Hello and welcome to the final session of Maths and Ideas with Reza Negrestani. I'm going to pass the mic off right now. Thank you so much, Theodor. Okay, so just in continuation of what we were discussing in response to Philip, yes, this is exactly what Stakemuller's point is, but of course he is not interested in the kind of Marxist historical perspective on the history of science. For him is this idea that the rationality of this theoretical science is historical, but this history is not progressivist in the sense of a convergent progression to an ultimate unification or a revolution.
And in the sense that the rationality of science, which is the rationality of the construction the structuration of scientific theories lies in retrospective power of reason in understanding its conditions of realization. And hence, this retrospective recognition or retrospective understanding of its conditions of a realization is what allows it to realize itself within new theoretical structures.
And this is basically also very, very close to a much more recent study by Danielle Macbeth, which I will talk about today as a kind of a philosophical conclusion. Danielle Macbeth, mostly in her famous book that came recently, Realizing Reason, sees the same thing she doesn't talk about. He doesn't try to find the license for the rationality of reason and the rationality
of the scientific enterprise in some sort of macrological, progressivist, historical dimension, but simply in elaborating reason's power of knowing that moves through the concepts. A retrospective recognition that fuels reason's own realization. the macro logical progressive uh progressivist um sort of paradigm that you're talking about would be a progressivist by necessity paradigm right is that what you're suggesting
uh like in the same same way that uh well i'm hesitant to generalize but What do you mean by necessity? So, I mean, if I'm not a Marxist scholar by any means, but... Yes, you mean that kind of necessity. Yes. Yes. Yes. But the whole idea is that this is really not... So, the line of reasoning, very roughly speaking, goes like this. that's so we can't really talk about his scientific revolutions and that also includes mathematics for example from geometry to algebra is all about the
structure of the theory in which certain statements are being put forward and the The core of this structure, which is the structure of theory, is being constructed by various forms of axiomatizations and are comprised of certain class or type of sentences. And Esther Mueller tries to basically formulate a very coherent account of what these sentences
which he called them Ramsey Sneed sentences. They're a class of formal sentences that are required and necessary for the structuration of any theory within sciences. Now, these are the core of a theory, and without these cores, there is no such a thing as theory-ladenness of science, hence no science at all the thing is that this is the micro logical and within the micro logical you do not need really this kind of a coherent is influential web of how these sentences basically are connected to one another
you have a still it some inference at the micro logical level but it is not this coherent is in French of account that gives you a that gives you a micro logical hermeneutics of what this theory tries to state as a whole so when you come from the level of statements or class of sentences to a level of inferential relations between statements or class of sentences that's when basically you can in fact give that this theory by necessity succeeding a
previous theory that was previous to it by virtue that it's in French a web of of sentences were encompassing the same class of observations that the new theory is trying to encompass. Is this like, I mean, so in Russell's introduction to the philosophy of mathematics, he says there's like the two parts of mathematics. One is towards greater and greater complexity, which is typically what mathematicians are working on and the other is a process of working towards logical simplicity
to describe the mathematical structure that you're working with. that type of logical simplicity is that what you mean by necessity? like when you're saying by necessity we by necessity you see this necessity I think this is Stegmuller's point that this necessity cannot be defined scientifically this necessity is only a historical necessity that you only retrospectively can look at by looking at the most superficial,
superficial not in a negative sense, but being on the top, the micrological level of a theory, of a structure. But not its structuration. What constitutes the structure on scientific theory are really the class of its core statements. And that's why basically Stegmuller tries to say that if you give up on the idea of historical progression, historical progression in the convergent sense, nothing bad happens. That doesn't lead to some sort of firebrand-esque anarchism of science or relativism or irrationalism.
No, because that only happens if you put all of the emphasis on the theory ladenness of science on the micro-logical level, which is from a stegular's perspective, it's inadequate if not completely wrong. I have a couple more thoughts on Hegel, but I'm not sure if we wanna go down that path. We can discuss any question because this is, I mean, we are, I can easily talk about more algebra,
But if you just want, you guys want to ask questions, ask questions, and it would be great if you can just somehow plug the math stuff also into these questions. Can we do the algebra first and come back? Sure. I would prefer that. Sure. We can do that. Yeah, sure. That sounds great. Okay. Because I know that last session when we were talking about, you know, this algebraization of geometry, particularly the Muslim tradition, I kind of skipped on so many steps and I promise
that I will come back and unpack some of those examples. So how about that? I will unpack some of the stuff that we were talking about last session, this idea of some of the algebraic solutions that are studied in the Middle Ages, particularly by Arab tradition, And then later on being taken up by Descartes' analytic geometry in which the emphasis is being put on the algebraic notation that has already subsumed the geometrical diagrams
within it. So let's do that and then I come back with a bit of a commentary on this Danielle Macbeth book on looking at the history of mathematics as an example of reason's own cognitive power that fuel its realization. So okay, let me turn on the...
considered from a from a canonical history of mathematics perspective to be far more advanced than geometrical thinking of course you know let's not talk about this idea that this this position has been fundamentally challenged in contemporary philosophy of mathematics nevertheless the main features of algebra I thinking as we were talking about is one operational symbolism we looked at or as me and Tabitha Pnokoros early research in algebra that's you could encode every dimension of the geometric of the geometrical
diagram in an operational symbolic notation. For example, we started to look at for example, parallelogram or square and then you could encode this simply in form of the squares, algebraic notations that have square components. You could decompose in fact every diagram of that sort into a ultimately a quadratic equation. So this was one, which main feature of algebraic thinking. One, operational symbolism. Two, preoccupation with
mathematical relations rather than with mathematical objects which relations determine the structure constituting the subject matter of modern algebra and this is very much also a point that Descartes makes that algebra is about laws not about particular objects and laws are a relation between objects they are derived from invariances mathematical algebraic invariances between objects that can be encoded that can be might be geometrical they can be might be topological but nevertheless can be encoded algebraically and three
freedom from any ontological questions and commitment and connected with this abstractness rather than intuitiveness again back to the idea that algebra is symbolic de-semantified through and through you do not need the intuitive contents of any mathematical object in order to do proper mathematics all you need is the de-semantified level of relations between objects namely algebraic laws algebraic rules and precisely this de-semantification is a power of algebraic formalism in the sense that once you
have attained the level of de-semantification, then you are capable of applying these algebraic rules or relations to other contexts and other objects otherwise hidden from the perspective of simple you know geometry which deals with the particularities of objects so then as you remember uh we talked about a little bit about al-qar azmi's method and how he
classifies different types of reduction to a quadratic equation now what i'm going to talk about today uh you know in again in continuation of somehow a recap of what we have been talking about in a previous session is the idea of geometrical algebra that is becomes quite popular in Muslim world and later
on being polarized or to different approaches to mathematics. Then during Renaissance, particularly after the Galilean Revolution, the work of Descartes and other works you see that's this geometrical algebra is being purified of its geometric component
and algebra becomes the hegemonic paradigm of mathematical practice but also becomes also in that very sense it becomes a hegemonic paradigm of mathematical practice it also provides philosophers with a new paradigm of rationality fundamentally different from the paradigm of rationality inherited from Aristotle in Middle Ages throughout the Middle Ages so as we have been seeing geometrical algebra is part of algebra as well as part of geometry. Now it follows that one can arrive at geometrical algebra by two different
rows. One can either start in geometrical problems concerning rectangles and squares and solve these problems by means of theorems. One can start with algebraic problems such as the solution of quadratic equations and reformulate them in geometrical language, writing rectangle instead of product. And we saw that, just even looking at, you remember that example from ancient Indian manuscripts that we looked at, but also, Kharaaz means, geometric algebraic solution, is that
You could do either way. You could start from a geometrical diagram and then go to the algebra or looking at just algebraic relations and move toward geometry. Let's start with, go a step back and see how this two-way account of the relation between
algebra and geometry is possible and you because this is already as we talked about or implied was already the case in Indian mathematics but also much more prominently in Greek mathematics and this two-way looking of geometrical algebra as I said moving from algebraic implicit algebraic relations to geometry or moving from explicit geometrical sorry explicit geometrical objects to implicit algebraic notation relations sorry so let's look at again go back to
Greek mathematics Euclid's elements and and look at one example so we have a a rectangle This is proposition one, reads in the translation, Euclid, he's translation of Euclid as follows.
If there be two straight lines and one of them be cut into any number of segments, the rectangle contained by two straight lines is equal to the rectangles contained by the uncut a straight line and each of the segments now geometrically this theorem very briefly it means that every rectangle can be cut into rectangles by lines parallel to one of the sites. So this is obvious that everyone sees it by just looking at the diagram. Within the framework of Greek Euclidean geometry there is no need for such a theorem. Euclide in fact never makes use of it in
his first four books. However if one begins with the algebraic operations of of addition and multiplication multiply the terms of the sum by a and add the results in elementary sorry however if one starts with the algebraic operations of addition and multiplication of numbers and ask how does one multiply a sum by quantity a the answer is multiply the terms of the sum by a and add the results so in elementary arithmetics this a
specific rule is needed all the time if this rule of computation is translated into the language of geometry proposition to the one that we read to one is the result in other words proposition to one in Euclid's elements furnishes a geometrical proof of an algebraic rule of computation and within Euclid's elements proposition two two and two three by two I mean
elements book two are just a special cases of this very proposition once more from the point of view of geometry there is no reason to formulate these trivialities as theorems For example, proposition 2.4 in Euclid's Elements, it says that if a straight line be cut at random, the square on the whole is equal to the square on the segment and twice the rectangle contained by the segments. Now geometrically this means if we take a point z on the diagonal
of a square and draw lines through Z parallel to the sides of the square. Let me make this for you so you can... This is 2-4, proposition 2-4. for this you a very before before moving forward I just saw that question I only read the
first part of it well because you see we are still in the bounds of geometrical algebra we are simply trying to show that how this back and forth and coding moves forward but you see the algebra and this is the car legacy now like geometry where our geometry is being subsumed with algebra is not simply expounding or exposition of geometrical objects, because algebra is not about objects. Algebra is about relations, it's about laws. And precisely because it's about relations, it doesn't need the required particularity
of any geometrical object. You can see that it basically moves from a level of particularity to a level of relations between particularities. And everything can be expressed throughout these relations, and precisely because these relations are de-semantified, namely they have been di-rested from the particular intuitive contents of any geometrical object they can be reapplied to other kinds of objects whether arithmetic so on so forth So, back to this, back to our...
So probably one more time, proposition 2.4 is saying that if a straight line be cut at random the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments which means that if we take a point Z on the diagonal of a square and draw lines through Z parallel to the size of the square this square will be divided into two squares and two rectangles now from a geometrical perspective this is trivial
and euclid hardly ever uses this theorem anywhere else but what makes it non-trivial is from the perspective of algebraic computation it defines it can define different computation So, as we've seen, the same diagram of a square divided into a squares and two rectangles without a diagonal, which is not necessary, also appears in Al-Khar Azmi's three types,
Al-Jabraq al-Muqabaleq. Now here it occurs in its natural place, something that is trivial in Euclidean geometrical treatise is non-trivial in Khwarezmij's algebraic treatise. Khwarezmij needs it, in fact, to justify his method of solving quadratic equations. In this case, we can see why Khara Azmi inserted the diagram in his book, because he is using it precisely because from an algebraic perspective, it gives a non-trivial computational theorem. If we assume that the author of the book Second of Elements also started with an algebraic
tradition to which a rule for a squaring a sum belonged, we can understand why he formulated the theorems 1 to 4, just as he did. But if he came actually from geometry, which actually he did, then we really can't understand what is the purpose of this theorem other than simply a geometrical triviality. Let me get some water, one second, I will be back.
Okay, any questions before I move forward? As Jake is saying, you see, this whole idea of equality in geometry and algebra is a really fundamental concept and actually is one of the main reasons of the difference between the two. Equality in algebra is equality of terms, algebraic terms, which encapsulate mathematical
rules, namely relation between objects, whereas equality in geometry, either from some historical perspective doesn't exist, or if it exists, is simply triviality, congruence, as we saw. This may be a tangent, but there's this bit in the Prolegomena where Kant is talking about the congruence of his hand and a hand in the mirror. And there'd be no way to tell the difference if you were to sort of mathematically illustrate his hand between the real hand and the hand in the mirror.
But clearly they aren't geometrically congruent. Yes, this is a very famous Kent mirror experiment. Kent and Kent seeing the mirror image and Kent's mirror image, namely the thing in itself, sees Kent. And the whole idea is that the co-constitution of the subject and object is more like a horror in which you think that you are seeing your mirror image, but what if it is only your mirror image that sees you? And this whole idea that we are simply the co-extensivity
of subject and object, namely Kant and his mirror image, is an imbalanced mathematical symmetry. It is not congruence. Two different terms are being bunched together under one One new term, Kant, seen himself in the mirror. And this is the whole idea of this imbalance or the dialectic between equality and inequality, which can only be put forward robustly and soundly in algebra, not in geometry.
that demonstrate a reliance that algebra has on geometry or am I missing something you see algebraic geometry or geometrical algebra yes I mean geometrical algebra sorry not algebraic geometry geometrical algebra the one that we were looking at here yes because this is proto algebra what algebra as such like modern algebra which purely works with numbers and hierarchies of numbers, no. There is no reliance on geometry. So this question...
It's pure formalism, namely it's completely devoid of any intuitive concept. Right, right. So but then what is its import into the world? Well, you see, this is the whole idea that reason or mathematical reasoning doesn't need to be built prima facie or initially or originally on some relation, purported relation with the world. It makes its own world and only retroactively sees how the mathematical world can be connected. I think I understand this, but I... And this is really the problem of encoding a structure.
You see, the relation between mathematics and world is not based on the givenness of sensed atom through intuition, but only through how mind can reflect the structures, which are the structures of its own and not the world onto the world and make it intelligible. And that's exactly the problem of mathematical physics. Right. But if I can just say one more thing. the formalism demonstrate its, I don't know, its worth or its truth through application.
Like if you had a formalism that... Only, only, you see, only from the perspective of physics, but not from the perspective of mathematics. Mathematics doesn't give a shit about the real world. That's the whole idea. Mathematics is not about sensible connections or intuitive connection with the world. Mathematics is about the structuration of mathematical meanings and mathematical truth. This is its own universal. I almost want to say that's almost like making mathematics seem like artistry. Yes, it's an artistry that is not whimsical. Let's put it that way. Okay, yes.
Mariam, or yes! Exclamation mark. Okay, continue. I'm sorry. No, it's okay. Okay, so, we talked about this, that, you know, already this two-way roads between algebra and geometry in Euclid's elements and it is exactly in these trivial theorems those really stupid why do fucking we need that theorem why do we need to know it geometrically it's
just a triviality but once we understand it algebraically as far as me did we see that they are non-trivial. They might be geometrically trivial, but in fact they are providing an algebraic method of computation that is non-trivial within the field of algebra. So this back and forth, road between algebra and geometry is already existing in Greek geometry. And Khara's simply accentuates this. So, as I mentioned, another great, even though obscure from a, you know, canonical historical
perspective of mathematics is Tabithep Nekora was a contemporary of Khara's meaning he did numerous contributions to the field of geometrical algebra he was a geometer and astronomer who had for his entire life he had devoted his mathematical studies to Euclid elements and propuse commentary on Euclid.
Now, in an obscure treatise by Tabith, he points out that the solution of the three types of quadratic equations according to the algebra people is equivalent to the application of areas with excess or defect as presented by Euclid. The example that Tabith provides has been usually misinterpreted in the history of mathematics
in the sense to show that there is in fact no... these examples have been shown that algebra is has no connection with geometry. There is no such a thing as these two directional paths within geometry and algebra. So I'm going to give a different kind of reading of these examples provided by to be to in fact show that in the field of geometrical algebra we can truly see how there is a non-trivial relation between from geology to algebra algebra
to geometry from formalism of relations to particularities of intuitions and vice versa as you remember those of you who have who actually looked at the proclos commentary on Euclid did you know that's in his commentary on Euclid proposition 44 Proclus says these things says you the most are ancient and our
discoveries of the muse of the Pythagorean I mean the application of areas they are exceeding and they're falling short maybe excess and defect it was from the Pythagoreans that later geometers took the names which they again transferred to so-called conic lines, whereas those godlike men of old, the Pythagoreans, saw the things signified by these names in the construction in a plain of areas upon a finite straight line. For when you have a straight line set out and lay at the given line exactly alongside the whole of the straight line, then they say that you apply the set area when however
you make the length of the area greater than the straight line itself. It is said to exceed, when you make it less, in which case after the area has been drawn. There is some part of the straight line extending beyond it. It is said to fall short. The idea of defect. Now the same terms application, exceeding and falling short are used in Euclid book 4. Euclid says to a given a straight line to apply a parallelogram equal to a given rectilinear figured and deficient by a parallelogram similar to a given one.
And then on proposition 29 it says to a given a straight line to apply a parallelogram equal to a given rectilinear figure and exceeding by a parallelogram similar to a given one. Now, from a historical perspective, the term parallelogram was introduced into geometry at the time of Eudoxus, long after the time of the Pythagorean. In Book 2 Elements, which is due to the Pythagoreans and where basically Euclid or whoever the
author of elements is, completely cover the Pythagorean dimension of the Greek mathematics, Only squares and rectangles occur, because this is the whole idea of the Pythagoreanism that we talk about, that's like the core philosophical, mystical, mathematical dimension of Greek mathematical practice, precisely because rectangles, everything from a Pythagorean perspective can be decomposed to rectangles and but also from a philosophical mystical
standpoint rectangles and squares reflect perfection the one not probably in Platonic sense but definitely an Eleatic sense so when the Pythagoreans invented their application of areas with defect or access the defect or access was probably required to be just a square not parallel similar to a given one now if Euclid's diagrams are simplified by assuming a square access or defect the resulting diagrams are essentially the same as Euclid's a diagram that we just drew proposition book 2 and proposition 6 book 2 also the
single steps in the proof proofs of proposition 28 book 4 and proposition 29 before are just generalization of the single steps in the proofs of proposition 5 book 2 and proposition 6 book 2 so basically what I'm trying to say is that from from the you know looking at individual books in Euclid's I mean, different books, different books, one, two, three, four, five elements is that
from the, individually from the perspective of each book alone, these theorems are trivialities as we talked about. But once we see that how they are being in fact incorporated to other books, to other theorems, there are non-trivialities. You can see it and the way that there are non-trivialities across these books rather than within individual books is precisely when they are being mobilized as algebraic methods of computation. For example, decompose a long rectangle into one square and the excess of it, and then
divide that excess into two equal rectangles, and this is a method of computation as we saw. It's simply a method of computing a quadratic equation. So this is from a historical perspective, Greek mathematics also has already been looked at inadvertently or unconsciously doing algebra. It is not that algebra is being invented in a specific timeframe, either by Indians or by Arabs, but it is simply, as we see, it is necessary for connecting different dimensions
of mathematics to one another. As we see in Euclid's case, even though these three propositions, geometrical propositions that encapsulate algebraic computations are trivialities in individual books. Nevertheless, across different books of elements, they are not trivialities. They are required for an ongoing construction of the mathematical universe. So let's go a little bit into nitty-gritty of how this works.
One second. So, oops. Proposition. proposition six look two these two propositions were just the theorems the Pythagoreans needed for the solution of the
problems of application of areas with defect or excess the application of a given area C let's see this is our area C the application of a given area C to align segments a B with a square defect
the rectangle aq oops the rectangle aq is required to be equal to a given area c and the defect or or excess of BQ, this one, the one with dashes, is required to be a square. Now, in modern notation, the two problems may either be written as pairs of equations with two unknowns, and that would be this.
Now compare this with with Euclid data proposition 85 and 86. You can find the data in the Google Drive, by the way. Or as a single equation with one unknown X or Y.
last session are just three types of mixed quadratic equations that were that were introduced by both Khara's me and Tabitha Kekora so you see that how Quite interestingly, geometrical trivialities of Euclid encapsulate general algebraic computations, algebraic relations, but only when these geometrical trivialities are being seen
not as geometrical particularities, but relations required for linking in a constructive Euclidean sense different geometrical objects. question before I move forward when you say linking different geometrical objects though there it's linking them through similarity no no by thinking as I mentioned constructively in the sense that how for example we were you know the whole idea by linking I meant maybe linking is
that a good word by by integrating them in the sense that you remember that for example how can you see no you remember in the construction of for example a point that was outside of a straight line when you had to create from that point outside of the segments you had to create or construct a new line that can be said to be equal with the segment you remember throughout this construction
you linked objects, particular objects to one another by constructing a triangle, a circle, another circle, and then that creates a linkage between them in which the pop-up function of the diagram was activated in the sense that a line at one time was a side of a square and another time was a radius of one circle, another time a radius of another circle and hence astute in a specific geometrical relation with other components in that sense. That makes perfect sense, thank you.
So as we saw that one of the purposes of the pair of, for example, Proposition 5 and 6 in Book 2 was to justify the solution of the problems of applications of areas with defect and excess. Now, confirmation of this view is obtained by considering application of the theorem book two to the solution of the problem, for example, 11, proposition 11 in book two. Now as I mentioned, that famous historian of mathematics, Onguru, who has written on
algebra has noted that proposition book 2 was used in the solution of the problem 2, the problem 11 book 2, which reads, to cut a given straight line so that the rectangle contained by the hole and one of its segments is equal to the square of the remaining segments. if the given a straight line let me turn on the thing again if they given a
straight line is called A and the remaining segment X the problem can be formulated as an equation so again listen to this carefully to cut a given a straight line so that the rectangle contained by the hole and one of its segments is equal to the square of the remaining segment so if a given a straight line is called a and remaining segments is called X then the problem can be formulated as the following equation which is of course is equivalent to this
can be solved by applying an area A2 to the line segment A with a square excess. In this figure that I just drew, I've reproduced Euclid's diagram. perpendicular lines AB and AC AB and AC are made equal to the given line segment a the additional segment X equal to AF is constructed in such a way that
the rectangle fk f k this the rectangle fk is equal to the square 8d this is square. Now, the text first describes the construction of the segment AF, and then next proves, by means of Proposition 6, Book 2, Euclid, that the rectangle Fk is indeed equal to the
Rectangle FK is indeed equal to rectangle to the square AD. Now, after that, the text shows that the segment x constructed also satisfies the original condition 7 and this is that condition
which we reduced it to a simpler form. Now in a modern notation, we would say that subtract Ax from both sides of x2 plus ax equals to a2. So in modern algebraic notation,
we say that subtract ax from both sides of this notation and obtain seven. This one, A, parentheses A minus X, parentheses close, equals to X squared. Then the text proceeds to do exactly doing the same. It says that FK, the rectangle FK, is equal to the square ad. Let ak be subtracted from each, exactly like what we did in the algebraic notation.
Therefore, fh, which remains, is equal to hd. This is hd. This very operation that we just did. Wait, did you mean FK equals H? You're sorry, you said FH. FH, therefore FH, let me just, this is such a mess here. here let me get this F H is equal to so F H which remains is equal to HD of
Of course, from my bi-diagram, they absolutely do not look equal, but from following the proof, they are equal. So this very method, which was basically can only be constructed by, you know, subtracting AX from the science of the equation or, subtracting ak from the figure is exactly what Kharaazmi calls the method of reduction,
And this analysis showed that proposition 6, book 2 was in fact used by Euclid in order to solve a problem of application of an area with a square excess. So which brings to the point that I was trying to make, that there is, within a geometrical algebra, there is in fact, from a sort of perspective, a back and forth movement between
algebra and geometry but this this cannot be done unless we see the geometrical trivialities those trivial theorems of Euclid in a new light not as a standalone but as methods as procedures as procedures of constructions and these procedures of constructions are invariant operations properly speaking there are algebraic computational methods and here it was the method of algebraic reduction questions
So do you mean that, just to reiterate, Euclid's propositions become non-trivial when relations are established between the propositions rather than within? No, when propositions are being understood as constructive relations within problems
in Euclid's books. that basically they encapsulate some method some general method some invariant method which from today's perspective we would call it algebraic computation when they encaps when they encapsulate that procedure, that method, or that invariant relation as in fact as an operation. No longer they don't tell you, describe you anything useful about some particularity, but once we encapsulate them, we see them as encapsulating an operation that is required to solve problems,
then they become non-trivial, which means that they can only become non-trivial when you no longer understand them as some pieces of intuitive information about some geometrical particularity but as an operation. An operation that encapsulates an algebraic computation, a method. Which in this case is like the decomposition and recomposition of rectangles. It's the ability to cut rectangles. Yes. Excess, yes. The whole idea of a notion of defect and excess, yes. If you have, for example, if you're oblong rectangle,
cut off the square from its short, make it a square, by taking its shortest side of the oblong and make a square, cut it off. Then what is left, the excess, then cut it again to two equal rectangles and then move, create an algorithm out of this. And here the same thing. We see that once those trivial propositions are being understood as a form of composition or decomposition that can be shared across different contexts and problems in different Euclid books of Euclid's elements, then they
become non-trivial. So I'm just going to try and re-articulate what I understand. The algebraic method proceeds essentially adhering to its method and forming continuities through its method. continuity in the sense that I would say that the continuity we need to be careful to using them because mathematical continuity because it might confuse is mathematical continuity as you see but algebra has both discrete
components and continuous components because it can equally deal with magnitudes in their continuity and with discrete quantities mainly numbers now the thing is that but if we take this idea of continuity as the idea that there are generalities and these generalities are operational or computational methods that can be shared across different contexts and problems then yes so my fundamental question is how do you cross-check the methodology well checking cross-checking the methodology is not something that you
can at this level of geometrical algebra you can do it it's only when you properly move to post-Renaissance modern algebra, where basically you have two things, theory of magnitudes and theories of quantities. Quantities that can basically be expressed even outside of not only positive quantities, hence the idea of negative quantity, but also can be expressed outside of the real numbers. This is where ultimately the power of algebra is.
It only comes into place where you can actually make these kinds of analysis when you have complex magnitudes and complex numbers. Where basically you can create, you can see that what is the fundamental theorem of algebra. We still don't know what algebra is, we are just simply blindly using it. And this is exactly what Euclid does, so as far as me. And so the way this idea of an algebraic operation being what connects the different problems.
So like in the case of the excess and defects, like the issue of a remainder, then the remainder, if we always denote it like lowercase a, then what is what it counts as in different problems is what drives the specific implementation of that operation as an algorithm for solving it. So like it's this idea of a variable, the a that might equal any of the. Yes, yes, absolutely. It is the whole idea that basically it infects geometrical cosmology with the plague of algorithm. And basically it can decompose everything into algorithm precisely by virtue of its remainder.
So you can always attach this method. Of course, we are just talking about very simplistically about this, again, geometrical algebra, not algebra proper. But even in this area, we see that you can always have a remainder. And then you can, in fact, apply the algebraic computational method, quantum algorithm, algebraic algorithm, again, to the remainder. So there is no part of geometry that can be unaccounted for. Everything can be approached by way of algebra. And this is not just about the formalism of algebra, as I was trying to say. It's that algebra gives an operational understanding of mathematics.
This operation is being expressed in terms of relations between particular objects, geometrically understood being provided by intuition. Well of course algebra can add its own objects. Which is why formal equality or the equal sign and also like you know zero as Maria was talking about are so essential because the sign of equality is the sign of being without remainder. It puts remainders on each side in the form of variables like into an economy or into a closed system which can be represented by that single algebraic sentence.
Yes, you see, whenever you have zero, this is the whole idea that I tried to very briefly mention in the previous session. The introduction of zero was about the idea of scaling. When you have a scale you have idea of balance and disbalance right now even if you have zero you can take precisely because zero allows you to express everything in terms of equality and inequality then you can take one part of your algebraic terms or sentences or statements to the other side of the equation, hence disbalancing it, and then achieve a further balance.
So you see, this whole idea of introduction of zero within algebra allows you to constantly see mathematics within that fundamental pair of concepts, balance and imbalance. Thanks for reminding me of the zeroing of the scale. That's like a really, really helpful metaphor. I'd forgotten about that. Or not metaphor, but analog, I guess. There is no time, but any of you know anything about the manuscript that Jean-Robert Argonde wrote on imaginary numbers, a complex plane?
I'm not familiar with it. Can you write it in the sidebar, too? Yes. Are you talking about an argon diagram representing complex numbers? Complex plane. So basically what he does in his original manuscript, so he introduces, and this is not something that he came up with, but basically it's not only called a ground plane, but also from a historical perspective, it was Wessel who actually came up with this idea, and he
elaborated its so it's called what's a log run plane this idea that so you see for example for a quadratic equation something like this X to plus or equal you see in the field up real numbers you don't have a solution to this equation. So you need to come up with a new object, with a new mathematical object and these mathematical objects cannot be derived geometrically, can be expressed
geometrically but cannot be derived from geometry, you need to have algebra. So this was a motivation behind Agrand's work and complex plane in the sense that creating a new mathematical object, namely an imaginary number, to which he can form a complex number, complex in the sense that it has both a real component, a component that is a real number, and a component that is imaginary, extended beyond the plane of the real numbers. So he can in fact solve
this solution within algebra. So when he wrote his manuscript, the actual reasoning behind it and his illustrations were actually he used the idea of zero as a scale, but a scale which contains on different sides real numbers and imaginary numbers. So this whole idea of zero is not really a metaphor, it is actually quite, it is precisely as you say it's an analog that's real in algebraic understanding of how you can
create new mathematical logic and the complex plane is simply something like this by the way if you those of you want to know so you have this this is called the reals the real number and these are imaginary numbers so the complex number essentially is something like this So this is your x axis and this is your y axis.
So this is the x part of it which can be expressed on the plane of the real numbers, on the axis of real numbers and this is the imaginary part that should be expressed on the imaginary axis. i something called imaginary unit imaginary unit in the sense that square i is minus one imaginary you see as a real number you can never have a square of anything to be a negative quantity so this is this is the whole idea of that he he created this complex
plane by way of algebraically creating this complex plane he managed to come with a new algebraic object otherwise inexistent both in geometry and an an algebra that is simply tethered to natural and real numbers. And of course this whole thing can be expressed geometrically as we see here. you can simply express x and y geometrically by way of trigonometry as x being, here if
we say that for example this is a, this segment here and this segment is b, it's going to be a2 plus b2 and y is going to be arctangent of y over x, that's it. so yes so this whole idea that I mean this is this is so also as an answer to theater you see
that this is this is the kind of an object that can't be really you can't have it in geometry yes here he ex uses a again a geometric plane and some geometric relations but these geometrics relations for this geometric relations and this geometric plane to in fact yield this algebraic object it needs to be subsumed already within the field of algebra because this is not something that you can in fact begin to think about it within the particularities or the intuitive particularities of the field of geometry.
You need to have an algebraic relation under which you now subsumed the entire field of nucleotide geometry. And this is like very Hegelian dialectical reason, as you see. Through the subsumption, you create a new transcendental plane. So I keep coming back to this skeptical problem, sort of compulsively, of verifying the methodology though. And so you're saying algebra is proceeding through forming new types of, essentially just forming relationships, right? Yes, forming relationships that are essentially basically the only criteria that warrants,
you know, that the methodologies, algebraic methodologies are sound and valid is based on the coherency, consistency of between algebraic notations. So how important is it to algebra or just how important is it in general to be able to, I guess, verify the objectivity of the results of that methodology? Can't or could we potentially sort of imagine formalisms that have yet to be
sort of subsumed or authenticated through algebra's method? and or and this whole idea of objectivity back to the idea that you see there are two levels of objectivity there is the objectivity of mathematical cognition which what we are dealing with here and this is properly speaking thoroughly being determined by criteria of consistency and coherency of formal relations and the other one is the objectivity of knowledge so you have one objectivity the objectivity of mathematical cognition a priori erken says in Canton sense and the other one is the
objectivity of knowledge core wisdom but knowledge for what knowledge to two different things now the whole idea as you say the ultimate question is that that how you really can commensurate the objectivity of erkensis, of mathematical cognition, with objectivity of knowledge. This is of course, this is basically something that I genuinely think that there is no final answer to this. This is exactly what I was trying to say, that the power of knowing starts from erkensis, from cognition and forward. But it is only historically, retrospectively,
it can understand, it can know how was it was, in fact, correct, could be applied to the objectivity of knowledge. And this is an ongoing project. And we need to see it not as a final answer, but it can only be answered historically. And retrospectively this is after war this of reason Um, and I'm not sure if this question makes sense, but it's you're you're saying that the the method of algebra proceeds through this criterion of consistency But that that criterion of consistency just is the method of algebra, isn't it? So so I mean how how do you even
begin to Take that first, right? Yes, but you see, again, one thing that you need to know, this is the criterion of consistency of axiomatics. So the consistency, I can understand what you are trying to say. You are basically referring to this problem, which of course I would say that Y is not a problem. the problem that from the perspective of the axiomatic of the foundation of algebra you have a consistency or coherent account of the algebraic methods but what really gives license
to this method so hence maybe we should step outside of algebra see it yes now yes I believe that yes that from a mathematical perspective the the universe of mathematics so to speak we need to step outside of algebra but the stepping outside of algebra does not mean that we are stepping outside of the universe outside of the mathematical universe into the correlation between in mathematics and physical universe. Because that would require, again, basically betraying the idea of what the criterion of mathematical truth is. The criteria of mathematical truth
is not in a correlation with the cosmos, with the universe of being, but with beings of mathematics as such. So from this perspective, yes, we can, and this is exactly what contemporary mathematics is, to show that in fact these methods of algebra are rooted and can in fact are being licensed by higher generalities of mathematics, topological, algebraic, geometry, categorical, shift theoretic, topos theoretic, so on and so forth. But this is very very very different from saying that the the war on for these methodologies needs to be found in the physical universe I I mean my just reaction is that the you're
saying you need in order to double check the validity of the claims that algebra makes we need to step outside of algebra but we step outside the next into the you know the next corridor which is mathematics but then my my I'm sort of pressing to say well why why are you why are we saying that we have to then remain within that realm of mathematics which proceeds by certain methodologies as well yes because because you see the truth you're only when we are talking about this we are only talking about the criterion no than the criterion with the structure of
mathematical truth and the structure of mathematical truth belongs to the structure general more general structure of what what you take something as a true within a particular domain of discourse this idea of truth is really important that true right you can we can only talk about truth when we are talking about two things one the idea of what it means to take something true one two what it means to take something through in a particular domain or universe of discourse. So essentially truth belongs to two things, to the norms of taking something through and
the universe of discourse. Because there is no such a thing truth as outside of a universe of discourse. That would be just fall and relapse back on the idea of preconceptual truth, which is simply metaphysical dogmatism yes yeah I I I think I'm in agreement with that but I immediately and of course as you say and let's say that let's say that's not mathematics and this is exactly when there was this event on the altar I asked this question to Fernando which for another didn't agree to me and I I don't want to press this thing and yes absolutely the methodology of mathematics can in fact be by mathematics I mean the whole fucking
universe of mathematics can in fact be seen in the light of far more general norms of taking something to be true and this is exactly what is happening today in theoretical computer science, the so-called Holy Trinity, which is the fundamental correspondences between logics, between computation, and between mathematical structures, namely between the question of proof, types, and structures. And this is, yes, and this is exactly, I think, where we are headed. Sorry, I'm not trying to be super antagonistic, it's more out of curiosity, but then when
you're saying like, okay, these are sort of, the only way we can talk about truth is if we establish sort of semantic fields. But is... Syntactic, semantic, pragmatic field. Yes, because these are, because basically these are the transcendental dimensions of something to be true but I immediately someone would notice that establishing that semantic field is already drawing a boundary of like who is in who's out this is the intersubjective community that we've established right yes but you see this idea of semantics semantics this is exactly I mean if you yes if we say that semantics we are just restricting it to some you know very
trivial natural linguistic dimension yes but this whole idea that what does it take for the semantic dimension to be constituted what kind of logical fundamental logical computational behaviors needs to be in place for something like that to be emerged and I'm referring exactly to those to those computational properties and behaviors and this is exactly what I meant by by this that's semantics we shouldn't see semantics as as established rules and this is no one I think even Brandon doesn't treat it treat them like that as
if we need to know the role of semanticity in order to you know move on with the game of giving and asking reasons but to see that there are some rules and these rules are quite axiomatic and not the sense of axioms as such but in the sense of some fundamental logical and computational behaviors that need to be in place for you to in fact have anything resembling semantics pragmatics and syntax and these are kind of computational logical behaviors and properties that of course are we all implement we all
implement regardless of who we are and how we are approaching semantics and how the very fact that we can talk about something right is the implementation of these fundamental behaviors and properties that can be couch in logical computational terms yeah I um and please if someone else would like to talk just interrupt me otherwise I'll keep going um but my oops I lost my connection one second one second okay go on I was just gonna say we can acknowledge that you know we're establishing essentially communication through the building of these semantic parameters.
Forgive my sloppy phrasing. Yes, semantic parameters which themselves are what you might call to be the... Under scrutiny. Yes, and under scrutiny precisely because they are emerging behaviors of some more fundamental logical computational properties see now I want to what I want to say is you can still establish the fact that those semantic parameters are necessary for communication but remain completely skeptical no they don't remain skeptical in the sense of you know some kind of the idea that there is no such thing as
knowledge or there is no such thing as semantics, they only remain skeptical in the sense that they are under an ongoing field of investigation, a skeptic. This is exactly the meaning of a skeptic is investigation. And this investigation ought to be understood as the power of rationality, which is historically and socially mediated. Because otherwise, then you fall into what Plato calls the Menno's paradox. Then how do you know that you in fact are a skeptic of this? If you have in fact, because otherwise if you see a skepticism as something, then how
do you, as if you know already something that we don't know. But that's already, that is really the, basically Plato's super acid of rationalism that he, you know, unleashes against the skeptics. Right. The Nietzschean madness. Yeah, yeah, yeah. But I, I don't know, I'm not convinced that the, I don't know, I'm still not convinced that just because of Mino's paradox or just because we're exposing the sort of lunacy of Nietzsche, I mean, it still seems...
At what point do you commit to the alternative? Well, you see, this is the whole idea of rationality rationality that you can only commit to reasons power of knowing understood as a retrospective recognition of its conditions of realization. Because commitment, precisely because it can, because this is the reason, is really the idea of a skeptic in the sense of an ongoing investigation, commitment to anything outside of this ongoing labor then becomes leads to Nietzschean lunacy. It's going to be debilitating. Either you expect, you
accept this idea that yes my knowledge right now is insufficient but nevertheless I have to commit to this minimal things in order for me to understand what was the case that I was wrong in the past. And maybe I was dogmatic. Or you are going to say that I know the result, and that's basically you abort the whole project of an ongoing labor. I mean, it's sort of Sisyphean because there's no guarantee that you find that peace. No, no, no. This is the whole idea of the alienation. This is a very alienating power. either you live with it or you die of lunacy right and this is the very idea
that sellers put forward that let's take sharpen this alienation to that extent that we actually feel at home within this very alienation yeah I mean the media sellers quote that I think of is like science proceeds as because it's able to jeopardize parts of itself but not the entire thing all at once you know yes and then you change the new chain madness it says throw out the entire thing sort of yes yes yes otherwise I am completely I do not have any and I have come to this conclusion that absolutely no objection against epistemological skepticism or the idea of a skepticism of the investigation this bit and believe me I think the whole idea of epistemological skepticism goes
far, far beyond this, simply the domain of theoretical discourses. Right. And I really think that one of the most corrosive components of epistemological skepticism is really the problem of induction. Right. He never solved it. He never solved it. No one has ever solved this problem. It's just keep coming back in different guises. Othnam tried to solve this, but his solution to the problem of induction was just simply, again, a kind of, I would say, even a more soupy version of Hume's solution. Thanks for talking.
Absolutely. So any question? Yes, yes, yes. I was reading, because of your recent paper, I was reading from the Critique of Judgment where Kant is discussing, what is the name of the chapter, objective purposiveness in nature. And he's describing the fact that moral laws have to be assertive as commands rather than facts, acts, the register of ought to be rather than is, in order to preserve, because of the necessity of assuming our own unconditioned autonomy, in order to make choices about intentional acts. And so is this sort of like similarly, we have to accept, so I guess like insofar
as to take something as true, like in some sort of pragmatic, fully meaningful sense, to commit to it, is to act upon it, is something beyond simply the formal, that doing this in response to an insufficient thing, like knowledge that we currently have, but which we know may turn out not to be real or complete or whatever, in order to continue to move forward has this same structure as Kant's moral law in the third critique, that it has to be a command. The order of reason is the command of reason, not the already existing structure. Yes, yes, yes, absolutely, yes, yes. And the thing is that as you see that the thing is
that it's quite actually very interesting and I still am thinking about this, this whole idea that it's only the power of transcendental ideals are the ones that warrant the intelligibility of an ongoing labor. And the transcendental ideals, if you really see them, and this is of course, you know, again, might come under the skeptical acts that Theodore was talking about that then what was warrant the power of transcendental ideals themselves. I think This is really the whole idea of transcendental, that the transcendental ideals are transcendental
precisely because they are the minimum, general requirements for you to have in fact any intelligibility, even the questioning of these transcendental ideals. So what you might call to be, they are general ends from which particular ends and their respective courses of action can be inferred. Without these general ends, there is no such a thing as intelligibility of anything. Sorry Theo. Unlike the conditions of objectivation, like identity, more like causality but less
like identity, quantity, quality and so forth, this is like a second order condition. Yes, those quantity, quality stuff, the pure concepts of understanding, so categories, which include modalities, quantity, quality, so on and so forth, are absolutely not a... They are a priori, only in so far as they are required for you to have, you know, experience of anything. But nevertheless, in Kant's philosophy of mind, they are being derived from the manner by which the mind organizes the manner. mind organizes sensible given materials. Hence they are still coupled with sensibility. Hence
they can in fact be vastly renegotiated and that's exactly what science tries to do. Contemporary science, what it does, it vastly revises, drastically revises categories. But those transcendental ideals as you say are on a whole different level they are completely what you might call in a hegelian platonic sense live in a different kind of universe they don't know anything of experience i i want to sorry jake if you were going to say i was just going to uh say who who would take the Sisyphean gamble if they weren't like sort of yeah I mean what what incentive
does someone have to take up this you know dialectical burden of working towards reason if there isn't any certainty from the get-go of its rewards you see it's this is the whole idea that there is no certainty when it comes to reason because reason is not about certainty the reason about reconstitution every constitution that can be trained as a problem of freedom or emancipation that's exactly what canton hegel tries to do that so the ambitions of reason are not in the epistemological certainty, but in the reconstituting power. Understanding
is something that we can talk about certainty, but not reason. And this is really the whole idea of Hegel's critique of Kant. Reason is not about the ambitions of reasons are not ambitions of understanding. I mean the freedom when talking about Hegel immediately like I think in my mind makes me think of sort of this image of God as the self-moving mover that it's completely free of any antecedent causes yes absolutely that is true but you see the the causality of transcendental it is a formal cause not a substantive cause
whereas for actually theological God you have a substantive cause it is this the The whole idea of the causality of reason is, and this is Plato's attack on Eliatism, begins the gesture of philosophy, that the difference between thought and being is not substantive but is formal. The causality of thought, the causality of reason is in its formality, which is negative, the negative power of concept. Now, yes, from this perspective, it has that kind of causality. But the thing is that we as subjects, simply as subjects, we are not under the power of reason in a full sense of causality of reason.
Reason is impersonal. Reason is not constituted of persons. It's that the whole idea and the project of freedom in Kant and Hegel is that how we can bring ourselves under this formal causality that is freedom. With the understanding that the subject we are is not just a rational subject of understanding, but also the sensible object of experience, of intuitions, which is heteronymous, through and so sort of this might tie back into how algebra proceeds through how
algebra moves forward through forming these relationships but in Hegel system it the systems proceeding through contradiction but the the way to proceed through contradiction is through resolving contradiction through integration that's my understanding of people and I'd be happy to if someone information you see this is integrated I think this idea of integration needs to be a little bit fine-grained this idea of integration is not essentially can't idea of synthesis. This integration is only integration in the sense of suspension and
mobilization, which is the idea of sublation. Well, I guess my... So this is the whole idea of subsumption is that you have basically, if you want to think about this very geometrically, in Kant's idea of synthesis is that, so you have different data on one level, for example, on a level of intuition or imagination. And then a power comes either from below or up levels, from either reason or category pure concepts of understanding, or from down, and fuse them on one level. Whereas in Hegel, the idea of integration is like this, that you get suspension on one
level and assimilation on another level. See that to me strikes me much more like Deleuze who proceeds not by the resolution of contradiction but simply by difference. There's no reconciliation between the two elements whereas it seems to me that Hegel's end-all goal is a is a version of like a holistic image of like the self-moving God and Deleuze's element keeps these two elements separate from each other. No, I don't think that this is really an accurate assessment of Hegel because ultimately what Hegel, the power of absolute knowing is the intelligible unity of the world in
its objective alterity and subject in the and reasons of formal spontaneity so you will still have not this holistic idea that you everything has has been fused but it's the whole idea that how they they form an intelligible unity and in this intelligible unity is always at the verge of disbalance in a very algebraic sense. In a sense that this is what Hegel calls history as a path to despair. That if you can't really understand the intelligible unity and why is this intelligible unity,
power of this balance between world in its objective alterity and its rational spontaneity, then despair is inevitable. I mean it seems like that conceptual continuity between subject and object in Hegel though just is that holism that I'm talking about maybe there that I'm trying to press at I guess. I think, I would say that it's not a continuity, but not in the sense that, and it is holistic,
but not in the sense of what you might call, you know, that they are being parts of the same whole. It's that, it's this whole idea that there are, in fact, an intelligible unity, unity in the sense that they are necessary conditions for the intelligibility of one another. Thank you, sorry for harming so much class time. Oh, no, no, no, absolutely not. I mean, these are a fantastic thing.
So very, very briefly, so in this whole session course, basically what we are really looking at was that we saw a form of progression of knowledge from sensibility to intellection. Well, of course we had to stop somewhere and where we really stopped are the Descartesian Cartesian implications of algebra in which mathematics starts to basically reject the
sway of sensibility over thought. But nevertheless, this is, you know, if we think it about broader scope of philosophical reflection that can encompass the history of mathematics, then we see that precisely after Renaissance, with the advent of natural philosophy, advent of modern algebra, Kant's critical philosophy, then on you know Frigge's concepts of logic, revolution in logics and mathematics in 20th century and then further in contemporary mathematics and contemporary
logics and computer science. particular Descartian worldview of mathematics in which algebra becomes the hegemony is also being fundamentally challenged. And this challenge is tantamount to precisely that philosophical challenge in which sensibility and the autonomy of thought are being rendered commensurate. No one holds sway of the other one. From the perspective, everything, but the whole thing is that even though they are both
necessary, it is ultimately from the perspective of mathematics and from the perspective of post-Cantian philosophy is the power of concept that comes first, namely the a priori acts of cognition. But the a priori acts of cognition are also dependent on individual senses, just exactly like Frigge's analysis of algebra via the logical sense and reference, that Yes, you have the reference, but the reference is also dependent on the sense. But then you see that in the Nofregan analysis, the challenge is that then precisely because
the individual senses, their material, their content, their meaning is provided by sensible materials. So there is essentially a dialectic within this history of mathematics that we were looking at, which ultimately comes to dialectic between concepts and sensibility. And as mathematics is moving forward, so as philosophy, you see that there is a priority of the power of the concept. So this power of concept is not being laid out in Cartesian solipsistic terms. But in fact we see that the power of concept is being attempted to be elaborated in terms
of how different senses of a particular reference are being integrated with one another. And that's, you know, we see this is exactly where logic and mathematics are being headed, in linear logic, ludics, you know, theory of structures in mathematics, like homotopy type theory but also in philosophy the school of in French you'll is there so on so forth and very very very quickly those of you who are interested to read this
entire course simply as a as a kind of a very narrow perspective into a broader projects of rationality a broader picture of reasons power of knowing which constitutes modern sciences read Danielle Macbeth realizing reason and And also a good commentary on that would be Ray's recent essay. The title of that piece by Ray is Comments on Danielle Macbeth Realizing Reason and Narrative of Truth and Knowing. Has that been...
Comment by Ruthledge. Oh great. Yes, it's out, yes. So just so everyone knows, we have about 20 minutes left of this session still, so if questions fire away, I guess. And I will also, I know that Philip asked for what about the other pair of fundamental concepts. We haven't been able to review those things, but I will put all of them on the classroom page in the next couple of days with some references for each one of them. So for those of you who are interested to look further
into this time window from antiquity to early Renaissance. Jake, Maria, Philip, Theodore, anyone? I just had a question about zero as used in a balance and Aurasme and graphs, just to see if I'm linking up the various points in this constellation well. So if your problem is that just putting rock A and rock B on moving plates like separately,
right, you can't, there's no way for you to index or to make meaning by themselves of like how far it sinks or how far it rises, things like that. But when you put these things against each other, zero and zero them to begin with, then every change in one becomes correlated to the opposite change in another in such a way that you can draw this like diagonal line. And you know what this is exactly? This is the notion that Kant calls real opposition, as in contrast to logical opposition. And in mathematics we call it magnitude. Okay. This is what magnitudes are. Magnitudes about the relativeness. There's been equalities inequalities and this is exactly Herman Grossman's canonical definition of
magnitude in mathematics something that can be said to be equal or in equal to something else within one and the same subject chance example of this would be exactly a ship setting sail on the ocean and its movements then its magnitude its behavior is being expressed in terms of two oppositional forces that there are both real one for example of risk westbound and one like a northeast bound wind coming and interacting with one another in terms of equality and inequality in one and the single same
subject the ship and that's what exactly negative quantity for Kent is and so is the ability to do so in the sense that the algebraic statement is self-contained and is able to be productively self-contained because of the equals is it the constitution of that subject of that one in the same subject in which you can put two sides and deal with real trade-offs yes yes yes and you see that so from from this perspective introduction of zero to mathematics and hence you know the bootstrapping of algebra algebra was necessary for you in fact that mathematical universe precisely because without the dialectic
of inequality and equality all you have is a dialectic between continuity and discreteness but once so so by the introduction of zero we come to that diamond of mathematical universe the dialectic the you know the four sides of the dialectic between equality and equality top and down and discreetness are continued on the sides and then you have different domains of mathematical of a structuration at each side, like functions, like combinatorionics, like theory of numbers, so on and so forth. Where is that diamond somewhere in particular that I'm not remembering it from? You don't remember it?
I think I drew it for sessions. Cool, I'll go back then. My other question, if nobody minds me, just sort of going on is this issue of the aleatory. And so like, this is just sort of like multiple strands of thought over the class so far, but that, so when you're talking about like the progression of science and the way that macro logically speaking, it does demonstrate this progressive continuity, but which at a micro logical level is fragmented over time. So you have all of these individual experiments that do or don't work out from, minimally from the scientist's perspective in an aleatory fashion. And then also from the perspective to various extensive
the procedures that he uses allow for, whether it's at the level of false positives or it's just at the level of like how strong of results you get from this or that population sample, this aleatory stuff occurs. And it seems like differentiating or creating points of the aleatory within which data points have a meaning you identified them within a spectrum and they can be given certainty and so forth. The construction of dye as opposed to like throwing grains of sand on the ground with no ability to discern what that resulting aleatory distribution is. Is that kind of the other side of the development of reason is like the production of a space in which permutation is possible so that eventually you can put together enough, you see what I'm saying? Like I'm trying to find what the space. Yeah, yeah, yeah, what you might call
to be the combinatorial mix of reason. Yeah, yeah, sure, yeah. Cool. Yeah, because essentially that opens divergent paths. And, I mean, so in the sense that the act of putting those things into question as to their truth in some register is this purely formal causality of thought. is the correlate or whatever is the material equivalent material causation the cause of how some materialis there may be then that aleatoriness that the production of chance in the same way yes yes and that essentially comes with
with the with the interaction of the inductive and deductive ports of theories. You know, the inductive is the one that is purely aleatory. And so I guess, so out of the different, it seems like there's more than one way to weave these things together as regards our knowledge production specifically, one of which is that all of the categories of the understanding and then maybe even higher issues of concepts are the result of hundreds of millions of years of
evolutionary advancement and behavior, which at each point is still this discontinuous aleatory, do I get this mutation or not, do I survive or not, sort of thing. So there's a time section of environmental exposure and aleatoriness that's present to the other time section of conserved traits and so forth at each point. So that comes up to us and it comes into contact with the system, with the materials for the formation of knowledge or the incentives to do so, and you bind those things together there. I'm thinking of the issue of the simultaneity of causation with its effect and the fact that there are multiple vastly different scales, temporal scales, of causation that are both becoming simultaneous with the production of a knowledge effect and are by it are getting zipped
together and so i can see like a sort of formal description of this as like something that is reason and arises from like linguistic and semantic and all this activities and then another one which is like our evolutionary cognitive history gluing itself to our present technical challenges or something like that yeah which would be the evolutionary history but then you see that the the whole idea again comes falls back to that idea that the very fact that you really you are going to talk about that evolutionary history did the second path presupposes in fact whether implicitly whether you want to agree or not the presupposes at the employment of the other dimension in which you have the
causality of self sufficient causality of reason the formal causality of reason precisely because if you don't have the inferential theory of concepts the concept of concepts now how are you going to talk about causes with the understanding that these evolutionary causes are not given to you in advance in sensibility or intuition you see that this is basically you need to have both and this is really exactly that again comes back to this whole idea of the intelligible intelligible unity of the world in its foreignness as
air can say subjective air can says in its spawn causal formal spontaneity in the sense that you have these but what is always you need that as a point of presupposition and that's how using pitching one against one another while one is already always needs to be presupposition otherwise this whole idea of this balance is not going to work out this way is that you can drastically by hitting these two against one another you can drastically revise your extant categorical understandings. Exactly in the same way that science shatters our categorical intuitions.
Okay. So what's at stake here is really, is not so much whether reason is a structure that was already present and operating in pre-human living history, or is some sort of spontaneous emergence from the backwards perspective of which the rest of history can't be thought without it at all, all of that, the correlation is circle. What's at stake is really the techne of making the second separate out from unity or from the question of the first in a way that is productive on, manipulates, disrupts the pre-given unity of our evolutionary history, between our evolution and our evolution.
Yes, absolutely. The whole idea, power of knowing, starts with reasons, basically a realization as the ultimate assault on the givenness of anything, including its own realization. this is uh this uh pippin puts so much emphasis on um self-consciousness in his reading of uh hegel's idealism um as being uh i mean he really makes it an autonomous entity in some ways uh it's
not yeah it's a not caused entity yes but you see this is the whole idea that the causation of reason exactly if we start to see this causation and the same thing about algebra and geometry in the historical perspective that we were talking about if you look at this causation as a substantive cause, then there is no such thing as autonomous reason. There is no such thing as autonomy, in fact, in the universe. Because there is no such an autonomy in nature. But if you do, and then comes basically if you endorse this, if you say that, okay, there is no such thing as substantive autonomy, you fall into Mendel's paradox.
Then what would be the solution? solution that the autonomy of reason is not really substantive the causality of it is not so substantive it's formal causality yes then it would take you to elaborate what it means to have a formal a spontaneity a formal cause out if you understand cause out the reason as a formal one rather than substantive the thing is that in order for you even trying to say that there is no such understanding a substantive autonomy in nature, that nature is thus and so, you require, in fact, an implicit presupposition of the formal causality of thought.
Because we can't claim anything. This is the whole idea that the formal causality of thought, or reason, whatever you want to call it, is a presupposition of saying anything. I think that you need to take this presupposition very seriously. Then what does it mean? What is exactly in this presupposition that requires for any claim to be presented by way of this presupposition? We're already part of the script.
Yeah, well, Frigia might call it concept script. well I just want to thank you for sort of standing up to all my questions really the kind of question that I have said they're so difficult and I think it's not just we can only every now and then work on a specific aspects of them because because they are actually quite daunting and says I mean the whole idea is that I don't think that philosophy or science or any of these fields can ever answer these questions in one broad suite it's impossible okay everyone I have
to be somewhere so any feel free definitely write your questions any questions and I will also put some of those extra materials on the classroom and then hopefully we will say remain in touch and think about some of this stuff together. That sounds great. Thanks so much Reza. And I think Jamal was asking if we could get the research. Yes. Okay. I will put it in the Google Drive. Yes. Thanks a lot Reza. This is great. Thank you so much. Thank you everyone. Thank you so much. Take care everyone. Bye. Bye.