all right welcome everybody to maths and ideas from antiquity to the Renaissance with Reza Negar-Estani I'm gonna hand it off to you Reza since we're starting a little bit late okay thanks Jake okay hello everyone so I'm going to continue the introductory session on the kind of overall idea of what you might call the broad evolutionary perspective on the possibility of mathematical thinking. And as I said, by evolutionary thinking I mean both in the broad sense of natural evolutionary
cognitive affordances but also cultural artifacts. So before I start, let's hear if you have any comments on the reading text or anything from last session and then I will start talking for a while and then we can open up the discussion again. Questions? Any comments on the text or… Can everyone hear Reza right now?
Well, I'll make a comment. It seems to be a… In one way, one can read it as a response to the early Wittgenstein's view of mathematics. And that's just that comment. And I found it very interesting, the idea that you can derive the continuous from the notion of walking on a line and the discrete of taking steps along that line. And the notion also that when you do a mathematical proof,
this sort of aha moment goes in, is in fact, the kind of, as I read it, the re-experiencing of this pre-verbal intuitive sense of mathematics and that the simple notion of a thing that's reversible that goes back and forth or some kind of algorithm misses both mathematical discovery and the discovery that people go through when they're connecting the steps of that proof to something in their pre-verbal experiences. It says, aha. So I think it's interesting, the choice as to whether one puts visual diagrams in a mathematical text
or not, as opposed to just making them simply set theory and the symbolic. As I say, I've always found that visual elements, tracking back and forth between the visual and the symbolic is very, very useful. Yes. I think with the comment, I think people like Bernard Tazier, Longo, Gilles Chatelet, it seems that there are two trends, at least when it comes to this mathematical thinking scenario. One is the emphasis, one is what you might call being in the extension of the general program
of intuitionism and intuitionistic logic, you know, since the time of Brouwer, and then sharpened by people like Poincare and Riemann, and then coming to the 20th century by people like, I don't know, like Stanisla Derhan or Giuseppe Longo or Jean Petitou. And the other one, what you might call to be coming from a kind of a formalist school of mathematics by people like Hilbert, by Cantor, which is actually I have been only recently starting
to finally read Hilbert's own lectures. And I can see that, so when you read the history of mathematics, particularly commentaries on history of mathematics, you always get this feel that Hilbert is absolutely anti-intuitionistic. But the more I have been reading his lectures, I also find the book. It's a really fantastic book, I highly recommend it. You see that he is not really an intuitionistic mathematician, but nevertheless it seems that this, what you might call the rift between intuitionistic program and the formalist program ultimately becomes two approaches to mathematical thinking that I would say that they are not
essentially diametrically opposite but more complementary in the sense that the intuitionistic program in 20th century at least becomes underlined in terms of mathematical creativity, simply mathematical construction, the logic of mathematical discoveries and constructions. Whereas the other one, the formalistic school, you see it's being underlined rather than as an opposite to the intuitionistic program, simply as an approach that first and foremost pertains
pertains to proof theory and construction of proof. I think I generally, the more I've been reading, I don't see that kind of classical opposition between these two schools of thinking. It seems that they are both correct, but they can only be correct once you see the domain of their emphasis. one mathematical construction and the other one proof. So, of course, you know, proof, particularly when you see in like development, recent development in logics, people like Jean-Yves Girard or Georgi Joparitsy, people who are at the intersection
between computer science, mathematics, and language, you see that they also want to emphasize the constructive, namely creative, in an intuitionistic sense, dimensions of proof as well rather than just the formalistic aspect. But nevertheless, even the creative aspects of proof using the intuitionistic resources, they are I think very, very different from the kind of creative aspects of mathematical discovery that are dominant in construction of mathematical objects, the ones that you mentioned in terms of pre-linguistic intuitions and pre-linguistic gestural thinking.
I will come back to this particularly when I talk about Euclid's elements and also introduce a little bit what we mean by gestures and how they are connected to diagrammatic thinking. But yes. I could ask a question. Sure, absolutely. Well, so, I mean, one thing that, I kind of wrote this down, I'll just read it. Yeah, I mean, I was, so this seemed, this was like, there were a lot of really interesting ideas about how mathematical creativity happens, but I was left wondering about how you would
account for the relationship between mathematics and an external, either physical world or mathematical world that outstrips the relationship or the sort of space between perception and language and so like that this is kind of famous really article that unreasonable effectiveness of mathematics and the natural sciences yes and yeah or that like any like structural realist position would be hard to defend from this view maybe or I was just curious about that yes actually there is a response to that article by Longo it's called the reasonable effectiveness of mathematics rather than the unreasonable effectiveness
of mathematics. I think, well you see that again the idea of reasonable versus unreasonable effectiveness it comes back again to the very much you can see it as the extension of the controversy between the school of intuitionism and formalism. Precisely because at the core of the intuitionistic program there is essentially a Kantian program of intuition in the sense that there is a continuity between what you might call mental acts or cognitive machinery that is pertinent to the architecture of mind and the kind of
what you might call the structural continuity that moves from the mind independent world from the physical world and extends into the mind and that's basically the manifold of intuition, the manifold of intuition or the unity of manifold of intuition, particularly the part that Kant calls primitive intuition, the intuitance. It seems that there is, he tries to say that there is in fact a continuity, that there is at that level, even though the intuitance are not by any means mental acts, and that's I want to make it clear. Nevertheless, so he is obviously against the Descartes' idea that sense impressions yield
knowledge. He doesn't want to say that the intuitives yield knowledge of any sort or they have any epistemic traction. But nevertheless, there is a causal continuity between the intuitives, between the manifold of intuition, the manifolds of sense impression and the causal structure of the world. so far as the intuitives or the manifolds of intuition are being manipulated by the productive function of imagination via what you might call the second and the third synthesis, the synthesis of imagination and the synthesis of what you might call bringing its imagination
under the concepts. And then Breuer talks about this that's precisely because of this causal continuity and in so far as also intuition plays a role in the mental act we can say that there is a minimum sense of effectiveness in the sense that there is already some continuity between the causal structure of the world and the causal structure of cognition. But But as you say, I think it is still really doesn't answer because if you're overemphasizing this causal continuity and Kant is very careful about this, Sellars is in fact very, very adamant that we need to be very careful about this.
You see that we will fall into some sort of what you might call correlationism. So this is the story of intuitionism in terms of its reasonable effectivity, reasonable in the sense that there is a minimal causal continuity between manipulation of the manifold of imagination, manifold of intuition via conceptual imagination and the causal structure of the world. Whereas the school of formalism, I think it's a different story. It depends on what kind of really vision of mathematics you are presenting. Are you presenting what you might call a rather mystical idea that there is in fact a mathematical
structure in the world and we are simply tapping into this mathematical structure or going to a different route which is the school of arithmetization or the classical Albertian school that's simply all it tries to say is that mathematical structures precisely in sense of arithmetic are adequate, fully adequate for encoding the physical structure of the world. But then again comes back to this idea that what is exactly that allows for this effectivity
to emerge. I genuinely, I don't have any answer to this because I think it's a really open question. But I tend to agree with the school of intuitionism that at least we can see that rudimentary mathematical thinking over which later more advanced forms of mathematical thinking are built at this rudimentary level I think we need to think about mathematics in terms of cognitive affordances that are pre-linguistic and as I said they have their roots in evolutionary processes of course again this opens up to another
can of worms and that's the problem of induction ultimately I mean if you have read the Bernard Tazier's work or Longo's work in the sense that they are trying to anchor some of the what you might call cognitive scaffolding of mathematical thinking in evolutionary processes particularly those which pertain to the nervous system you see that's precisely they are talking about some sort of heuristics some sort of inductive heuristics and we know that the problem of induction is quite a slippery thing, in the sense that when it comes to induction, we neither can prove it deductively nor prove it inductively.
So ultimately, following Hume and later people like Nelson Goodman and Hilary Putnam, we have to resort to what you might call an ultimate philosophical cop-out, in the sense that well, precisely because we have no proof of the induction, whether deductive or inductive proof, then we have to abide simply by this solution that inductive has been given us through evolution. And that's enough really. But I think this is again sweeping the problem under the carpet of evolution.
Just because something is reliable and evolution is about reliability doesn't mean that it has an epistemic legitimacy. Evolution might be reliable but its reliability is very different from an epistemic birthright. So it's like the structure of our cognition and perceptual systems must have evolved in accordance with objective reality in some way, but part of the problem is that causality itself is imminent to our cognition.
And so what is that causal nature between what Maddox is describing and... Yes, absolutely. And as I said, I think the more I have been looking into this, it seems that ultimately it's, you know, the Hiumian revenge is coming back in the sense that in order for us to really investigate this causal relation, causal structure that extends from mind independent world into the cognitive machinery is that we have to account for the problem of induction. And the problem of induction is something that for a good reason people have stopped
thinking about it precisely because it's like some sort of, it's really, it reignites epistemological skepticism in its most corrosive form, which is not essentially a bad thing. But nevertheless, it can also lead to quite a huge crisis in modern sciences. Ultimately, you use inductive reasoning in order to investigate the causal structure. And insofar as really that mathematical traction on physical world again boils down to the problem of induction, inductive reasoning, and insofar as we really do not know what induction is to begin with, that is already, you know, can be said to be, has the potential
of fundamentally undermining the legitimacy of mathematical sciences. A great article that I really suggest, even though its connection with mathematics might not be immediately clear, but nevertheless talks about the problem of induction, is a seminal essay written by Hilary Putnam I think in the 60s it's called the problem of induction and the degree of confirmation one last thing on this topic would it be taking things too far field if I ask what your comments are on
on Quentin Mayasu's answer to the problem of induction? I think, okay, I might be, I don't have any sophisticated answer or any kind of non-trivial answer to this, but the more I have been thinking, it seems that Mayasu, in fact, on the surface it appears that he radicalizes the problem of induction. But I think he does not in fact radicalize it. It makes it a little bit more mediocre than what it actually is. Precisely because he ontologizes the problem of induction. But when you read a person like Putnam,
his later book, Representation and Reality, even at the level of epistemological investigation, The problem of induction is something like a super acid. You don't even need to ontologize this to make it worse. It's already bad enough and it's already radical. Even at the level of epistemology, I think the problem of induction is quite a fundamental enigma that, of course, Kant tried to make sure that he is solving this, but nevertheless, I don't think that he ever solved this. So I think, I mean, again, very briefly put, I think Meosu, I wish that he had only emphasized
on the epistemological side of induction and contingency rather than ontologizing this problem. Because ontologizing this problem makes it almost irrelevant, turns it into some sort of metaphysical, mystical scenario for me. But of course, this is far more subtle than the scenario that I just described. But I want to distinguish the epistemological side of the problem of induction, which to me is quite corrosive already, and the ontological side of it. Thank you. Welcome. The name of the other book is Representation and Reality.
Reza, I had a question, so following up on that. In terms of the problem that this poses for mathematical sciences themselves in terms of undermining their results or methodology, does it make sense to say that it's a problem? One caveat here before you move forward. We need to be careful. Not their results in their own mathematical side, but when they are combined with physics. Right. When it pertains to the physical world. Yeah. That was where I was going was mathematical physics. And I was just reminded in terms of this evolutionary account of matching perceptions, intuitions, and whatever manifold with objective reality.
that like in the case of the brain and perception that there are actually all of these you know heuristics and not necessarily in edge cases reliable patch-ups like you know over the blind spot in our vision temporal and spatial blind spots in our vision and you know various other places and that a lot of those are actually inductive filling in gaps of various kinds through different kind of ad hoc inductive processes and I'm just wondering if there's any potentially if that as an analog to issues of bad induction in mathematical physics creeping in because of that reliance on an induction that can't be interrogated, like for lack of a theoretical framework that solves the problem of induction fragmentarily or... Yes, no, absolutely that is correct.
I mean, yes, this is definitely the case. I mean the whole idea is that as I said just because evolution war it is it can trust it as a source of epistemic legitimacy and the same thing about induction with from an audio I don't know if anybody else got it but I missed the first like sentence or so sorry oh I was I was saying I was basically saying that just because evolution always works and evolutionary cognitive processes are always reliable, this is very different from them being source of epistemic legitimacy. And
yes, you can see this, that Hume for example, Putnam in a different way, Nelson Goodman, the stove in a different way. They tie the problem of induction at different levels, for example, to the structure of the memory, the structure of some of the perceptual mechanisms. And precisely in so far as the memory, our memory at least, is highly contingent, I mean contingently structured in a specific way, that already creates a, you know, quite a fundamental blind spot. And it's really, you can say that the reliability of inductive reasoning, whatever its source,
evolutionary source may be, is already the main blind spot for mathematical physics. And the first person who really looked into this was Boltzmann. This is essentially Boltzmann's development of statistical physics is really to mitigate the problem of induction. But nevertheless you see that it creeps out, creeps up again in his own theory of statistical thermodynamics. So yes, this boils down to the past problem, right? why is the past more highly ordered, right? Or at least it's connected to it in the sense that we can reliably consider ourselves to be in the most recent instance of an ordered
pattern. Yes, yes, I mean, well, the problem of induction can be, you know, laid out in different ways. It can be, but nevertheless it always has something to do with how we extract law-like regularities or patterns whether through the use of numerical asymmetry and what is really numerical asymmetry it means that for example if the observations in the past followed a specific pattern and every observation that came corroborated the previous observation we tend to turn this into a law-like regularity in the sense that we use this in order to predict that whatever happens from now will follow the previous observed occurrences.
But this is simply a numerical argument from numerical asymmetry of past observations. It's just an argument from the perspective of reliability, not from the perspective of epistemological truth. Right, that makes sense. So, you're probably going to discuss this down the road, but I was thinking about Euclid's parallel postulate, for example. And the story that's usually told around that, I think, is the notion of how somehow mathematics is grounded in empirical reality epistemologically, but that mathematicians create fictional worlds around that
that are just lying around, many of which remain fictional, but some of them are picked up by physicists years later and seem to perfectly describe the universe very well. I mean, how would you frame that process in terms of the Tessier approach to mathematics? I think with Euclidean geometry is, I think, a different story altogether. There are still so many controversies whether, in fact, Euclidean geometry is mathematics or not, precisely because it's not an axiomatic system. so many people treat Euclidean axioms as axioms like in the mathematical
definition or logical definition of axioms but they are not really axial formal axioms they are not logical axioms they are intuitive axioms as you say axioms that have roots intuitions of reality and yes basically you know neuroscience and people like Alan Bertho, the neuroscientist that I mentioned last session, they have done a huge deal of research to verify that yes, there are in fact the neurogeometry of our vision corresponds to some of these mathematical intuitions are in in fact responsible for them, the ones that are available in Euclidean geometry.
But I think Adolf Gronbaum is a very famous philosopher of science in 20th century, is known as the great rejecter, because he was very, very adamant to a stave of the threat of mysticism and ineffability in philosophy of science. And he has a monumental book called The Problems of Space and Time, The Fundamental Problems of Space and Time. Again highly recommend it, it's really probably the best book ever written on the problems of space and time in physics. And he says something like this, I can't remember paraphrasing that paragraph, but
But it says something like that if we are saying that, for example, our neurogeometry of our vision, the causal structure of our vision and our nervous system have been evolved in the sense that they afford these kinds of mathematical intuitions, then the first obvious trivial question, which is also a very, what you might call a pertinent question, be then why is that it took so many thousand years to people in fact come with something like Euclidean geometry or if you say for example and it is really the case that after Pouankara so many neuroscientists in 20th century
start to say that well you know when even when it comes to hyperbolic geometry we have identified some hyperbolic, you know, neurogeometry of vision that in fact affords this hyperbolic geometry. But then again, Grönbaum says that, you know, why is it then again 4,000 years or 2,000 years or 3,000 years it took for the most geniuses in mathematics to develop hyperbolic geometry if the affording machinery was already in place? So I don't think that we can simply resort to our evolutionary structure to look into the evolution of mathematical thinking.
Yes, I think we can talk about this, but I think, as I said, we need to be also cautious of simply eliding the distinction between mathematical norm and causal structures provided by evolution. In addition to that, the question I had is, what's the difference between intuitionism and Platonism in mathematics? I always thought of Platonism as the notion that the universe has a mathematical structure that matches our mind, something like Roger Penrose, for example. that intuitionism would be something that suggests
the procedures of mathematics are different than formalism in terms of discovering things and constructing things and even understanding them. But maybe it's agnostic about stating what the universe is. Yes, as I said, I mean, the only comment, one caveat I want to point out is that when we talk talking about Platonism, we are not talking really about Plato. because in fact Plato is very, very much, his idea of mathematics and logics very much in tune with what you might call the intuitionistic program. But nevertheless, commentaries of Plato, precisely because Plato is one of the, really one of the most misread philosophers of all times.
So the idea of Platonism is in fact being equated with Plato himself. But yeah, so Platonism is really this, is the ideality of mathematical objects being already encoded in the structure of physical universe. Whereas the intuitionism is the idea that there is in fact a causal continuity between the structure of mind-independent world and the structure of what you might call, not a structure, the conditions necessary for the possibility of mental cognition. So called cognition. So there is this continuity and where is this exactly continuity? In Kant it's really the manifold of intuition. Plato's, any person who is interested to look at Plato's
Plato's, any person who is interested to look at Plato's theory of mathematics, which as I said, very much in tune with the intuitionistic program, particularly intuitionistic logic, right before the Allegory of the Cave, the so-called analogy of the divided line. And one of the best commentaries given on the analogy of the divided line in Plato is a recent book that just came last year by one of the most, you know, famous and I think
most insightful commentators on Plato, Rosemary Desjardins. What's the title of that book? It's called Plato and the Good. Also another person, I think, let me check the name of...
I can't remember her birth name but I think it's Ori Harari. She is someone who has written extensively on Plato and Aristotle and also Euclid, and particularly this issue of what is exactly Plato's position in history of mathematics
and what kind of school of mathematics he's really defending. She's very good in fact, this person. In fact, I was thinking that maybe I should use one of her texts for one of the sessions that we are talking about Euclid. Any more questions? Do you perhaps have a list anywhere of mathematicians you think are cool? I'm interested in a long suit.
You go along with a lot of what my intuitions are leading on. Sure. Sure, majority of mathematicians are cool, but yes, I mean the problem, the only problem would be that, you know, I think it would be kind of text that is not essentially all technical because, you know, the cool mathematicians are also usually the most abstract pure mathematicians, which can be, you know, basically it's like extremely out there even for people who are really mathematicians. Right, because most people don't usually think about how the cognitive apparatus developed with the object.
But I noticed that with, for example, the intuition in school a little bit. Well, okay. I think William Lover is a very good example. and let me... William Laverre is a good example. Alan Cons is a very good example. Who else? For now these two. Alan Cons. Cons? Is that K? Double N, C. Oh. A and N?
O, O, instead of, yes. Alright, thank you. Sure, I mean, there are so many good mathematicians who are really coming from this direction, but as I said, he wants to go for a mathematician who actually is capable of writing non-technical philosophy of mathematics on his own practice. And these two are really two of them, those kinds of mathematicians. Particularly William Loeber, he is in fact someone who is extensively invested in philosophy, particularly German idealism and specifically Hegel.
Who, Alan Kahn? No, William Lover. Oh, William Lover. All right, thank you. Because, I mean, because I think Heidegger makes a big deal of how in the preface of Cheek of Pure Reason, he erased, so to speak, the part on originary synthesis. And Heidegger really emphasizes that. And I think it's like how you put it where someone who's very deeply investigated in their own school, it's because only at that point when, you know, there's a branch developing that do you get that originary synthesis that improve mode of cognition or new objective reality.
Yes, well there is a, if you want to read basically the most, what you might call the most thorough book investigating this by someone who is a mathematician who is also what you might call, even though he's not really a neuroscientist but nevertheless he's a philosopher of neuroscience is Jean Petitoub and particularly in a book called the Neurogeometry of Vision. There is in fact a short, you know this book Neurogeometry of Vision is like
2,000 pages but there is in fact there is like a small essay of a condensed version of this in the form of essay online I can't remember the exact title but it's something about eidetic variation and John Petitou is this Okay, I have one other kind of like very specific question about the text, which can you hear me? Yes, yes, absolutely. It's maybe kind of a naive question in a way, but so when he, he talks about the Poncarré-Britzos isomorphism, and so that on the one hand,
have a vestibular line on the other hand we have the visual line and that we're able to sort of transfer transfer a structure that that's how a lot of um that's what part of what makes a lot of mathematics possible and um i was i'm interested in the fact like what that transfer of structure is or how it happens like like he sort of takes it almost as a given like oh yeah humans can you know there's a vestibular line and there's a visual line and you can just transfer them you But what is it, if those two aspects of cognition aren't sort of isomorphic already, how, I guess you would say, well, we just evolved the ability to do that, but still on the basis of what? What is it that's similar between a line of flow and a visual line
that makes us able to have that metaphorical... Sure, sure. Berzo talks about this quite a lot in his book, The Brain Sense of Movement. It's the idea that there is nothing really in common between them. It's that basically they are three different forms of invariance. And actually there are three. One is the vestibular, one is the what you might call the visual line which is Which involves the operation of the ocular system or what's called the retinal imaging And the other one is really the saccadic jerk
The saccadic jerk is in fact has also the is a third invariance and these three invariance is what you might call to be the rudimentary forms of three things, the curvature, the orientation, and basically what you might call the stabilization principle. The vestibular, so actually this, the continuity, the inertial continuity, which pertains to the vestibular system, the saccadic jerk which pertains to orientational trajectory, and
the visual line which is basically the curvature and that also pertains to the retinal imaging. So the thing is that Umberto was a neuroscientist working for NASA and he did a lot of research on sensory motor and the problem of gaze stabilization in humans and animals. And gaze stabilization is simply this, that's, you know, this idea that something happens and we turn our head suddenly, the first thing that happens is that the psychotic jerk, which
is one of the fastest movements, takes place. And the movement, there's an incompatibility in terms of rate of speed or velocity between the movements of the neck and the creation of what you might call a continuum of images that are seamlessly integrated to one another. So the whole idea is that if between these three lines or between these three invariances afforded by the vestibular system, by retinal imaging and by the psychotic jerk, if they are not in place and if they are not stabilized or integrated with one another,
there is something happens and it's basically what you might call, you get a glitchy, simply an image as you turn your head in the sense that as if these retinal snapshots are not fully being stitched together. So this invariant, this total invariance that is based on the integration of these three rudimentary invariances allow you to see the picture, the image, particularly the moving image, whether it is your head is moving or moving or
an object is moving outside, it appears as a continuity, as a seamless image that you can in in fact, stabilize your gaze on it. So this is exactly, you know, Bertho's, and it goes through really, really thorough details about the architecture of this. But nevertheless, as I said, three rudimentary forms of invariance being integrated with one another. And actually, the two regions of the brain are responsible for this integration. is called the Wernicke and the other one is called Broca's area and these are really responsible for integration of vestibular signal, a spatial
transformation and visual signal processing. And of course Longo talks talks about this in his book called Physical Singularity of Life, Mathematics and Physics, where he brings the role of memory also into the equation in the sense that the very fact that this abstraction, this proto-abstraction of a continuous line, which is the integration of these three forms of invariance is being preserved through generation is precisely because it is always coupled with the model of anticipated action. And model of anticipated action, I mentioned last session, you know, in the deepest evolutionary
sense can be thought in terms of reward driven models of action that you need to in fact capable of making a model of these forms of action in order to anticipate how your situated navigation in environment might unfold. Whether you are a predator chasing a prey or you are chasing a mate in terms of sexual activities, evolutionary sexual activities, etc. But yes, so I highly recommend this book by Alan Bersow if you are interested to see the
way that he talks about the specific neurobiological architecture and neurophysical processes are responsible for the integration of these three forms of a spatial invariance orientational trajectory inertial continuity and curvature well so the transfer of structure is actually sort of an integration that's taking place all the time not just in the special case of of mathematics. Yes, yes, because I mean the whole idea is that it's not just about visual signal processing.
Are you familiar with the idea of receptive fields? No, I'm not. So receptive fields are these what you might call to be modules for processing sensory information. And the thing is that receptive fields share some of their modules with other receptive fields. For example, in terms of acoustic signal, auditory signal processing, some of the modules of auditory processing are being shared by visual and spatial information processing. So you can see that these receptive fields precisely by virtue of the shared modules
of information processing at different levels, the majority of the information that we think we get and we process, for example, visual sensory signals are in fact, are not just visual, they are fusion, sensory fusion of different receptive fields, haptic, you know, auditory, inertial, gravitational, so on and so forth. Thanks. That's very interesting. Okay. So last session we talked about the idea of this, what we were just talking about, this
what you might call neurobiological affordances for certain cognitive machinery that itself affords mathematical abstraction, mathematical thinking. And we talked about the construction of spatial invariances, precisely because spatial invariances are absolutely necessary for activities of the organism, for the construction of the model of anticipated action, for sensory motor locomotion, for any form of mobility.
And as I mentioned, at their base, all of the spatial invariances have something to do with a stabilization of motion. And in terms of people like Bernard Taizier or Joseph Pellongo or Jean Petitou, they, also as Stanisla Duhan, they argue that precisely because of the prevalence of these spatial invariances, in the sense that all of our cognitive abstractions are coupled with spatial
invariances. They are dominant in our mathematical thinking or in mathematical constructions. For example, Stanislav Drian in his famous book The Number Sense, he shows that the idea of for example numerosity, the counting of numbers in elementary counting of numbers is also being done by resorting to the use of these spatial invariances. And in fact, the very fact that we can recognize cardinalities between, for example, two different sets is also again tethered to what you might call spatial invariances and spatial transformations.
geometric invariances and geometric variations. Now today I'm going to talk about precisely this looking a little bit into some of the research that has been done in developmental psychology in pre-linguistic infants and emphasized on on the relation between the role of spatial invariances or spatialization activities in development of higher order but nevertheless still rudimentary mathematical abstractions,
Yeah, yes. So in a number sense, Stanislaw Drahan starts the book with this thesis that So, behind numerical mathematical thinking, thinking with numbers, there is also again a fundamental evolutionary cognitive affordance, and that's what he calls the number sense. And the number sense, he details this via different case studies and different kind of both neuroscientific and also cultural studies of this number sense in different
cases to show that this number sense is actually quite simple. It pertains to what you might call a rudimentary capacity for counting numbers. But when I'm saying rudimentary capacities, because this counting activity can only handle small numbers, usually up to five. And the interesting thing that Stanisla Duhane starts to investigate is that there is a correspondence
between this rudimentary form of counting that you instantly realize the numbers are less than 5, you know the difference for example 3 from 4, 4 to 2, 3 from 1. You are capable of making these comparisons for any number that is under 5. But the thing is that he notices that there is in fact a correspondence between this rudimentary activity of counting and what you might call sense of a space. So there is a correspondence between the number sense and sense of a space. how these numbers or groups that they're between the numericity of numbers their
cardinality and their spatial grouping their localization in the space this is one of the examples. You see, in the number sense, the way that he talks about this, he particularly looks into, as I said, to pre-linguistic infants and how they recognize the difference between numbers that are less than five. But one thing that happens here, and that's an indication that this number sense is correlated with our sense of a space is
this experiment that you have four dots on the above line and four dots on the lower line and these four dots standing one-to-one relation with with one another now as you see if you condense the lower line and add two more dots the child no longer is capable of understanding these transformation as an addition whereas in the first case the child could say that the four dots on
the above line the the above line and the lower the top line and the lower line are equal in the new transformation where you added two more dots but also you condense the distance between them the child would say that the top line is larger in number precisely because of the the transformation of the sense of space. It occupies what you might call a longer line, whereas the lower one that you add two more dots is now more condensed. So the sense of space affects the number sense.
This is one experiment. Another experiment is the... Let me try to make this a little bit... To prove that infants discriminate the numerosities 2 and 3, they are first repeatedly shown collections with a fixed number of items, say two left. Following this habituation phase, infants look longer at collections of three items right than at collections of two items. Because objects' location, size and identity vary,
only a sensitivity to numerosity can explain infants' renewed attention. So it seems that how children or what you might call pre-linguistic humans differentiate different cardinalities, different numbers from one another, at that rudimentary level, at the level of number sense, it can only be done via spatial invariances, size, location, occupying for example a larger expanse versus a smaller expanse, so on and so forth.
You can see this again in a different way in the evolution of number systems, what you might call modern number symbols. You see, I mean, in the advent of human civilization, the first instances of people marking numbers are usually done by way of these straight line markers or notches. So for example, if you have 50 kettles, you create corresponding notches for each sheep.
But the thing is that keeping track of the notches, adding to them or subtracting them, or in fact counting them, becomes extremely hard. Not only in the visual sense of tracking the notches, but also in the sense of what you might call the computational cost of the memory. So the first technologies of numbering or counting that are being invented at the dawn of civilizations are essentially technologies for grouping. So in order to keep track of these notches, like 50, 60, 100 notches that are standing
next to one another, is that you need to group certain amount of these notches, for example let's say 5, into one group. So instead of you drawing them next to one another, you start to put them in the groups of five. So you see you already compress information. And in so far as you are compressing information, the mode of a spatial encounter, visual encounter with your numbers becomes different. But also the computational cost of memory is significantly decreased.
Another example of this is really what you might call later on is Roman numerals. Roman numerals, why is that they became obsolete when the Arabic numbers arrived? Does anyone know? Because Arabic numbers have information encoded in the place of the number, the place of the digit, and that was more efficient? Yes, yes, absolutely. I mean, the idea is that, as you say, basically each number is represented by a different symbol and all you have is a very limited source of symbols in order to for you to be
capable of combining them whereas with Roman numerals for example you can see that writing for example number 58, that becomes extremely a laborious task. But when it becomes extremely difficult, almost impossible as a number, is when you are trying to add or subtract from it using again a different Roman numeral. So you see that in the evolution of numbers, at least the number symbols and the process of counting, there is a tendency to use different abstract spatial invariances to make this
process of numbering or counting more smoother and more computationally friendly. Another example of this is a problem of what you might call visual illusion. Let me get this for you. That while a spatial, the rudimentary spatial invariances
or sense of a space allow us to manipulate numbers or have a better number sense, but nevertheless they also create some troubles. Sorry, I have a problem sharing this screen with you. Okay. You see particularly this center, what you might call the solter illusion, you have already
given these dots, the white dots and the solid dots, a spatial configuration. So you would say that precisely because they have been spatially ordered that makes counting easier. But that's far from truth. In fact precisely because the solid lines are... there is more space between solid lines than the white solid dots and the white dots, it appears that the white dots are more than the solid dots, but the number of both are equal.
So you see that using rudimentary spatial invariance or sense of space, what you might call a spatial organization can also create problem for numbering and counting. Again, another example. These two disks, the one that is in this array and the other one that is, you know, has a disk form they are both equal the number of dots but we usually first of all the first thing is that while it allows us the one that has a disk format fully organized while it
is easier for us to count it than the one that is completely disorganized but also it It creates an illusion that in fact the number of dots are more than the one that is in this array. So that intuitive spatial organization, while permits rather effective manipulation of numbers, are less than five as a total of numbers grow it creates more problems than effectivity so there there is so a Stanisla Dehien's argument is that in
the history of mathematics we see a constant struggle between development of of new spatial organizations or senses of space and sense of numbers. Whereas rudimentary senses of space or spatial intuitions might not be adequate to handle might not be adequate to handle higher cardinalities or complex numbers. More complex spatial intuitions or geometrical organizations of space are not only adequate,
only necessary but also adequate for handling complex numerical structures. This is again an argument raised by Gilles Chatelet in the sake of the mobile, that this ultimately boils down to some sort of reinforcement effect or entanglement effect between arithmetics and geometry in the sense that when we look into the history of mathematics wherever there is a revolution in the domain of numbers we also we see a
a corresponding revolution in the domain of geometry, development of new senses of a space. And that's basically what ultimately geometry is, the art of organizing a space in different ways. Now, Now, but the question is that, so before going toward that direction, people like Bernard
Cortesier, Longo, Jean Petitou, they propose that when we look into the history of geometry, we are not dealing with pure mathematics. But we are dealing with a specific subgenre of mathematical universe where you are capable of using your what you might call evolutionary instantiated cognitive affordances and mobilize
them to create more complex, more abstract senses of a space or a spatial invariance, simply new tools for organizing other space. And then these new tools for organizing other space, namely the tools of geometry, have played an extremely significant role in the history of mathematics, particularly in conjunction with development of algebraic and arithmetic structures. We will look into this correspondence geometry, algebra, and arithmetic, particularly between that time frame that I mentioned from
antiquity to end of Renaissance, to see how analytic geometry is in fact used even in its most intuitive framework to construct not only new mathematical objects but also to construct new mathematical domains particularly you know its influence on algebra linear algebra what you might call developments of theories of magnitude etc Now, before we move forward, I want to talk a little bit about, as I remember I concluded
my comments in the last session about the difference between two forms of, two general forms of spatial invariance. The ones that are given to us by evolutionary processes, afforded by evolutionary processes, things like what you might call a proto-abstraction of the continuous line, and more complex forms of spatial invariances. They are complex precisely in the sense that they are being constructed out of these more rudimentary evolutionary afforded senses of space.
Now for the rest of today and also next session I'm going to talk about this transition from as I said, rudimentary senses of a space and spatial invariances and the complex ones, the ones that are being constructed. For people, again, for people like Longo and Teziers, one of the main, what you might call, mediums that allows in fact for the evolution and we are no longer talking about natural
evolution for the evolution of construct complex spatial invariance out of simple spatial invariance is something that they call gestures that you might see that gestures are in that nebulous zone between natural evolutionary afforded cognitive processes or proto-abstractions and fully complex mental abstractions. Now before I start to define what gestures are and how mathematicians use these gestures,
Again, gestures need to be understood in conjunction with two things, sense of space and sense of mobility. Before I go to this domain of defining gestures and how they have played a role in the construction of some of the more complex forms of mathematical abstraction, I would like to ask a question. Does anyone know what gestures are, even in the most intuitive sense of the word, gesture, and how possibly it can be connected with mathematical abstraction?
I think it would be the, in a semantic perceiving context for essentially a piece of information to indicate what type of information is to reveal where its trajectory is. that hint, that intuitive sign is what ends up producing the invariant of mobility that we perceive. Excellent, yes. Yes, yes, definitely. Anyone else?
well it's something that's geometrizable at least like there has to be some way to spatially represent what it is and it's a form of abstraction because whatever the gesture does isn't simply like the spatial movement that it can be encoded as it represents something it does some kind of work like the gesture of like hefting two different things to determine their relative waves like there is some sort of functional abstraction that accompanies a spatializability and it's an operation. Yes okay yeah so they need to be geometrizable or simply being open to a spatial different
forms of a spatial organization but then what would be the difference between a gesture and a diagram? I'm not sure. I'm just kind of guessing based on what you're saying, but perhaps a gesture is something like a gist, and that there is like an affective opponent to it. Yes. Can you elaborate a little bit on this affective component? Well, that it's in a way felt and sensed, but it is not like epistemological or ontic and that it has a like generative power sort of like a driving towards the urge or like driving towards abstraction or I don't know something like that.
I mean again I'm kind of just... Sure, yeah, yeah, definitely the generative generative feature of it is extremely important, yes. How about, is it something like an active construction in the sense of energyometry you would use a straight edge and a compass to do something? In other words, you're creating space when you do it? Yes, yes, absolutely, yes, creating a space, yes, definitely. Yes, basically, this is, yeah, sure. Maybe a diagram is like the semantic form and it's merely implicit in the gesture.
Implicit in the gesture. Well actually it is the other way around that the gesture is usually implicit in the diagram Um I was thinking first of all about just gesturing in relation to counting and arithmetic like with digits bodily digits I'm also thinking I'm thinking in terms of like art practice really the difference between a gesture and a diagram but gesturing seems to be a little bit maybe more experimental or like less locked down than a diagram, which seems to be like diagrammatic ultimately once it's finished. So maybe gestures are a mode of navigation that are always retractable.
Stuff like that. It's more of a creative space. This is my instinct. Yeah. Well, that part that you mentioned that before being finished, when they are finished, they become diagrammed, that's again an excellent point. Yes, basically, you see, it's really hard to really define gestures, precisely because they are not mathematical objects. They are actions. And for the most part, there are actions of mobility open to what Jake said, a spatial encoding or a spatial organization.
There are also generative in the sense that they can, what you might call, there is this line by Paul Valery I think that they can create ancestry or a dynasty or dynasty of other gestures they can evoke one another. So in a very, I think, there is a, so when you're looking to particularly in new approaches to philosophy of mathematics, the topic of gesture crops up quite often.
But then there is also a suspicious sense of mysticism about gestures, you know, precisely because they resist fixed formal mathematical definition, they are also susceptible to ineffability, some sort of gesture mysticism. I have seen this among so many mathematicians, a friend of mine, that when you are asking what a gesture is, they say, well, gesture is being defined by another gesture and so on and so forth. So there is a... Gestural regression in defining gestures.
Daniel, can you mute yourself? Can you mute yourself? Um... I think I can hear my voice from one of your... Loud speakers. Loud speakers. Sorry. Daniel, I'm having trouble muting you from my end for some reason. Can you mute your... Can you mute your mic? I think it's coming from there. There seems to be sound being generated when it's turned around. when it's turned around. Yeah, stop for a second. Okay, yeah, awesome, yeah, thank you. Okay, sorry, Reza. No, no, no, fine. And one of the best critiques of gesture
has been given by Brian Rothman. I can't remember the name of the book, with something called Beside Ourselves, but I think you can check it online, Google it, Brian Rotman. He's a mathematician. He precisely points out to this mysticism of gestures. But nevertheless, I think there is, there's definitely a grain of truth to importance and the significance of gestures in mathematical creativity, particularly in the sense, in the constructive sense of mathematics
rather than the proof or formal sense of mathematics. I think we need to, rather than defining it, at least single out some of the features and look at it by way of some examples, you know, what exactly counts as a gesture. So in a broader sense, I think gestures, as I mentioned, are in that nebulous zone between embodied form of heuristics, you know, evolutionary afforded heuristic processes, and conceptual
abstractions, you know, more complex forms of abstractions that can be said to be mental acts. It is when, now the difference between them and simply those simple forms of spatial invariances like a gesture of the predator mobilizing its body toward the prey, is that these gestures are moving in the opposite direction, in the sense that mental acts are being reconnected with the heuristic processes, rather than heuristic processes, in fact, being mobilized toward
mental acts. It's what you might call to be a de-stabilization of the stability of mental abstraction by way of heuristic processes, particularly as Hunter said and a number of you mentioned, embodied affectivity, embodied cognition in an activist sense. A good example of this of course is Archimedean principle of buoyancy. I mean Archimedes is quite famous for being a mathematician who heavily used gestures
in order to arrive at new scientific discoveries. utilizing, for example, resources of physics to make a new mathematical invention using a mathematical resources of a particular mathematical domain in order to, for example, create an engineering machine. So again, gestures not only have these characteristics, but precisely because what you might call them are forms of actions or action abstractions that can be deployed in different fields precisely
because they pertain to some basic spatial invariances or basic forms of spatial organization that are beneath the structure, the domain of physics, the domain of engineering, the domain of mathematics and other disciplines, other mathematical disciplines. Now the famous example of Archimedes is, as I mentioned, the principle of buoyancy, the The idea that, you have heard the story that Archimedes is being given a votive crown by
the king of Syracuse. The king is suspicious that the votive crown might not have pure gold in it. That basically the goldsmith has put silver or some cheaper metal into the votive crown. Now the problem for Archimedes is that he knows the principle of density, he knows the measuring of weight, he knows what a volume is, so he has access to certain range of physical mathematical abstractions, namely for example the relation between density, weight and volume.
So he has the concepts. The problem is that the vortic crown has an irregular shape, so he cannot represent its volume and hence he can't really accurately determine the density of the metal used in it. We know that gold is denser than, for example, silver. So if he can extract or if he can measure the accurate volume of the votive crown, he would be capable then to say if this votive crown is pure gold or it has some cheaper
metals in it. Now as I said, in so far as the shape of the votive crown is irregular, he can't do this. can't measure the accurate volume. The only solution would be for him to melt the crown into a regular shape, but then that would completely ruin the crown. So he conducts a mental experiment which according to the historical accounts or the legends around him, he arrives at this mental experimentation in quite an accidental, arbitrary way. He notices, so as I said, I want to distinguish this movement in the opposite direction, rather
than moving from those embodied evolutionary heuristic processes toward conceptual mental abstraction I want to say that gesture in fact is moving in the opposite direction from linguistic mental abstraction toward those heuristic processes. So he notices that once he when he goes to the bathroom and he goes to his bathtub he notices that an amount of a certain amount of water is being displaced as and then he starts to think that there
is a correspondence between the amount of water being displaced and the weight volume and density of his body. Effectively, his thought experiment is that he puts himself in place of a gourd of water. He imagines that he is, in fact, a portion of liquid in this system that has been displayed. And his body, namely the displaced water, he starts to think about this as this virtual space where he can replace his virtual body, namely the gourd of the water, with the votive
crown. So he essentially makes these connections by way of some sort of what you might call embodied interaction with its environment, with the system, the bathtub. And precisely he starts to see through these new points of connections that are being emerged through this embodied thought experiment, he sees new connections between what he already he has, the kind of abstraction that he already has, namely the volume, the concept of the volume, the concept of density, and the concept of weight.
So that's how he manages to come up with the principle, and the principle of buoyancy, to start to measure the volume of the votive crown using the weight of displaced water. Well, this is an embodied thought experiment, an embodied thought experiment in which the embodied affectivity destabilizes the existing relation between given conceptual entities, volume, density, and weight.
Through this process of destabilization, parameters between these concepts are being reinvestigated and reformed. And this is essentially a form of gesture, in the sense that the nature of this thought does a few of those things that you mentioned. It generates a series of new problems otherwise inaccessible from the perspective of conceptual abstraction alone. It does redefine the parameters
between existing conceptual abstractions. It mobilizes the current abstract resources that Archimedes has using what you might call all pre-existing, but nevertheless uninvestigated forms of heuristics. But most importantly, what makes this a gesture is that a new form of
A new dialectics between a stability of thinking and de-stabilizing creativity or construction emergence. In fact, Archimedes uses this dialectics between a stability of mental abstraction, current existing mental abstraction and destabilizing effect of embodied heuristics to go back and forth and create a number of principles, the principle of buoyancy, the principle of density, a new relation between density and weight, a new relation between volume and weight,
so on and so forth. The history of mathematics is full of these, in fact, this kind of process of mobilization of gestures or what you might call dialectics between a stability of current abstract resources and the de-stabilizing effect of heuristic, intuitive methods. Now, as I mentioned, in all of its forms, gestures has something to do with mobility,
with precisely this generative aspects of this dialectic between stability and instability. Whereas diagrams try to simply capture the finished part of this mobility, the semantic explicitness of this mobility once it's already being concluded, the mathematical creativity, at least the gestural mathematical creativity that mathematicians often use to arrive at new methods of constructions involves with what you might call melting the frozen, the
semantic frozenness of the diagrams and again go back to that domain of non-equilibrial dialectics between the stability of current abstractions, conceptual abstractions, and the destabilizing effect of heuristic, intuitive processes. You can see this in every diagram in mathematics. I mean, one of the most famous examples of this, that mathematicians often try to see past the diagrams, simply tap into the domain of this non-equilibrial dynamics, which is
the domain of gestures, you can see it in another, for example, mathematical objects devised by Archimedes, the Archimedean spiral. So what is really Archimedean spiral? Archimedean spiral is simply a two-dimensional picture of a spiral that starts from a center and diverges from the center and moves around. This object, this diagrammatic object, doesn't tell us anything about how it is constructed. a mathematician like Archimedes and later on other mathematicians in fact looked into
the this this spiral and came up with different methods of construction is that they look they try to see how what what sort of mobility or what sort of inestability this mathematical object or this geometrical representation represents. From at least Archimedes' way of thinking about this spiral is that in its frozen two-dimensionality, this spiral can in fact be said to be constructed by a point that diverges from the center
in a uniform accelerated pace along a central axis. So you can imagine this, the mobility of this point that makes this spiral as a point that starts moving like these dots that are condensed, like dashed marks that are condensed, And as it moves toward the edge of the spiral, the distance or the gap between these marks or points are being increased. So Archimedes in fact use this in fact to talk about what you might call a proto-account
of uniform acceleration. That in fact Archimedean acceleration can be understood in terms of protokinematics, a point that moves from a central axis to uniform acceleration. Other examples of this would be… One second, let me… And this is the Archimedean spiral, as you see.
It represents a form of mobility of a point that moves away from its central axis at a uniform speed. Now Archimedes essentially starts to, instead of looking at the concluded semantics, concluded abstraction that this spiral represents, he starts to look at it as a form of mobility. And the mobility here, the mobility of the point that moves away from the central axis at uniform acceleration, can be seen as a mode of construction for this spiral.
Depending on how you interpret this form of mobility, or how you can represent this mobility of this point, you can in fact come up with new modes of construction for this mathematical object, namely this spiral. You can see to the left of the Archimedean spiral two different methods of making a circle. So when you are looking at a circle, that's just a diagram. It doesn't tell you anything about how it has been constructed. it. Once you melt this diagram, once you gesticulate toward its, what you might call, forms of
mobility, then you can in fact come up with new modes of construction of the circle. So in the first one, you see a homothetic construction, namely construction of a circle by means of dilation and contraction but that only works if you have the principle of mathematical homothesy or self similarity in place and we know that self similarity or a scale invariance is a main feature of Euclidean geometry so So this method of construction, dilating a circle or contracting a circle, in the sense
that its internal properties are being left intact, can only happen in Euclidean geometry, namely a scale invariances. So the left one, you see it, that the curve you simply dilate or contract it, and that creates going back and forth between different scales of a circle whose internal properties are unchanged as you scale it up or down. Whereas to the right you have a completely different source of mobility. You can create a circle using a curve, but imagine that your curve was something like
an elastic band and you can squeeze it or pull it to make the circle from this elastic band so this represents a different form of mobility a mobility in which is based on not a scale invariance of Euclidean geometry but the continuity of functions so you have a topological transfer or topological transformation of your elastic band that you have pulled and some of its internal properties have been changed and some are still unchanged the first one is
something that can happen only in Euclidean geometry. The second one is essentially a homeomorphic function and you do not have homeomorphic functions. You cannot make this circle if you do not have something like topology. So you see depending on how you interpret the mobility responsible for the creation of of a certain mathematical object, in this case for example a circle or a spiral, you can in fact develop new forms of mathematical abstractions, mathematical modes of constructions, which in turn also elicits new mathematical fields.
The difference between Euclidean geometry and homeomorphic functions in topology. Questions before I move forward? What was the length between scale invariance and Euclidean geometry specifically? Like why is it that scale invariance only arises in Euclidean geometries if that's what you said? Yes, scale invariance, we will talk about this when we are looking at Euclide. Scale invariance is really the main feature of the parallel postulate. What is parallel pasteolate? So you have two different versions of this. One is that two parallel lines never intersect, that's a kind of modified version of Euclid's
own pasteolate. Or given a line and a point outside of it, you can only draw one line that could be said to be parallel with the first line. This is John Playfair's version of parallel postulate. Now, when you look at the parallel postulate, precisely because of these properties that two lines never intersect, or you can only draw one line going through a point outside of a, you know, you're given a straight line, and that's any other, that can only be parallel to it. can only be parallel to it, you see that this amounts to this idea that the kind of constructions
that the parallel postulate allows are all about preservations of scales and properties. of properties given the invariance of a scale. So for example, think about this, think about that you have one point in one of your parallel lines and then you create from that point on one of your parallel lines, you can create infinite triangles by connecting it to points on your lower parallel lines. Okay? So you are basically changing the scales of this triangle.
But the thing is that precisely because two parallel lines, the scale between them is always invariant, the distance between two parallel lines remains the same, this creates actually a very strict form of preservation of structural properties. In Euclid's elements you can use parallel postulate to create all sorts of mathematical objects and that's the whole point of it. But insofar as parallel postulates in Euclid has been framed in a way that the distance between two parallel lines is always the same, that imposes fundamental constraints on how
structural properties are being preserved under transformations. We will look into this in terms of different kinds of constructions that are basically based on the parallel positively. But all you need to know is the idea that a scale invariance ultimately boils down to the scale invariancy of two parallel lines in the sense that no matter where you go, how you extend these two parallel lines, the distance between them will remain the same.
distance that never changes, and that's really the property of the Euclidean geometry, as I said, preserves structural properties in a very strict sense. And you only have, as I said, you only have the preservation of structural properties in this sense, in the sense of a scale invariance in Euclidean geometry. You can imagine, you have a Euclidean, for example, triangle, you scale it up and down, okay? You see that essentially all of the part-whole relationships in your triangle remain the
same. Even though, for example, the sides grow, the sides of the triangles grow, the angles might grow, but the part-whole relationships, and that's what I mean by a structure of properties, the part-whole relationships between the idea of a triangle in Euclid, its sides and angles remain the same. Simply you have congruence, you have equivalence. So in short, in a space that's not curved, you can transform an object without distorting it? Yes, yes. You might imagine Euclidean universe is like this universe that conveniently, if you zoom in and out of an object, that object always remains the same.
You simply, it forms in the sense of part-whole relationship never changes. You simply magnify it or dilate it or contracts it. Hence, the idea that the structural properties, part-whole relationships, do not change under scale transformations and object always is equivalent with with the moment before the transformation this is the whole idea of congruence in Euclidean geometry the idea of equivalence and in the case of parallel lines is that like is that
logically related to the uniqueness of the construction of the second line of of the parallel? Yes. Yeah, so those are related, that uniqueness and . Yes, yes, yes. I mean, that's why I mentioned John Playfair, precisely because John Playfair makes this explicit, that if you have one straight line, and you have a point outside of it, you can only draw one line through this point that can be said to be parallel with the previous line, hence the uniqueness. Right. Cool, thank you. Welcome. Oh, I can see that what literature on Archimedes and Euclid would you recommend?
Well, there is this book called Genius of Archimedes by Springer that came a few years ago. It's a fantastic book. I can find it for you. I think I have a PDF of it. I can put it on Google Drive. As for Euclid, I think with Euclid it's kind of strange. Because there are so many commentaries on Euclid, but they are all similar. I think the best commentaries on Euclid have been given by those two female philosophers
that I mentioned. One is Ori Harari and another one more recently by this fantastic philosopher named Danielle Macbeth. And Macbeth has a number of fantastic essays on Euclid, but also she has a book that came a couple of years ago, it's called Realizing Reason. It's about diagrammatic reasoning from this time of Euclid to the notational system of Frigge and and beyond it's a very very good book
who was the first female you mentioned I I will get the exact spelling but I I think, how are we, this is her name. And yes, the other one is Danielle McBeth. All right, thank you. I'm kind of curious about the modal status of gestures. So like, are they necessary or contingent or contingent or they sort of have this like synthetic. I think they are counterfactual. They are what you might say that they are,
you can approach them as counterfactuals. I mean, the whole idea is that gestures are really tools for hypothesis construction. Okay, okay, assuming. There is a good actually book on this if you want to look at it. It's a book by Lorenzo Magnani. Lorenzo. It's called geometry. Again, I can put it under Google Drive,
where he actually talks about the, you know, the, what you might call, modal interpretation of gestures gestures and their role in hypothesis construction and counterfactual playing the role of counterfactuals. I mean, Chatelot talks about them in terms of metaphors. And he has a famous line in the Stake of the Mobile that he compares gestures with Trojan horses. in the sense that they mobilize the resources of one domain of thought and they deploy these resources in a new domain of thought, exactly like a Trojan horse that smuggles the soldiers.
And once they deploy the resources of the old field in the new field, they instigate new problems. They basically create points of tensions. And these points of tensions, Chatelet shows that they are usually rife with opportunities of construction or what you might call creative forms of abstraction.
Can you recommend a couple of books that give some sort of synthetic overview of the connection between various domains of mathematics. I mean, I notice you have conceptual mathematics by Laver, but I was wondering, beyond category theory, are there any books that you like that just try to look at the relationships within the various domains from a modern book? Sure. Sure. I mean, the simplest book on this front is by Klein. The two books on mathematics considered from... Yes, yes, but also
there is also, you know, a different version. I mean, it was like he wrote so many books. But yes, I mean, anything that he wrote was really like, you know, really... Although the mathematics that he talks about are... you might not call them modern mathematics by any sense but nevertheless at least talks about some you know important connections but also I will find so I have yes I have some some more stuff I can share who was this guy again you cut out right when you said it Klein? Felix Klein. All right. Okay, I will, I will, I have some, some books. I will
differently put them in a Google Drive. These and the ones that I mentioned. I mean I could ask more. I mean like the relationship between or the relationship between gesture and affect is kind of mysterious to me. like how a gesture is effective? I think affective, probably affective, we need to, we need to kind of like define the affectivity
or restrict the affectivity of a gesture. I think what you might call the affectivity of gesture is affectivity in the sense of embodied cognition. Simply, again, remember, so it's interesting that I think what these people talk about, gesture, they are really referring to the second function of imagination in Kant, the productive imagination. so what is really productive imagination so imagination has two two really two functions one is called the primitive function the intuitive at that level all
you have is that you are you are dealing with a singular representation of an object construction of a singular representation of an object qua an image so this is the primitive function of imagination this the second function of imagination namely the productive function of it is when you manipulate the singular representation by way of concepts by applying different concepts to this singular representation. So insofar as in Kant imagination and you know the manifold
of intuition is essentially a causal thing it had absolutely at the level of intuitive in Kant we are not talking about concepts we do not talk about categories of understanding it's simply a causal structure you might interpret it in terms of embodied cognition precisely in the sense that an activist talk about cognition at that point simply you know the physical embodied interaction with the environment so it's the idea that how this singular representation that has been afforded by no enacted interaction with the environment can be manipulated by different concepts this is the
function of imagination as a simulation engine it sounds a little bit similar to what purse would have called abduction yes yes absolutely yeah yes yes absolutely and you you mentioned before that there's something mystical about gestures or in the sense it's difficult rigorously to find them as why are enumerating characteristics and giving examples, can you talk about the limits in defining gestures? Yes, I think it's basically, you see this idea of gesture especially became,
you know, like what you might call a buzzword after, you know, the kind of domains of mathematics like category theory became pop mathematical domains and you see that basically category theory precisely because you can make any object via morphisms and morphisms are essentially pointers really literally pointers and these pointers you know again evoke the idea mobility in a quite literal sense again this this became a really like a hot topic the idea of gestures and that gestures have some fundamental ubiquity
in in the domain of mathematics but then you see all of these things that well you know you are asking that what is the gesture they say well gesture is you know something that points toward other pointers and that's basically something things that came through again category theory what is a point well point is nothing but a pointer endowed with a limit function or a limit case but then you say that what is a pointer they then tell you that any pointer can only be decomposed in terms of other pointers pointing to it so you get a form of regress the same thing with gestures that precisely because of its generative
and ambiguous mechanisms you never get a fundamentally you get never get the definition of the gestures but simply the regress of the definitions that a gesture is something that gesticulates and other gestures so This, I think, while by itself is not essentially a point of mysticism or susceptible to mysticism, in so far as, in fact, so many mathematical definitions are like this. and yes okay at the level of mathematical definition if we have circularities but the
circularities can be virtual circularities part of what makes these concepts maximally stable but if you use these this circularity in order to talk about really you know the the the fundaments of mathematical thinking I mean take them out of context then I think that there is a huge room for these mystical approaches to what you might call the evolution of mathematical thinking and I think even Châtelet with a fantastic philosopher of mathematics fall in this trap I think
more extreme case of it would be a friend of mine, Grino Matsuella, which again is a fantastic mathematician, but the way that he always talks about gestures, you ultimately get some sort of rococo romanticism of gestures ad infinitum. And I think these are essentially and I think these are essentially part of again that's you see that's idea that where what is really the fund fund was really the foundation of mathematics so in set theory you get precise definition of set being the founder and sets being the fundamentals of mathematics and then people are started to liberate themselves from this kind of what you might call pigeon-holed fundamentalism
but that led to some sort of again some you know anarchic approach to the question of foundation in mathematics via category theory essentially a bad interpretation of category theory and particularly the philosophy of gestures we are you know all of the newest studies about you know a spatial invariances and the deep connection between neurobiological structure and mathematical abstraction um that that leads me to like a little bit more of a
speculative question which is so a gesture is sort of the kind of thing that someone like Badoo or Deleuze would base a philosophy of the event on, right? So like a virtual problem or an event for truth procedure. And I'm curious the degree to which you feel like this kind of mathematical entity can be translated to problems of social philosophy and politics and ethics. or if there's sort of an obfuscation in that. Hard to say, I genuinely need to think about this. But yes, I mean, even though I don't want to make the, take this, I don't want to over determine this analogy
analogy between event and gesture but yes I can absolutely see the connection and this is really the problem of the virtual and you know the gestures of virtual entities as well but that's why I precisely because I still need to think about this I I think we need to suspend inflating the model of mathematical model of gestures or heuristic model of gesture to the more social aspects otherwise because because so many things can go wrong in terms of you know the definitions in terms of how we are misapplying something that has a even though it is vague even in mathematics but nevertheless you still can control you still have some intuitive understanding of what gestures
are about which are basically what you might call intuitions of mobility gestures are about intuition a spatial intuitions of mobility there's there's vague and then there's big yes there is a I mean you can see that on both sides of this panorama you have too many ambivalence so I think best not to connect and reinforce this vagueness. Right, to remain silent. Yeah, or at least I think there are there will be actually more interesting things that we can do with gestures and how we can think about them philosophically and the question of you know abstraction really the question of
abstraction essentially we still don't know what really abstraction entails. So I think in that specific domain I think it would be actually fruitful to utilize in fact the vagueness of the gesture precisely because the power of the gesture is ultimately in its vagueness but a vagueness that can be parameterized, that you can in fact control it. And again you can see a very, very actual connection between this, what you might call useful vagueness of gestures with the usefulness of vagueness that Peirce talks about.
What does parametrized mean? I know metric is in it, so it must be about measure. Parametrized, I specifically meant it in the sense that spatially parametrized. We know that mobility, based on what kind of spatial parameters you are providing, essentially the kind of geometrical encoding that you have, you can interpret mobilities differently. So it's a coordinate system that you can apply? Not essentially coordinate system, but you might say different forms of geometry
or different intuitions, geometric intuitions of a space. You see, let's think about this, and next session I will definitely talk about this, about gestures and more examples of this. The problem of force in physics and how you articulate force is essentially entangled with how you represent the space or the surface to which a series of force are being applied. For example, again in our Archimedean example, we are essentially dealing with a fluid system
and hydrostatic pressure. So you can imagine that basically the body of Archimedes disturbs the hydrostatic equilibrium of the fluid system. New forces are being applied. Imagine just as how Archimedes thought of his body being a gourd of water, namely just a virtual cutout in this fluid system. You have this virtual cutout in the fluid system. From different points, forces are being applied, forces of the liquid, of the fluid that has been disturbed are being applied to this virtual cutout. Now in order for you to formulate the new state of the fluid system and measure the
hypostatic pressure you need to be capable of integrating these forces are being applied to this virtual vacuum by simply representing the area the surface area of this virtual cutout of this virtual vacuum depending on the kind of geometry and the geometrical method that you are using to integrate these different forces that are arriving from different positions and squeezing fluid to this virtual cutout, you can in fact give different measures of hydro-static pressure in your fluid system.
The intelligibility of force depends on the sophistication of how you are geometrically representing this force. Let me... Does any of you know anything about the so-called Kashi integrals?
So, Cauchy integrals is a form of integral made by Agustin Louis Cauchy. And it precisely tries to answer this question. Let me see if I can draw something for you here. Mm-hmm. OK.
Let me... Okay, so can you see the white screen at the middle? Yes. So you have a fluid system. You have something at the middle. ways that it turned into a triangle all of a sudden so so this is essentially what Archimedes did you can see this as a system of fluid and this is the cutout
and this is the water or the fluid so what Archimedes did in this experiment by putting himself in the place of a gourd of water is that you can imagine and the amount of water that was displaced. You can imagine the body of Archimedes as virtual cutouts, which also equals to the amount of the body that has been displaced. Now imagine that Archimedes think of himself as a virtual vacuum being opened up into the
the body of this fluid system. Now before Archimedes went to the top, the hydrostatic pressure in the top was in the state of equilibrium. But once he introduced himself there, it disturbed this equilibrium. Right? So now if you imagine his body as this virtual cutout or this virtual vacuum at the middle, then you can see that what happens is that as soon as this virtual vacuum opens up, basically you get something like this. You get... Oops, sorry.
you get the liquid tries to go inside this vacuum in order for the system to retain and regain its equilibrium. So, essentially, these forces that are being applied to this virtual cutout, to the surface of this virtual cutout, need to be represented for us to understand what would be, what kind of transformation this fluid system need to undergo in order to regain its equilibrium. Now, this becomes the problem of measuring the surface area of this virtual cutout
cut out in order to represent the kind of forces that are in place in this disturbed system, this equilibrium system. Now Kashi tries to come up with a method of integral that essentially can measure the surface area of this virtual cut out. Remember this is just a virtual, there is no such a real vacuum in your body of water. Imagine it was just a virtual thing. So Kashi creates a complex point, I will introduce, I will, when I'm talking about gestures next session, we will look at it how Kashi makes this thing, construct this new point.
It constructs a point, a complex point, and an integral for it that can measure in fact this virtual cutout, and hence can articulate the kind of force that is in place for the system to regain its equilibrium. What was this guy's name? How do you spell it? Oh, it's Koshi.
Thank you. So even with the quarter of an hour late start, we're like right about at the end here. we have like a last question a couple of questions anything to tie up yes questions I had kind of a metal level one about the diagram like it's not something that you really would have to pull it back up but in terms of the nature of the surface of the cutout volume and the nature of a virtual volume like a cutout could be any surface or any have any shape be any
proportion of the volume of the fluid is that surface insofar as you mark it on the diagram sort of indistinguishably a physical representation and a mathematical representation or concept well strangely enough in middle ages using the Archimedes buoyancy they actually had a physical representation of this and their method of measuring this let me do this again the measure the way that they were measuring this is by way of the method of exhaustion so So let me, I have a problem here. My iPad is frozen for some reason.
Okay, so, okay, I will try to resurrect this iPad next time that we are going. But basically it's like this. that for example that virtual vacuum there is actually they were thinking in in terms of an actual geometrical representation like it for example a curve a coarse curve right now you can in fact solve this problem using the method of exhaustion for example what is the method of exhaustion which is actually an Archimedean again way you can introduce a big a square or a rectangle into this curve that covers the majority of it of this closed curve then add more
try and rectangles on top of this so that as the these rectangles become a smaller and smaller they also cover the nooks and crannies of this curve so once Once you have all of these rectangles, then you can calculate the rectangles and come up with the area, the surface area of your curve. This is the method of exhaustion. But nevertheless, it is fundamentally inaccurate and that's why Cauchy had to in fact construct a new point that represents any of such curves. Okay, and that's the transition from like a numerical limit method to calculus to an
integral. Yes, yes, yes. Thank you. Other questions guys? Going once, going twice. To be brief, so basically the rectangle method would be a lot like the triangle works when computer graphics images are formed. Oh absolutely, yes. Yeah. Yes, absolutely. I mean, you basically, again, this is really the whole idea of method of exhaustion really invented by Archimedes. For example, in order for you to make a curve, you can use triangles, as you say, and make
these triangles ever smaller, and to the point that they become so small that they can in fact cover and create the curvature. But the thing is that this method, as I said, is highly inaccurate precisely because it does not have infinitesimals and does not have the calculus. It's kind of funny that it's called exhaustion, given that that word is like very specifically the exhaustion of the totality of all possible space to fill, but it's exactly just an endless like fractal series of places that are not filled. Yes, yes, absolutely, yeah. But nevertheless you would be really
surprised that we will get back to this method of exhaustion when I talk about Nicole Orem's representation first the invention of the first mathematical graph but also the representation of uniform acceleration something that Galileo almost 300 years after Orem reinvented again he used this yeah Yeah, absolutely. Basically, the thing is that now everyone agrees that Galileo really didn't do that much. He simply resurrected Aurim's method of exhaustion in order to represent uniform acceleration to talk about, you know, a different picture of cosmos.
Yeah, so that's like, I remember seeing that diagram of Aurim's on your blog at one point, but I didn't realize that's like 13th century 12th century yes yes yes yeah I mean Orem is like the second Archimedes really I mean in the sense of he's capable of using some sorts of a highly intuitive way of thinking to go back and forth within engineering between optics mathematics kinematics and create these absolutely phenomenal you know inventions or discoveries in fact is you see so many people is still think believe that the
invention of classical perspective was due to painters but in fact Orem is the first person who came up with the idea of a classical perspective almost 80 years before any artist who did it in Middle Ages. And just refresh me the classical perspective that's before the vanishing point right? Vanishing point is modern and then classical is... Yeah but in fact Aureen classical perspective has vanishing point and you you would be surprised that it's basically when Aurim saw a connection between the optics and diagram of uniform acceleration.
And you can easily intuit this, how this might be connected to the diagram of classical perspective. It's like the idea that you drive in a road in two parallel lines, will meet each other as you accelerate forward. forward oh so like the zoom is acceleration there is like visual yes uh-huh yeah yeah oh that's very cool uh okay sorry everybody i mean i could just keep doing this for a long while when we're supposed to be um so are we so we're still 2 30 again a week from today are we skipping the holiday or yeah sure cool And I and Jake can you remind me
Sometime today tomorrow, so I upload those books the Google Drive Yeah, I'll definitely remind you I'm also gonna pull out like try to extract the list of the stuff that's been mentioned in the sidebar Excellent excellent All right great all that stuff will get sent out over the next few days guys, so stay tuned Unless you have anything else that you want to do Reza Are there any business you want to finish up with? It seems that everyone is saying goodbye. A democratic decision, something like that.