rational response to these sort
of questions?
NB: Futility…well, I suppose it depends on what your
measurement scale is. So if you do assign equal value to
all life, or all humanlike beings, and you think that a nonfutile action is some action that changes the total value,
then you run into problems if the universe is infinite.
Although there are technical issues there, which my paper
Infinite Ethics covers.
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C: The use of non-standard analysis.
NB: Yes, and it gets very difficult. But suppose you have
a different criterion of what’s futile or not. So you might
think something is worthwhile if it makes you happier,
and some people you care about, and people on earth are
not harmed, and perhaps helped a little bit, and a few
hundred children in Africa are surviving. I mean, these
are quite big outcomes from one perspective.
C: But the question still remains of the independence of
philosophical thought, and whether in this endeavour
there are certain assumptions that have already been
made, illegitimately. For instance, if you were to produce
a paper which said simply, the rational thing to do is to
despair, there’s no point in doing anything, that would
obviously be bad for the Institute – aren’t you automatically disposed to a sort of philosophical optimism?
NB: No, I don’t think so. Certainly if someone came up
with a good argument that it doesn’t matter what we do,
or that we’re all going to go down the drain, then that
would certainly be a paper that would seem very relevant. I mean, I’ve written some papers which, although
they’re not quite saying that, could…the Infinite Ethics
paper doesn’t quite say that, but it’s going in that direction. Existential Risks 4 doesn’t say that we’re going to be
4. At http://www.nickbostrom.com
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extinct, but it certainly takes that possibility very seriously and argues that the probability is significant. The big
picture is difficult. We haven’t really thought about it,
and one shouldn’t be surprised if there are some rather
unexpected things once one has started to think about it,
that would come out of that.
C: Is it possible that the more interesting, the more
demanding ethical questions will arise precisely from the
position of despair rather than from the assumption that
the human race is a good thing, that it’s a good thing to
live longer? You’re asking a tougher question if you say
that the sun will expire, and with it all human life – now,
how should one act…it’s a more profound ethical
question.
NB: Well, I think the sun will expire, and we want to
know how we should act now, so I suppose that’s the
question I’m asking in a sense. Although you want to
make it more specific to be able to get something out of
it. But I think in general whenever you do some kind of
reseach project you have some kind of starting point.
Maybe one starting point and then you see where you
end up. Now for an Institute you can have more than
one starting point, for each paper you write you might
have a different starting point, and each person might
have a different set of background assumptions.
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C: The pessimistic view would be that for evolutionary
reasons, there’s a fundamental mismatch between human
beings’ requirements, their psychological requirements,
what they would need to have to make them happy, and
reality: Freud basically says he’s a pessimist because, as a
result of our evolutionary, our biological legacy we are
doomed to be perpetually dissatisfied. Would that have
ramifications for you, given that you say that a superintelligence would be a superoptimiser, would such a
superintelligence be able to overcome this mismatch
between needs, requirements and the capacity of reality
to meet them?
NB: With a superintelligence that could be one scenario,
you could have a ‘Santa Claus’ superintelligence scenario,
where you have some wish-granting thing. You might
have a similar thing without superintelligence.
In a
certain sense that’s what technology is – it increases the
ability of humans to realise their desires. Things sometimes backfire, people sometimes have conflicting desires,
and you get war and you get pollution. But there is also
something more specific, which is that even if, and to the
extent that, that’s how humans currently are – we’re not
designed to be happy, but to maximise inclusive fitness
on the African savannah, which is very different – it
doesn’t imply that that’s the way we will, or should,
remain forever. Which is the transhumanist perspective,
that human nature itself is something that humans can
change. And certainly this points towards an area of
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interesting questions.
C: If humanity is going to actively shape its own future,
its own destiny, it would have to presumably make some
choices, make some selections about the kinds of
desirable characteristics it should have in order to change
itself so that it can be happy. This goes back the question
of the ‘transhumanist values’: those values have obviously been selected on a partisan basis, they relate very
strongly to a certain political position.
NB: I wouldn’t say so much a certain political position,
in the sense that there are people with all different politics
in transhumanism, from social democrats to libertarians.
But it’s certainly true that the transhumanist perspective
is one distinctive take, a specially-valued take on some of
these things, and not everybody does or should agree
with that. But in terms of humanity choosing, that’s one
scenario where you would have humanity as a big entity
making co-ordinated decisions. One might question the
desirability of that, but even more perhaps the probability of that, in the sense that right now there are lots of
groups of humanity that are making their own decisions.
C: The conception of humanity acting as an agent.
NB: Yes, it’s very problematic, I agree with that. If you
think like that, you will miss out certain, maybe quite
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likely, possibilities: there are these effects that arise when
you have competition between groups; we might do
something that we all know collectively might lead to
disaster, just because it’s difficult to co-ordinate. Arms
races are a good example of this, they’re good for nobody
but they’re hard to avoid. So you might get a different set
of possibilities if you had a world government or some
kind of very powerful UN in the future that could make
co-ordinated choices than if you have a situation where
you have multiple nodes of power that act independently.
*
C: Some of the scenarios you’re talking about are the
stuff of science fiction movies. To what extent are you
yourself engaged in a sort of science fiction? Is it a
matter of our having to tell ourselves these stories which
extrapolate from the state of technology in our society,
the things we know we can do, in order to come to terms
with the massive risks that are the necessary corollary of
every new technology?
NB: I don’t know, I’m not a great science fiction reader,
or watcher, I’m afraid. Clearly a lot of people find
inspiration in science fiction. I think it’s both good and
bad. It’s good in that it might make people aware of
possibilities or of future worlds that they hadn’t thought
about, and it can serve to stimulate people’s imagination.
On the other hand I think it might distort people’s
intuitions, because there is a constraint that all science
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fiction you read or watch has to satisfy – it has to tell an
interesting story, so you often have heroes rescuing the
world, you have dynamic struggles with human-like characters playing a key part, and determining the outcome.
But that’s not necessarily the most likely way for things
to happen. If you’re going to constantly read these, it
probably shapes your expectations. If it’s all filtered
through what I call the ‘good story bias’, then that can
actually warp your thinking.
C: Writers like Greg Egan have dealt with biotechnology,
AI, and so on, in a way which certainly doesn’t conform
to that stereotype.
NB: Yes, I think he’s certainly one of the exceptions, I
should read more of his work.
C: Science fiction does indeed stimulate people to think
about the world and about situations they hadn’t thought
about; are you not providing a similar service to people
who possibly don’t read science fiction, providing the
same necessary service of widening the scope of people’s
everyday thinking?
NB: Maybe, I don’t see that as the main thing that I’m
aiming for, but maybe that’s a good effect. Widening
people’s thinking, to me, has the feeling of a cliché.
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I don’t know if there is a systematic bias towards
narrowness in people’s thinking. There might be in
specific cases. But maybe some people’s thinking ought
to be more focused rather than widened.
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COLLAPSE I
On the Mathematics of Intensity:
A Logic of Self-Belonging
Thomas Duzer
“One must be a rigorous logician or grammarian, but at the same time
full of fantasy and music” (Hesse)
*
“I am speaking, obviously, of philosophy and philosophers, of those who
force themselves to see, to know, to prove as many things as possible in the
course of their existence” (Chestov)
*
“Intense codings are connected and sometimes severe mathematics are
required since, in effect, the price of victory over the shadow cast by a
slumbering but perpetual captain—against the depredations of this
hippocampus, then—reveals itself to be eternity and the absence of chains.
Nevertheless, for the benefit, and for good reason, of those unhappy souls
who remain trepidatious before it, let us recall that where self-belonging—
which is substantial freedom— is concerned, where the universe is concerned, no-one is bound.” (Anaximandrake)
‘To be in being,’ is to be being according to the mode
corresponding to one’s proper singular essence. The
term ‘being’ is used out of convenience, for its neutrality,
thus indicating no prejudice as to possible different names
of being. We will see that the baptism of being is the
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ISBN 978-0-9553087-0-4
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supreme activity of ontology, which for that reason
cannot be ‘first philosophy’, despite the position of
Aristotle and of the scholasticism that followed him. This
position however differs from that of Levinas.
Necessity is the only modality of being; thus freedom
is the effectuation of its own necessity. As Spinoza says,
‘freedom is not opposed to necessity but to constraint.’
This effectuation corresponds to what is commonly
known as grace, or the state of grace. Each instant is the
good since it can only be justified by itself. This process
is in fact creation of being – that is to say, becoming. It is
a question of the expression of the Mystical, of that
mental zone (qua primary determination) that is not
linguistic, but purely intensive. This expression or logic
of the Mystical is called the mathematics of intensity. The
unconscious is in fact a true physical topography (specific to each singularity) each of whose dimensions corresponds to a zone of intensity. It has a thousand plateaus.
The integral production of the mathematics of
intensities has a preliminary requirement: the conquest of
the unconscious by any means available. In fact the
unconscious is first of all in chains, in a state of servitude.
It is under the empire and the influence of diverse
instances of the Socius as the power which, through the
mediation of the law, separates the individual from what
he can do, that is to say from his power, hence from his
singular essence. These instances are grouped under the
unifying aegis of the Superego which systematizes them
in the form of the Ego. It is a question of the imposition
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of a coding filter on the singular profile of the unconscious, a filter which only allows certain passages
between certain points of its territory. The rest remains
in the shadows, and so we are etymologically correct to
name it the unconscious. The unconscious is spoken, the
unconscious speaks. Ça parle (It/id speaks). Every attack
against a bastion of the coding filter manifests itself
through a specific emotional phenomena: anguish. It is,
moreover, by way of this sign that we can recognize that
it is indeed active. Each bastion that is conquered is a
step towards the annihilation of the filter. Obsession is an
intermediary phenomenon; the bastion is not annihilated
but besieged.
Such a parasitized, colonized unconscious (one which
thus regards itself as a theatre) presents two centres of
gravity. One is virtual; it is the centre of gravity
corresponding to the essence of the singularity. It is
virtualized by the second centre of gravity imposed in a
continuous fashion by that structural effect of transcendence which is the Socius. This centre is called Superego
if it is not identified through consciousness. In this case,
the unconscious is master, the master is unconscious.
Otherwise, it is known as the internal tyrant or monster
(composite being) as soon as it is detected. This detection
correlatively compels the localization of the singularity’s
own centre of gravity and the quasi-complete distinction
between the Self and the non-Self. The Lacanian concept of the Other is thus dethroned in favour of a concept
of negation which is internal to mathematical logic (and
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not to a Hegelian dialectics). This is a corollary of the
destitution of the Ego in favour of the Self. At this stage,
the former remains no less virtual. Its actualisation passes by way of the precise localisation of the internal tyrant
or monster and the destruction of its citadel (which is not
the strongest but the most strategically well-situated) permitting the metamorphosis of the Ego into the individual.
The centre of gravity of the individual is the ego which is
the informational point of compression of preceding
configurations of the unconscious; it is therefore in
perpetual becoming and in incessant movement in the
unconscious. The ego, the unconscious and their diverse
configurations form the Self.
It is a question of carefully distinguishing the ego, the
Ego and the Self. Let us note that the speed of thought
is all the faster in so far as there is no Ego, that is to say
no Other, but the operator ego which generates the
instantaneous synthesis of apperception and produces
reflexivity as a result. Meanwhile, it is necessary that it
produce a sufficient unconscious to have a memory available, that is to say in a form that is compressed and not
deployed, as of a ‘picture’. The ego as the engine brake.
Yes, good narcissism, qua success of the mirror stage, that
is to say, non-psychoanalytically, refusal of identification
and thus comprehension of the reality of the illusionism
of the image, is the constitution of an evolutionary story
whose global coherence is suspended at the instant, open
to the hard purity of the real event. The ego therefore, as
necessary and variable fiction: matter and chronology
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and egoic form, which are separable instances, each
capable of subsisting for itself. But the Self cannot be
reduced to it. Bad narcissism proves therefore to be a
belief of a transitive type: the cult of the fiction, of the
fixing, of the Ego. Thus, the narcissistic phases must surpass the Ego, that is to say in passing by way of the
intellect which is thought in act, that is to say, thought
without images. In fact, the Self is not constituted on the
basis of a Lacanian Other, that is, by an avatar of the
dead God, but by the symbolic exchange with another
rational being. It is only by way of the symbolic that
experiences can be connected and integrated with each
other, even whilst they remain perfectly heterogeneous in
reality. Yes, first and foremost, the Self is real coherence
thanks to the symbolic – contrary to the Ego which is an
attempt at the imaginary conjuration of the circulating
“empty case” which drives the signifiers.
Primary narcissism – that of the Ego – is mysticism of
the All, that which it is not abusive to designate as the
desire for fusion with the mother. The Other here is but
the imaginary disguise of the Same, that is to say, the
obsessive negation of every other real in an auto-parturition with no issue. The Ego is thus only a local illusion,
a reflection of the absolute in a lateral pool. The Thing:
stasis and abolition. It is, moreover, this rejection of the
symbolic which posits the psychotic as such. Lapsus
mentis : psychosis conflates objective genitive and subjective genitive. On the contrary, and far from the totalitarian affirmation of the Ego, the auto-referentiality
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conquered thanks to the symbolic instance proves to be a
remedy against infinite regression. Because the only
movement in which one understands all is that of death.
The movement of nous, that which produces the opening, which constitutes the Open, when it ceases, gives
way to a bounded, deployed, unfolded totality. It is
indeed the mystery of All which must be clarified – in
other words, that which makes a totality a totality. And
it is in psychogenesis that it is appropriate to seek the
solution: first alterity encountered, neoteny of the human
animal. This is moreover why the language of the human
being qua social, since language-using, logically produces
the Ego. Let us note in this regard that the substantial
Ego is nothing but the perturbation of the Self’s relation
to Self (that is to say of reflexivity), and this evidently for
the purposes of subjugation. It is therefore not so much
that the All is an unattainable mirage, an illusion specific
to the suspension of senses. The problem is rather that
the All is in fact homogeneous with a repose modeled on
death, a death which is the truth of all transcendence.
However, madness will never be mad enough to render
the ontic and the ontological indistinct. Transcendence
versus chaos? No. Let us be clear: the wisdom adequate
to chaos, that is to say to a truly inhuman madness, is
discernment.
We should in fact realize that the point of being proves
to be junction of the real, the imaginary and the symbolic. No precedence of any one of these dimensions over
any other. Otherwise, transcendence qua asymmetric cut
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arrives as a hallucination provoking real effects. This
caesura, heterogeneous to the symbolic, is the arrival of
the void as such, that is to say, of non-being as phantasm.
Because it is indeed castration which is the origin of the
imaginary’s real takeover of power. In effect, the sovereign image is the negation of the puncta coacta which
constitute it, and thus the affirmation, by denial, of the
All. Denial of castration? In fact, it is the delirium of the
All itself which testifies to castration. The real of
castration is delirium, not as process but as legitimation
of the takeover of power of one of the dimensions of
being. Yes, the symbolic caesura differs from castration
because it is the condition of connection and disconnection, that is, of the formation of alls. In fact, there is no
all which is not fragmentary, and thus relative. An all is
not all unless there exist possible components which it
does not totalise. Yes, the pure multiple is untotalisable.
Were this not so, the absolute, formalised, would annul
itself infinitely. This is in fact the mistake of all religion:
faith as strict inverse of doubt, correlate of the exclusion
of their subsumption.
In short: the All is the index of transcendence, that is
of coalescence between the real and the imaginary, thus
of the refusal of the symbolic, that is to say of the other,
and thus, as we will see, of the Self. Differences, feared,
are aligned under the category of opposition. It is
remarkable that it is the rejection of the other that gives
birth to the Other, that is to say to the very possibility of
alienation. One sees then that it is a sophistic tropism
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that makes possible voluntary servitude, then denied,
hence perfectly interiorised. But no, all is not in language, and the “All” is only in language. Yes, since it
reflects nothing, being appears as chaotic, void, or eternal
multiplicity disseminating itself sempiternally. Thus,
presence qua eternal is the assumption of time as event of
pure difference, this paradoxical substance. But to attain
the All – the end of time – in excluding the other, that is
to say the Self, amounts to joining with the Other – that
is to say, with nothing.
But, let us be specific. We have said, necessity is that
of the singular essence, that is to say real essence. There
is no global necessity unless there is a One-All, therefore
an outside of the universe, that is to say if 1=2. This initial paradox tries to conjure up freedom, the unpredictability of the living, and notably as to the issue of the
struggle until death. For a judgment, second illusion,
which is triplopia, it would have to be that 1=3. In this
case, the delirium goes even further. Desire in an
impasse, construction of sin ad hoc, the big Other. Ne uter.
In achieving the paternity of the One, one would lose in
it the immanent distinction in the indiscernible, this
mixture escaping the dialectic of the void and the plenum.
This is to say that one would lose the act of the intellect,
the movement of nous. One would thereby lose singular
individuation in favour of a regular cloning. And we can
be sure that this would indeed be a shame, so long, of
course, as one allows this anti-wager, which is an authentic risk, hence one which no law probabilises in advance.
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Thus one and one only entity or instance, the Self, is
“conscious” of “its” contents via the intermediary of
reflection between individuals. The obstacle, which is
considerable, is that one finds some who take themselves
for Egos. But the Ego is a false unity, a false coherence
which, in order to subsist qua illusion, must mortgage the
future in the form of a quasi-cancerous growth, by
destroying – that is, consuming – possibles. It is in fact
its internal contradiction which forces it to submit in
order to be able to project its contradiction, and make of
the other the imaginary suffering appendix of its fictive
Ego. But to make an Ego, you need at least two bodies.
Even the dialectic of master and slave, that servile caricature, requires two human beings, for it deals with desires
and consciousnesses. Not that of the Ego. In fact, it is a
question there only of a relation between an object and its
possessor. This is patently obvious, for example, in
fetishism. Recall Federer’s thesis, reprised by Bergler
(the typical neurotic): the slave dedicates himself to the
master in order to escape the panic to which his freedom
exposes him. But by the same token, this master is only
his instrument. Ellipse.
Consequently – and this is no solipsism – there is a
Self, diversely modified into spirits and bodies which are
images, that is to say, figures of the agency of the imaginary. The Self qua intensive is nevertheless traversed by
the symbolic dimension of language, that is to say,
perceived as an Other which affects the Ego.
Nevertheless the Self is one and “unreflectable”. One
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cannot “see” it, since it is not an Ego, an image, but, on
the contrary, that place from which one sees. Here we are
far from the form of the Ego. The Self is that “planomenon” (Deleuze), that paradoxical substance where the
polyphony of intensities circulates in pure immanence;
intensive grammar itself in perpetual differance since it is
connection, syntax of differences. “To be in being”: here
one is indeed involved in a logic of self-belonging.
But to believe in the Ego or in the Unique (Stirner) is
much more than not to believe in anything, it is to believe
in nothing. And as Weininger says: “cowardice is a way
of taking nothingness to be something”. For their part, in
fact, the nihilists preach, whether silently or not, and
along with others, the following: do not expect anything.
But this is to expect nothing. In fact, the nihilist always
expects something but it is nothing that he desires. Even
alerted by Blanchot that the apocalypse disappoints, he
waits, and that is all. He does not want himself living, for
that would be to risk, enjoyment certainly, but above all
suffering: he wants himself dead. Now, this nothing
would not be lack unless there was the possibility of
something in its place. Certainly, certain halfwits, subtle
rhetoricians that they are, will not hesitate to object that
lack is precisely something. Which is true, indubitably
so. Isn’t it in this way that we can understand the demon
of the malicious Descartes’ Cogito? But let us reassert that
“nothingness has no properties”; and refuse every
“effect”, every mystification, as base. For you will only
find the plenum there where it is not lacking, which is to
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say, there where it is. It could only be lacking where it is
not if it were there. Which is to say that it is logically
contradictory for it to be lacking. In effect, it is not lacking in its place because the place does not belong to it.
Certainly, there is void where one looks for the plenum.
But the question is: in whose interest is it to put your
desire, which lacks nothing, in an impasse? Who could
want to set up contradiction at the very fundament of
ontology? Those who satisfy their desire through your
impotence, those who rule by and on the basis of
anguish. Because, and this is the only axiom of their
“science”, they know full well that anguish is the link to
lack. Here, the secret of tyrannies reveals itself, all the
more blinding for it. Being is no longer the knot of the
real, symbolic, and imaginary, but that which is put in
quotation marks becomes lèse-majesté, and being is nothing
more than that which passes via the mouth. Because, just
as the Stoics knew – and this is only a paradox for the
excarnated of all descriptions – it is the incorporeal which
renders possible the real mixture of bodies. It is the very
condition of possibility for the event. Yes, despite the
egoic myriads, the symbolic is transcendental.
It is therefore clear that to recognise the instance of
the Other is, paradoxically, to misrecognise that of the
symbolic. To posit the Other is to render castration
unsurpassable, transcendent. But the instance of the
Other is only a projection of the imaginary upon the symbolic. This is indeed what leads us to claim that the Ego
only arises from the SuperEgo (qua effect of a certain type
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of Socius). Certainly, “something”, in the symbolic, must
structure the Id; but the SuperEgo petrifies it, i.e.,
produces the Ego. Another organisation of the Socius is
therefore necessary to resolve this problem. In fact,
castration, linked to the imaginary belief in the Other (the
uncastratable), induces the separation between the zones
of highest intensity.
Nihilism is therefore, as Nietzsche emphasised, the
touchstone. Its partisan is on the near side of the
intuition of the distinction between joy and sadness,
between augmentation and diminution of desiring-power.
“What sense? What sense?” chants this ridiculous parishioner, a sort of plebeian relativist Heraclitus, for whom
ascent and descent are equivalent. So, that place “where
we feel the whole nervous system burning like an incandescent lamp” (Artaud) is for the nihilist a place
unknown, or perhaps forgotten – but, at least, lost. It is
true, as Gabriel Marcel says, that this type of philosophy,
which, fundamentally, opposes to being an end not to
receive, is coherent, rational, “philosophical” in sum.
Likewise, Sartre concedes that the spirit can deny itself
and even possesses an infinite power of doing so. And
already, the cavalier Descartes, that Pascal from before
the Fall, finds the reason for that strange mania in the
disproportion between finite understanding and infinite
will. Point noted.
Thus, whether it is a case of the All, of God or of the
Ego, every philosophy of the transcendent One is indeed
in this sense a nihilistic and ancillary philosophy. Yes, in
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order to be able to reciprocate (whether explicitly or not)
being to the One, it cannot but differentiate them, but as
derived. But let us posit a precondition: Spinoza, according to the vulgate, is a philosopher of the One-All. We will
nevertheless retort that his system leans towards
Epicurism and Atomism. Because it is indeed rather
Neo-Platonism which is the philosophy of the One-All; it
is the doctrine of emanation which properly characterises
the latter. Good and Evil are its autophagous principles:
the absolute devours itself in it. Psychosis and transcendence. In complete contrast, Spinozism is pure immanence. The good and the bad are only ever said relatively here, but are nevertheless absolutely distinct. Here is
the difference: joy has no need of sadness. Joy as
augmentation of power is sufficient unto itself, whilst
sadness is servile, bound to lack, that is to say, to anguish.
Thus, it is with absolute sovereignty that Spinoza withdraws God from the grip of the theologians and their
creed, and makes of it the pure multiple which is only
multiple of itself. His work, and in particular the
polemical scolia of the Ethics, demonstrates this clearly.
Not cannibalistic bipolarity, but one axis only: aristocratic joy as augmentation of force. Chronos emasculated. What is power [pouvoir], then? It is merely the lowest degree of force [puissance]; it is force’s separation from
itself, self-separation and projection of this separation.
Thus: the One is pure alienation. Hence it is the instance
which allows the others to link themselves into a whole.
Thus there are ones. Which can also be said: there is
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only one and the other if One occupies the function of the
Other, that is to say the One qua alienated. Now, this
function of the Other, if it is a question of joy, becomes
that of negation. No dialectic. Negation of negation does
not yield identity. It is identity which denies itself in
denying. Being is not becoming, it is becoming which
creates being. What of their conflation? It is the Ego.
The non-Self is the site of transcendence alienated in perpetual negation but, therefore, and consequently, of itself,
that is to say of its Self which is illusion and its sole substance. For the One is indeed a pure generic difference,
but the latter is without relation, and consequently,
without Self. Self-sufficient image. But the image, irremediably, is punctured. It is thus the epiphany of the
perverse process in the pure state: the image identified
with being, the terrible synecdoche. Non-separation,
devourment, death fantasmatically denied, swallowed.
Immortality is promised to the perverse only in exchange
for their absolute alienation. But who makes it
necessary? For in fact joy is entirely other, eternity, hic et
nunc, that is to say the pure Self which arises from the
relation of pure differences. Yes, the Self is the indiscernability of the one and the multiple.
Thus the mathematics of intensities is firstly a
geography, a cartographical relief map of the
unconscious. But secondly, it proves to be a pure
pragmatics. In fact, thanks to nous, to the intellect, an
automatic computation is effectuated which mobilises, in
the Self, the full gamut of intensities, and extracts those
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which are adequate to the solicitations of the non-Self,
whether it is a matter of connections or disconnections,
that is to say of compositions or of decompositions of
relations, of links. Thus the domain of “us” can arise.
From now on, the unconscious is productive, that is,
capable of continuous creation. At this stage, it is
possible, by way of a linguisterie, to replace the term
‘unconscious’ by the term ‘in-conscious’. In fact, the
liberated unconscious is not conscious in the sense of
being re-presented reflexively in consciousness but in the
sense of being intensively presented in it. Then, and only
then, is it a question, for the philosophies of the multiple
and of the concept, of effectuating logical conjunctions
(connections and disconnections), that is to say, within
the immanence thereby conquered, of creating significant
liaisons in systems of relations which are external to
their terms. This is how the mathematics of intensities,
which is the logic of self-belonging, and hence etho-ontologic, can become a real pragmatics, that is to say, a
constructivist philosophy between free individuals.
Various tactical and strategic mappings, scattered and
principally offensive, disseminated in universal literature,
await an Esperanto. The science of these lacings and
interlacings has yet to be elaborated. This mathematics
of intensities is the real ethics of the event, the practical
science of manners of being, that is to say, an empiricism
at once superior (Schelling) and radical (William James),
and therefore transcendental (Deleuze). It is the
knowledge of desire since “desire is the very essence of
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man” (Spinoza). If we connect this theorem with
another Spinozist theorem, namely “essence is force”, we
immediately deduce that the force of man is desire. Being
being produced by becoming, ontology, the science of
being qua being, is logically secondary. In this way, ethoonto-logic becomes capable of supplanting the ancient
onto-theo-logy, since the coherence (of Logos in act, the
actualisation of Logos), which differs from the non-contradiction of classical logic, is no longer that of the Ego or
of God but that of the force of desire. Heraclitus, the
Stoics, Montaigne, Spinoza, Hume, Nietzsche, William
James and Deleuze (and perhaps even Wittgenstein) were
already closer to etho-onto-logy. But they conceptualised
an onto-etho-logy without naming it as such. To name it
thus proves that each of them actually taught the
inverted double of that which they did, that is to say, an
etho-onto-logic in act. They were individuals but
designated individuality without conceptualising it. This
is why Spinoza’s Ethics is written invertedly (incessant
shuttling, halls of mirrors, figures of light): substance
turns around the modes and not the other way around.
So it is that the name of being is that which, logically, gives
it becoming in immanence. Igitur.
Translated by Robin Mackay and Ray Brassier.
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Crowds
Keith Tilford
p262-3.
Untitled, 2005.Ink on Paper, 121.92cm x 177.8 cm
p264,265.
Details from the above
p266-7.
Untitled, 2005. Ink on Paper, 120.65cm x 163.83cm
p268,269.
Details from the above
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COLLAPSE I
Qabbala 101
Nick Land
I NTRODUCTION
Is qabbalism problematical or mysterious? It seems to
participate amphibiously in both domains, proceeding
according to rigorously constructible procedures – as
attested by the affinity with technicization – yet intrinsically related to an Outsideness through which alone it
could derive programmatic sense.
If there is no source of at least partially coherent signal that is radically alien to the entire economy of conventional human interchange, then qabbalism is nothing but
a frivolous entertainment or a fundamentally futile practical error. Yet unlike any kind of metaphysical assault on
'the noumenal', qabbalism cannot be definitively critiqued on a purely rational or formal basis, as if its mode
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COLLAPSE I
of 'error' was that of logical fallacy. Since qabbalism is a
practical programme, rather than a doctrine of any kind,
its formal errors – mistakes – are mere calculative irregularities, and correcting these is actually a procedural
requirement of (rather than an objection to) its continued
development.
It is the rational dismissal of ‘the’ qabbalistic enterprise that is forced to take a metaphysical stance: ruling
out on grounds of supposed principle what is in fact no
more than a guiding ‘empirical’ hypothesis (that signal
from ‘outside the system’ is detectable by numerical
analysis of codes circulating within the system).
Epistemologically speaking, qabbalistic programmes
have a status strictly equivalent to that of experimental
particle physics, or other natural-scientific research
programmes, even if their guiding hypotheses might
seem decidedly less plausible than those dominant within
mainstream scientific institutions.
Lovecraft understood the epistemological affinity
between natural science and programmatic (as opposed
to doctrinal) occultism, since both venture into regions
once declared mysterious, following procedures of a
rigorously calculative-problematical type. It is the alliance
between purely speculative metaphysics and common
sense that betrays such affairs of pure reason to futility,
since they lack the calculative traction to revise their own
conventional notions on the basis of their encounters.
Practices – however implausible their guiding
motivations – can know nothing of absolute mystery or
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metaphysical transcendence because their realm of
certainty is procedural-problematic and uncontroversial,
whereas their reserve of knowledge is empirical,
refutable, repeatable, revisable, nonmystical and accumulable.
There may be no ‘empirical’, procedurally approachable mysteries – or mysterious problems – of the kind
qabbalism guides itself towards. If so, it will approach this
fact in its own way – empirically, probabilistically,
impressionistically, without any logical, transcendental or
philosophical meta-discourse ever having been positioned
to put it in its place.
I. P OPULAR N UMERICS
Traditional gematria (whether Hebrew, Greek, Farsi
or Arabic1) have distinctive typical features: (1) They
substitute letters for numerical values, overcoding numerals where they exist. (2) They code for discontinuous
numerical values, typically 1-10, then 20, 30 ... chunked
in decimally significant magnitudes.
The ocean in which qabbalism swims is not
mathematics, but popular numerical culture. From a
mathematical perspective it remains undeveloped, even
ineducable, since it cannot advance beyond the Natural
number line even to the level of the Rationals, let alone
to the ‘higher’ numbers or set-theoretical post-numerical
spaces. Where counting ceases, qabbalism becomes
1. See Incognitum, ‘Introduction to ABJAD’, in the present volume.
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impracticable.
Socially, qabbala makes an implicit decision against
specialization, in order to remain virtually coincidental
with the entire economy of digitizable signs. It is essentially ‘democratic’ (in the most inclusive sense of this word),
even when apparently lost in its own trappings of
hermeticism. It is bound to the ‘blind’ undirected contingencies of pre-reflective mass-social phenomena, with all
the inarticulate provocation this entails in respect to
professional intellectuals. Wherever exact semiotic
exchange occurs, a latent qabbalism lurks (even within
the enclaves of intellectual professionalism themselves).
Deleuze & Guattari’s ‘Nomad War Machine’, within
which number is socially subjectivized, captures crucial
aspects of this qabbalistic fatality.
Historically, qabbala arises through epic accident, as a
side-product of the transition between distinct modes of
decimal notation. Its historical presupposition is the shift
from alphabetical numerals (of the Hebrew or Greek
type) to modular notation, with its resulting unlocalizable
(and theoretically indeterminable) confusion. This
transition provided the opportunity for a systematic
calculative ‘error’ – the mistaken application of
elementary techniques appropriate to alphabetical
numerals – simple addition of notated values – to the new
modular signs. This mistake automatically resulted in
digital reduction, by accident, and thus as a (theoretically
scandalous) gift of fate. Arising historically during the
European Renaissance - when zero, place value and
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technocapitalism finally breached the ramparts of
Western
monotheism – qabbalism (born in a semiotic
glitch and thus lacking the authority of tradition or even
purpose) was compelled to hyperstitionally generate an
extreme antiquity for itself, in a process that is still
ongoing.
Technically, qabbala is inextricable from digital
processing. Emerging from calculative practicality within
the context of blind mass-cultural metamorphosis, it
antedates it own theoretical legitimation, making sense of
itself only derivatively, sporadically and contentiously.
Its situation is analogous – and perhaps more than
analogous – to that of a spontaneous artificial intelligence,
achieving partial lucidity only as a consequence of tidal
pragmatic trends that ensure an integral default of selfmastery. Practical systematization of technique precedes
any conceivable theoretical motivation. Dialectical
interrogation of qabbalism at the level of explicit
motivation thus proves superficial and inconsequential,
essentially misrecognizing the nature of the beast. (It is
equally misleading to ask: What is a computer really
for?)
Politically, qabbalism repels ideology. As a selfregenerating mass-cultural glitch, it mimics the senseless
exuberance of virus, profoundly indifferent to all partisan
considerations. Indifferent even to the corroded solemnity of nihilism, it sustains no deliberated agendas. It
stubbornly adheres to a single absurd criterion, its
intrinsic ‘condition of existence’ – continual unconscious
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promotion of numerical decimalism. Qabbala destines
each and every ‘strategic appropriation’ to self-parody
and derision, beginning with the agenda of theocratic
restoration that attended its (ludicrously robed) baptismal
rites. Even God was unable to make sense of it. It has no
party, only popularity.
II. P RIMITIVE N UMERIZATION
Among the primary test-beds for qabbalistic analysis
are the numerolexic systems inherited from cultures overcoded by the modern Oecumenic alphabet. These
include the Hebrew and Greek alphabets (with their
Neoroman letter names and mathematico-notational
functions) and the Roman numbers (inherited as
Neoroman letters and still numerically active in various
domains). In this respect, the absence of names for
Neoroman letters are an index of their pseudo-transcendence – as ‘unnameable’ – within the present Oecumenic
order.
A discontinuity is marked in the alphanumeric series
(0—Z) by the fact that the numerals composing the first
ten figures in this series do have names, grouping them
with the letters of previous alphabetical numbering
systems from a certain qabbalistic perspective. This might
be taken as the residual indication of an ‘alien quality’ still
characterizing the numerals in relation to the Oecumenic
cultural order they now indisputably occupy, a legacy of
the cultural trauma attending their introduction.
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The qabbalistic provocation posed by the English
number names is conceptually comparable to that of any
other numerolexic system, while surpassing any other in
the intimacy of its challenge. If the numerals have names,
shouldn’t the qabbalistic processing of them as words
yield – at the least – compelling suggestions of nonrandom signal? If the standard numeral names emit nothing
but noise when qabbalistically transcoded, the attempt to
establish relatively persuasive criteria for the evaluation
of qabbalistic results suffers an obvious and immense
reverse.
What, then, would count as a minimally controversial
first step in such an examination?
Surely the most basic of all qabbalistic (or subqabbalistic?) procedures is simple letter counting – Primitive
Numerization (PN). As a reversion to sheer ‘tallying’ PN
has a resonance with the most archaic traces of numerical
practice, such as simple strokes carved into mammoth
bones and suchlike palaeo-ethnographic materials. If
anyone was to bother systematizing PN procedure for the
purpose of mechanization or simply for conceptual
larity, it would be most efficiently done by transcoding
(‘ciphering’) each letter or notational element as ‘1’ and
then processing the result numerically.
PN’s extremely tenuous relation to issues of modulusnotation ensures that it can only ever be a highly dubious
tool when intricate qabbalistic calculation is required. Yet
this utter crudity also makes it invaluable as a test case,
since it minimizes axiomatic arbitrariness and precludes
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any plausible possibility of symbolic conjuration (‘sleight
of hand’) while fully sharing the qabbalistic ‘deficiency’ of
sufficient anthroposocial or communicative motivation.
Common reason – sanity – insists upon noise as the only
PN output consistent with the general intelligibility of
signs (a pre-judgement applying rigorously to all qabbalistic procedures).
No message should inhere in the length of a word,
excepting only the broad pragmatic trend to the shortening of commonly used terms. It is immediately obvious
why this exception has no pertinence to the case in
question here, unless stretched to a point (for instance,
expecting the smaller numerals to exhibit the greatest
lexical attrition) where it is straightforwardly contradicted by the actuality of the phenomenon.
So, proceeding to the ‘analysis’ – PN of the English
numeral names: ZERO=4, ONE=3, TWO=3, THREE=5,
FOUR=4, FIVE=4, SIX=3, SEVEN=5, EIGHT=5, NINE=4. Is
there a pattern here? Several levels of apparent noise,
noise, and pseudo-pattern can be expected to entangle
themselves in this result, depending on the subsequent
analytical procedures employed.
To restrict this discussion to the most evident secondary result, not only is there a demonstrable pattern, but
this pattern complies with the single defining feature of
the Numogram2 – the five Syzygies emerging from 9-sum
2 . On the Numogram, see Abstract Culture 5:Hyperstition (1999).
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twinning of the decimal numerals3: 5:4, 6:3, 7:2, 8:1, 9:0.
In the shape most likely to impress common reason
(entirely independent of numogrammatic commitments)
this demonstration takes the form: ZERO + NINE = ONE +
EIGHT = TWO + SEVEN = THREE + SIX = FOUR + FIVE –
revealing perfect numerolexic-arithmetical, PN-‘qabbalistic’ consistency.
The approximate probability of this pattern emerging
‘by chance’ is 1/243, if it is assumed that each decimal
digit (0-9) is equiprobably allotted an English name of
three, four, or five letter length, with 8-sum zygosys as the
principle of synthesis. 7-sum or 9-sum zygosys are
inconsistent with any five or three letter number-names
respectively, and thus complicate probabilistic analysis
beyond the scope of this demonstration (although if
everything is conceded to the most elaborate conceivable
objections of common reason, the probability of this
phenomenon representing an accident of noise remains
comfortably below 1/100).
Partisans of common reason can take some comfort
from the octozygonic disturbance of the (novazygonic)
Numogrammatic reference. How did nine become eight
(or vice versa)? Lemurophiliac numogrammaticists are
likely to counter such queries with elementary qabbala
(since digital cumulation and reduction bridges the
3 . PN confirmation of the Numogrammatic Novazygons (9-Twins).
ONE + EIGHT = NINE + ZERO. (PN 3 + 5 = (4 + 4 =) 8)
TWO + SEVEN = NINE + ZERO. (PN 3 + 5 = (4 + 4 =) 8)
THREE + SIX = NINE + ZERO. (PN 5 + 3 = (4 + 4 =) 8)
FOUR + FIVE = NINE + ZERO. (PN 4 + 4 = (4 + 4 =) 8)
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COLLAPSE I
‘lesser abyss’ in two steps, 8 = 36 = 9, as diagrammed by
the 8th Gate connecting Zn-8 to Zn-9).
III. AGAINST N UMEROLOGY
Consider first an extraordinarily direct numerological
manifesto:
When the qualitative aspects are included in our conception of numbers, they become more than simple
quantities 1, 2, 3, 4; they acquire an archetypal
character as Unity, Opposition, Conjunction,
Completion. They are then analogous to more familiar
[Jungian] archetypes...
It is hard to imagine a more ‘archetypal’ expression of
numerological ambition than this. Yet rather than meeting this claim with docile compliance, the qabbalist is
compelled to raise a number of awkward questions:
(1) How can a numerological coding that proceeds in
this fashion avoid entrapping itself among the very
smallest of Naturals at the toe-damping edge of the
number line? If ‘4’ symbolizes the archetype
‘Completion,’ what to make of 127, 709, 1023, or similar
small Naturals? Do they also have analogues among the
intelligible archetypes? How would one ‘qualitize’
(2127)-1, or a larger number (of which there are a very
considerable number)?
(2) Is an ‘archetype’ more basic than a number in its
unsymbolized state? Does ‘qualitizing’ a number reveal a
more elementary truth, a germ the number itself
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conceals, or does it merely re-package the number for
convenient anthropomorphic consumption, gift-wrapping
the intolerable inhumanity of alogical numerical
difference and connectivity?
(3) Why should a number be considered ‘quantitative’ in its Natural state? Is it not that the imposition of a
quantity/quality categorization upon the number requires
a logical or philosophical overcoding, a projection of
intelligibility alien to the number itself? Quantity is the
decadence of number (while quality is its perversion), so
– since arithmetic provides no basis for a reduction of the
numerical to the quantitative – what is the supposed
source of this (numeric-quantitative) identification (other
than a disabling preliminary innumeracy)?
(4) If ‘1’ numerologically evokes ‘Unity,’ why should
UNITY not qabbalistically ‘evoke’ 134 (=8, its
Numogrammatic twin4) with equal pertinence? Can any
expressible ‘archetype’ avoid re-dissolution into the
unfamiliarity of raw number pattern? Numerology
might assimilate ‘2’ to opposition, but OPPOSITION = 238
= 13 = 4 (twice 2, and the Numogrammatic twin of (‘4’
= COMPLETION = 212 =) 5), while even if numerological
‘3’ as CONJUNCTION = 237 = 12 = 3 finds itself qabbalisitically confirmed (at the extremity of its decimaliza4 . Employing August Barrow’s ‘Anglossic Qabbala’, the basic tool of which is the
Alphanumeric Gematria. This numerization of the Neo-Roman alphabet , continuing the procedure now familiar from Hexadecimal, is a continuous nonredundant
system, supplementing the numerals 0-9 with numerized letters from A (=10) to Z
(=35), treating the 0-Z alphanumeric sequence as a numeral succession, corresponding to the numerals of a modulus 36 notation.
Thus UNITY = 30+23+18+29+34 = 134. 1+3+4 = 8.
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COLLAPSE I
tion), this is not, perhaps, in an altogether
comfortable mode.
Numerology may be fascinated by numbers, but its
basic orientation is profoundly antinumerical. It seeks
essentially to redeem number, through symbolic absolution into a ‘higher’ significance. As if the concept of
‘opposition’ represented an elevation above the (‘mere’)
number two, rather than a restriction, subjectivization,
logicization and generalized perversion, directed to
anthropomorphic
use-value
and
psychological
satisfaction. Archetypes are sad limitations of the species,
while numbers are an eternal hypercosmic delight.
Nevertheless, qabbalism is right up against numerology, insofar as it arises ‘here,’ within a specific biological
and logocratic environment. The errors of numerology
are only the common failures of logic and philosophy,
human vanities, crudified in the interest of mass dissemination, but essentially uncorrupted. The numeric-critique
(or transcendental arithmetic) of a Gödel (or Turing, or
Chaitin (or Badiou?(??(???)))) can be rigorously
transferred to this controversy, demonstrating – within
each particular milieu – that overcodings of numerical
relation by intelligible forms – ‘archetypes’ or ‘logics’ –
are unsustainable reductions, reefed on the unsurpassable
semiotic potency of number. Gödel has shown that there
is always a number, in fact an infinitude of (natural)
numbers, that simulate, parody, logically dialectize, paradoxically dismantle, archetypally hypervert, and in whatever way necessary subvert each and every overcoding of
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arithmetic. Number cannot be superseded. There is no
possibility of an authoritative ‘philosophy of arithmetic’
or numerological gnosis.
Qabbala assumes that semiotics is ‘always already’
cryptography, that the cryptographic sphere is
undelimitable. It proceeds on the assumption that there
cannot be an original (unproblematic) coding, providing
the basis for any solid definition or archetypal symbol,
since the terms required for such a coding are incapable
of attaining the pure ‘arbitrariness’ that would ensure the
absence of prior cryptographic investment. There is not –
and can never be – any ‘plain text,’ except as a naïve
political assumption about (the relative (non)insidiousness of) coding agencies and the presupposition that
communicative signs accessibly exist that are not already
‘in code.’ Since everything is coded, or (at least) potentially coded, nothing is (definitively) symbolic.
Qabbalistic cryptocultures – even those yet to come –
ensure that number cannot be discussed or situated
without subliminal or (more typically) wholly unconscious participation in numerical practices. Logos,
including that of numerology, is also always something
other than itself, and in fact very many things.
Qabbalism thus operates as an inverse or complementary Gödelian double-coding. Where Gödel demonstrated that the number line is infested by virtual discursive
systems of undelimitable topicality and complexity,
pre-emptively dismantling the prospects of any
conceivable supranumerical metadiscourse, qabbala
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demonstrates that discourses are themselves intrinsically
redoubled (and further multiplied) by coincidental
numerical systems which enter into patterns of connectivity entirely independent of logical regimentation.
The supposed numerical de-activation of the
alphabet, marking semiotic modernity (the era of
specialized numerical signs), has an extremely fragile
foundation, relying as it does upon the discontinuation of
specific cultural procedures (precisely those that
withdraw into ‘occultism’) rather than essential characteristics of signs themselves. The persistent numerical
functionalization of the modern alphabet – with sorting
procedures based on alphabetical ordering as the most
prominent example – provides incontestible evidence (if
any was required) that the semiotic substructure of all
Oecumenic communications remains stubbornly
amphibious between logos and nomos, perpetually
agitated by numerical temptations and uncircumscribed
polyprocesses.
At the discursive level, any ‘rigorization of qabbala’
can only be a floating city, with each and every definition,
argument and manifesto continually calving off into
unmasterable numerical currents and alogical resonances.
How could qabbala be counterposed to a code, to
meaning and reason, when CODE (= 63) finds duplicitous
harmonics in MEANING = REASON = 126? If qabbala positions itself discursively AGAINST NUMEROLOGY (= 369),
the echoes of its novanomic signature perpetuate
themselves even through such unlikely terms as
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Land – Qabbala 101
SIGNIFICANCE (= 207) and SIGNIFICATION (= 252).
Pronouncements that begin as projected logical discriminations revert to variations on triplicity and the number
nine, performing a base qabbalistic subversion of
philosophical legislation and its authority to define (or
delimit connectivity).
No polemic against numerology – whether conducted
in the name of qabbala or of Oecumenic common reason
– will transcend the magmic qabbalistic flux that
multiplies and mutates its sense. Perhaps dreams of
numerological archetypes even sharpen the lust for
semiotic invention, opening new avenues for qabbalistic
incursion. But this at least is certain: Numbers do not
require – and will never find – any kind of logical
redemption. They are an eternal hypercosmic delight.
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COLLAPSE I
Notes on Contributors
ALAIN BADIOU
Playwright, novelist, political activist and philosopher. Author
most recently of Logique des Mondes. Teaches Philosophy at the
École normale supérieure and the Collège international de
philosophie in Paris.
G REGORY C HAITIN
Researcher at the T.J.Watson Research Center, New York
(http://www.watson.ibm.com/).
http://cs.umaine.edu/~chaitin/
REZA N EGARESTANI
Theorist working in Shiraz, Iran. Author of ‘Cyclonopedia’
(Creation, forthcoming 2007), on the rise of the Middle East.
MATTHEW WATKINS
Honorary research fellow at Exeter University, UK.
Maintains the ‘Number Theory and Physics Archive’ at:
http://www.maths.ex.ac.uk/~mwatkins/
N ICK BOSTROM
Director of the Future of Humanity Institute, Oxford
University, UK (http://www.fhi.ox.ac.uk).
http://www.nickbostrom.com
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ISBN 978-0-9553087-0-4
http://www.urbanomic.com
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COLLAPSE I
THOMAS DUZER
Studied philosophy and mathematical logic in Paris.
Currently working on the non-dialectical passage between
Deleuze and Badiou and, in a wider context, on the
systematization of theories of the multiple.
KEITH TILFORD
Artist currently working in Seattle, US. Exhibits with the
James Harris Gallery (http://www.jamesharrisgallery.com).
http://metastableequilibrium.blogspot.com
N ICK LAND
Theorist and Journalist currently working in Shanghai.
ccru00@hotmail.com
Interview with Matthew Watkins conducted in Exeter by Robin Mackay.
Interview with Nick Bostrom conducted in Oxford by Robin Mackay and
Ray Brassier. Interview with Alain Badiou conducted via email by
Robin Mackay and Ray Brassier.
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