Hello and welcome to the first session of math and ideas from antiquity to Renaissance with Reza Negra Thani. I'm going to pass the mic off to him now. Thank you. Okay, hello everyone. So we are going to slowly start our class, our first session. Today I'm I'm going a little bit on the introductory side, start to introduce some of what we are going to do and how we are going to approach some stuff. So I think the best thing would be just give you a very brief rundown of what the course is going to be about.
So this session and next session, we just talk about a little bit some of the broad conceptual basis about the evolution of mathematics which can be also said to be relevant to what's going on in contemporary mathematics. Then from the third session we are going to start looking at very basic, but nevertheless important examples.
And these examples, each session is going to highlight a specific pair of concepts, fundamental to mathematical thinking. So I'm going to present at least a dozen of, you know, basically it depends on how we can move forward, give you some of the, what I would think are fundamental conceptual problems in the entire history of mathematics. I'm going to present them in the format of oppositions between two fundamental concepts,
for example, discrete and the continuous, balance and imbalance. We are going to see why is that they are important for mathematical thinking and precisely in far as some of these concepts, particularly in modern 20th century onwards in contemporary mathematics, they are extremely, have been transformed toward abstract sophistication. It would be, I think, really hard to, without any mathematical training, to underline the importance of these concepts via contemporary mathematical practice.
So that's why I decided instead of giving you examples from contemporary mathematics, which is of course also a disappointment, but nevertheless it's necessary, starting to give examples at a very, very basic level. So the kind of examples that I'm going to provide extend from what you might call the dawn of civilization, human civilization, from early antiquity to late Renaissance. As I said, precisely because as we move forward we know why is that, I particularly emphasize
this period, precisely because while mathematics is formal discipline, even not in the sense of formalism that today we understand, but nevertheless you have a minimal formalism. But there is also an intuitive mode of presentation that involves in the kind of mathematics that's going on in this time period that is extremely accessible. We still can, regardless of our background, we can still look into it and we can see what's going on. It's not fully foreclosed to the layman. man. And the reason, another reason is that throughout this period, particularly not from early antiquity,
end of antiquity to the late Renaissance, there is a dominance of analytical geometry. analytical geometry as we will move forward introducing Euclid's elements and stuff is it has a very intuitive diagrammatic representation of the mathematical universe. By intuitive I mean it both in a common sense intuition but also more specifically in the Kantian sense of intuition in the sense that it
involves with what you might call diagrammatic reasoning with visual notations so it's very easy to track the manipulation of symbols the manipulation of a theorem or how you solve a problem by way of a method that in one way or another has pictorial representation so that's one so basically this is this is the framework for this course. As I mentioned, the two first sessions are going to be just broad discussion about the
The origin of mathematical thinking and a rather broad understanding of if really this is where the mathematics is coming from or at least we think mathematics is coming from, then given these conditions, given these forms of abstractions or cognitive technologies that mathematics deals with, then what would be the future of mathematics? So this is the first two sessions and everything else after that would be introduction of a pair of concepts that as I said I think are, you cannot have concrete mathematical discipline
whether you are working in algebra or in geometry or in topology without at least a basic understanding of these fundamental concepts. So, today I'm going to talk about some of you I know that are familiar with some of these things. It's nothing new, I've been talking about it before. But nevertheless, for those of you who don't know about this, I'm going to talk about the cognitive origin of mathematics from a very, very broad perspective, starting with evolutionary processes from the evolutionary dimension of mathematics.
The first thing that we need to know about mathematics is simply a local mode of thought. And so far it's a local mode of thought that has been developed over centuries. We need to treat it in terms of certain evolutionary processes. By evolutionary processes I do not exclusively mean natural evolutionary processes, but also cultural evolution. With the understanding that the word evolution applied to these natural kinds and to the
cultural kinds are being driven by different forces, by different, and we need to treat them differently. Yes, there are continuity between them, but that continuity should not be an excuse for aligning the distinction between two different evolutionary constructs or evolutionary domains. Before I start, let's hear your stories that why you took this class and give me a little bit information about you know your work and what you are interested in and how you think
that this might be relevant with whatever you are doing because precisely I try to as we move forward try to customize this entire course in a way that at least it can address different aspects of your projects and your domain of research. So let's start with Jake and then everyone else. Jake, I can't hear you. Your mic is mute. Got you. Sorry, thanks. Yes, so I'm Jake Hamilton. I'm a proctoring the course, a certificate student at the new center. I've taken a few of Reza's classes before and I also
am a developer I just finished coding boot camp. This class in particular I'm interested in the way that such a highly formal and compact mode of reasoning which is like in many instances considered to be somehow the most basic to underlie even physics or to be a fundamental law of nature intertwines with its own cultural history and how that kind of revolves around specific needs for technological innovation or specific kinds of problem-solving at different points in that cultural history and how I can apply that both to sort of code projects and philosophical projects, sort of using mathematical reasoning to solve specific problems, new problems as they're encountered.
Thea? sure yeah I'm also a certificate student studying critical philosophy at the new center yeah I'm sort of just interested in the relationship between math and philosophy and math and sort of thought in general more quickly yes I mean what What exactly, what is the particular connection between math and philosophy that you are interested in? Or is it just simply the overarching connection between mathematical domain and philosophical domain?
Or is it like a specific connection? I think specifically the relationship of math as, I guess, I don't know, I guess I'm more interested in the axiomatic aspect of math. and then generally how math relates to thought I guess. Uh-huh. Uh-huh. Yep. Does Bryn want to go next?
Yeah, if we just wanted to go from left to right or something, we all see, I think, the same order down there at the bottom. So, Bryn, if that's okay, if you want to start or we can move on to Chagas. Yes, I mean, you guys go ahead. I'm listening. I'm also experimenting with my iPad to see if I can get this sorted out. Okay, it looks like Bryn's mic is not working, which hopefully we can sort of get figured out sometime soon. and you can sort of reprise that. But if we want to move over to the right. Are you there, Chagas? Oh, yeah, yeah. Sorry. I'm seeing me at the other end, so I thought I was going to be last. But, yeah, I just have virtually no background in math.
So I'm just really excited to see what Reza has to offer. And I'm just really excited just to see how much I can learn during the class. When you are saying that you don't have any math background, do you mean in the sense that's a higher education math background or you mean that you absolutely have no background in mathematics? Almost absolutely. Okay, okay. Basically as close as absolutely as you can get. Although I am really wanting to learn about different perspective and I'm extremely excited.
Yeah, but I'm not like, yeah, I have no higher education and yeah. Okay, excellent, excellent. Thank you very much. Christian, do you want to go next? Sure, I can go next. For me, I guess I'm interested in definitely like the long term, like the long term big time point where math develops over long periods of time and produces long term structural changes as it proliferates the modes of producing knowledge and information.
But as far as my project specifically, I'm interested in artificial intelligence economics in the long run. And I'm just now getting my feet wet. But, and also I'm interested in like diversifying web protocols and like hopefully making alternative social media and alternative, basically a post internet. Okay. Interesting. Excellent. Daniel, are you there? Do you want to move on? Sure, sure. I'm sorry, I couldn't see. The order is a bit different, I think, depending on...
Yeah, no. I mean, my background in mathematics is mostly functional. I think it's mostly up until a high school level and going through calculus, but not necessarily understanding a lot of the motivating ideas behind it at a conceptual level. So I think from that perspective I'd like something that's going to be dealing with this cross-section of philosophy and mathematics a bit more clearly, foregrounding the ideas even at a very rudimentary level. I'd like the ideological development of math and I don't know, I mean I think what I want to do with it I'm not quite sure yet, I don't understand necessarily the practical application. background is in more philosophy and literature. So yeah.
Which, specifically which trajectories in philosophy or literature? It's a bit scattered, it's a bit piecemeal. In literature I've done comparative literature studies, so different national literatures, with 20th century poetry thoughts. I mean with some of the mathematics stuff I know I can kind of understand why bad you might come up with set theory and some of the infinitesimal calculus with the moves but I don't understand that and and it's complex in any way or even when I read something like that I don't understand where this is coming from with some of the motivating ideas behind it sure sure so I kind of I'm looking at this as something that's going to give me hopefully the ability to deal
with some applications of mathematics and contemporary philosophy more so sure sure so thank you Jeff do you want to go next yeah sure can you hear me okay yes okay I'm a painter I'm an abstract painter and I've used math over the past 15 years to attempt to kind of disrupt compositional norms, especially my own compositional norms to try to push something into kind of other realm and and push past what I think of in
abstraction as a kind of will to power. So the past six years I've been working on a series of works that employ random walks and I'm at a place where I'm I I don't want to say I'm at a dead end, but I'm pursuing a kind of end game, as it were. It's like I'm sort of pushing toward the next thing or how to sort of expand beyond the canvas. I've done some experiments with sound, with video, looking for ways to kind of expand. And at the same time, on the other side of things, to better ground my understanding.
because as someone else described, I have about a high school age math background. I came to math simply because of artistic aesthetic imperatives. I was working on an idea of non-hierarchical composition, and that's how I came to using systems, and especially two-dimensional ways of arranging parts to, as I said before, try to subvert what the compositional norms would be. So that's where I am right now, and so I'm looking forward to working with you in the seminar. I just want to say also I just got your monograph that you did with Jean-Luc Moulin.
Is that it? Yes, yes. And just arrived today, I haven't sort of dived into that yet, but I'm looking forward to hearing what you have to say on abstraction as well. So. Sure, sure. Absolutely. So, Maria, is Maria next or I'm not sure. Maria, if you wanted to go. Oh, yeah. So I'm also a second-to-second student here. I'm actually in art and curatorial studies and I guess I'm somewhat of an artist. Not sure how this like math direction would specifically benefit what I'm doing, but I've been inspired to go further in this like in this direction in like method and math and
like scientific method from reading Deleuze's writing on Leibniz and also just digging into William Wimsatt right now. And so from some kind of like from these starting points and like and going further in this class I'm interested in like methods and starting locally and like extrapolating outwards and especially using spatialization as like a method in problem solving going deeper into that otherwise the only other math that I have like any like like philosophical math concepts are from I guess right reading your writings on it
like and yeah I'm really interested in really excited about support for this yeah Oliver you want to go ahead Hello. Yeah, hi. I'm on camera. Okay, yeah, I'm Oliver. I have a background in continental philosophy. Currently, I'm not doing very much, and I don't intend on using this course practically in any sort of concerted aspect.
like for business or whatnot. Actually, I'm kind of interested in the way it, the way mathematics can impact on, on the way I sleep, but that maybe that's just, that's just crazy because my sleeping isn't very good. But yeah, I think maths has a lot to do with, I mean, when I was younger, I used to have a lot of mathematical dreams and, And I would wake up and write lots of theorems onto paper for hours and hours and then go back to sleep. And I found that my sleep was sort of deeper at that point. But apart from that, I'm mainly just studying, reading Ernst Bloch right now.
But, I mean, that's not something special for me. I go through a lot of different books and whatnot. But yeah, I'm really interested in basically comparing the way that, and seeing the way that math can relate to practical functions, physiognomy. If that's too much of a stretch, I'm not sure. I suppose evolutionary math has something, the history of math has something to contribute there. Excellent. Thank you. Cool. Omar and the arbitrary listing I have in front of me, that would be you up next. Hi. Can you hear me? Yes, we got you.
All right. So I have, I think, almost no background in mathematics. And my background is literature. So I think as far as what I can say now, maybe the application of mathematical thought in an actual system in music, that would be my where I could maybe build upon what I learned from this course. But as we go on, perhaps my interests will further develop. Uh-huh. Interesting. Philip, you want to go ahead?
Or are we looping back already? Sure. I'm a writer. My background is in comparative literature, French, Latin American, classical Greek, English, and American. And I've recently kind of rediscovered my love for mathematics, which has been dormant for about 40 years. I did, years ago, go up through number theory and linear algebra, mostly on my own as I was doing things like acting at the time. I've been reading for the last six or eight months math books of all sorts, tracking back and forth between different modes of presenting mathematics, which vary so incredibly that I'm kind of fascinated by it.
There are the thousand page long calculus books, which bury you in every technique of calculation. On the other side, they're the overly logicized, to me, Borbaki books, where everything is presented in the language of text theory. So it's nothing but a set theory, nothing but symbols, which kind of bores me after a while. And I found that I tend to like things that mix a certain amount of prose with a lot of mathematical notation. And I find, for example, the books of someone like Serge Lang, who taught calculus in Columbia, where I went in the 60s, to be quite elegant and impressive and concise in presenting what are the key moments of any particular form of mathematics,
and a certain number of forms, but not overwhelming you in terms of calculative detail. My other favorite mathematics story is the idea of math in the Soviet Union at a time when under Stalin, for example, Stalin was the chief scientist and if you were a geneticist he would always have to look at what you were doing and decide whether it tracked or not with dialectical materialism, but that he didn't understand mathematics so he just left people and it didn't take much in terms of money because all you needed was chalk and a blackboard so we left these people alone in their institutes and when the soviet union fell there was such a
renaissance of russian mathematicians who were immediately gobbled up all around the world at places like princeton right away from gelfand on that it's just it's just wonderful that one can develop all that when left alone and what kind of castles in the air on the blackboard one can develop. And the last thing I'll say, I realize I've gone on too long here, is I intend to kind of incorporate my new feel and love for mathematics into a book I'm writing called Valparaiso, which is based on my time in Chile in 1973 during the overthrow of Allende, before and after, where I was trapped
in a hotel for about 10 days with other wanderers from around the world and we fell into this kind of classical literary trope, the Spanish Inn or the Decameron situation, where everybody sat and told each other's story is at night or the Saragossa manuscript is another famous variety. And one of the things I noticed that I really like in this literary form is the characters who tell stories are totally generic, and they often have names like the Kabbalah or the Astronomer or the Fugitive or the Mathematician. And so I'm envisioning writing a monologue that is by a mathematician But not the kind of mathematician that my uncle Albert was who was a mathematical magician
Who went into systems theory and became a defense theorist for the neoconservatives, so that's my complicated tale Well, I think that's everybody you want to Anything else you wanted to ask or should we go ahead? No, we can go ahead. Excellent. Fantastic. Okay, so I think, yes, because it looks like some of you have some familiarity with mathematics, but some don't. I try to not take for granted that when I'm talking about some even the most basic mathematical object or
mathematical concept I don't take it for granted that you might know about this I try to my best to to really elaborate what it is what it's supposed to say what it's supposed to do but nevertheless any time that I mentioned something that you either think it's vague or you absolutely have no idea what I'm talking about please interrupt me and ask me to elaborate also another thing as as we move forward we see that I mean this is not really exclusive to mathematics some of the terms that I'm using and I will use might be also terms
that we use in philosophy we use in whatever research domain we are doing we also use them in a completely common sense way those are I think the most dangerous one precisely because if we I don't elaborate exactly the mathematical definition of these concepts or these terms it would be kind of you know create some sort of cognitive for you in the sense that for example you know a good example of this would be
I don't know the concept of balance and imbalance that for example you know in ordinary life we have very kind of intuitive understanding of what these terms might mean. I want to make these distinctions so in order for us to be capable of making the connections with philosophy, with theory, with literature, or art, but nevertheless we should avoid making trivial connections in the sense that we just we should avoid from now on in this class try to whenever we hear
a word or a concept that we might apply it in philosophy or theory we should be suspicious that's what mathematics means by this concept might be thoroughly not essentially but most of the times might be thoroughly different from what we mean it in philosophy in order for us to really connect non-trivial the mathematics with philosophy with art with other disciplines is that we need to preserve at least the mathematical connotation or the sovereignty of the mathematical terms in order for us to be capable of connecting it with the similar philosophical lexicons or theoretical vocabulary that we have
that's one thing another thing would be as I said from the third session we start to look at very rudimentary mathematical problems just examples and And the first session that we are going to work on something that actually is a mathematical problem rather than just a conceptual basis will be just like something like division and multiplication. I will, I understand that, I anticipate at least that all of you have some grasp of just the basic mathematical operations. But in so far as the kind of examples that I'm going to present, while they are fitting
in this criterion, they nevertheless involve something more. So again, if you can't follow what's going on when I'm trying to present a methodology for solving a problem, whether I'm talking about Euclid's elements or I'm talking about some Egyptian method of division please again interrupt me and tell me to go over to the first step and start rebuilding the solution and also it would be great instead of for the first few sessions instead of like trying to read stuff like do some light homework and I let's not even thinking about that
you don't need to try hard to come up with the right solution but simply just engaging with a mathematical problem to a certain methodology that we have been reviewing in the class that I think really helps precisely because no matter how much we try to conceptualize about mathematics doing mathematics is an extremely different thing even at the most basic level I mean that activity is like writing, no matter how much you philosophically think about certain arguments or thoughts, as soon as you exteriorize it, putting it in writing, things start to become more complicated.
You see the connections, how the connections are emerging, where the flaws are, so on and so forth. The same thing about mathematics, and even more so. So like some sort of light homeworks would be great I think for this class. I try to keep them to the most basic rudimentary level. So I think that's enough in terms of what we are going to do and possible scenarios that might arise throughout the class. So today I'm going to, as I said, I'm going to talk about the evolution of mathematical
thinking. I think we can start to think about the origin of mathematics without some sort of cosmological story. I think we can flash back to at least 542 billion years ago. The time frame is Cambrian explosion. It's an evolutionary epoch in which the complexification of life starts to happen.
During this time frame, one of the most specific things that happens in organism is the evolution of the nervous system. And precisely in so far as mathematics is essentially a cognitive technology even in its most abstract form, but it is cognitive. We need to, as I said, we need to anchor the specific cognitive abstraction that mathematics deals with within the more general cognitive economy afforded by evolution. At its most basic scaffolding level, this evolution is an evolutionary epoch starting
around the Cambrian explosion almost 542 billion years ago. So let's start with, as I said, the complexification of life, the complexification of organic life begins with the evolution of the nervous system. What is exactly at its most rudimentary level nervous system is? What is, precisely because we know that nervous system is just a kind of, it's an organ really in holistically understood.
And so far it is an organ, it's, and that has been instantiated by evolutionary processes. Of course it addresses some problems in evolution, it tackles some forms of evolutionary pressures. What would be then the function, the basic function of nervous system? Does anyone know anything about the basic function of nervous system? At its most basic level. just forget about our brain, let's just talk about some minimal, yes, a response to a stimuli.
Yes, but a response to a stimuli for what reason? Why an organism needs to have, in fact, a response to a stimuli? To correlate its behavior with an environment or with its environment? So to enable environmental correlations? But why in fact even does it need environmental correlation? What makes nervous system in fact to address this problem of correlation? Is it correlation is as an answer, but what is exactly the problem? Survival, yes.
Reproduction, yes, absolutely. Yes, so the most basic functions of any organism, what you might call survival and continuity. Of course, you can put them in less fancy, innocent words and simply call them predatory and sexual activities. Simply by predatory we don't mean it in some sort of predatory, predatory, but simply finding food. Now it's interesting that without the nervous system, let's imagine that we completely get
rid of all the concepts afforded to us by our nervous system and try to see how an organism who does not have any affordances, any evolutionary affordances that has been provided by the nervous system, what would be, what kind of problems of survival and reproduction does this organism tackle? Precisely if you do not have nervous system, the first thing that happens to an organism is that it does not have a sense of a space. It does not have a sense of a self. These
are all capacities afforded by highly complex nervous systems. So we can't really attribute them to a time where nervous system in its most basic structure is evolving. If survival, If survival, if predatory and sexual activities are prominent evolutionary problems for an organism and if organism at this time does not have a nervous system, what would happen? kind of problems arise for an organism that tries to survive while it does not have even
at the most basic level something resembling like a nervous system. Any answer? Any thoughts? Well distant can't coordinate. Like it's a very literal sense of space in the sense that no piece of like a multicellular organism without a nervous system can communicate its position. Communication can only happen via diffusion in all directions and at a constant rate. Yes, I mean, yes, absolutely. In a nutshell
Well, what we mean by the sense of space is not given. At that level, organism absolutely has no sense of space. But why the sense of space is important? It's precisely because if you do not have a sense of space, a form of differentiating yourself from the background, you can't even differentiate yourself from something else. simply the idea that at this most basic rudimentary level for an organism that does not have a sense of space, it simply cannot tell the difference between itself, its food and its predator, precisely because there is no marker, no spatial marker that allow the organism
to make this differentiation. So what essentially happened at this level of evolution, you get the problem of autophagia. The organism simply consumes itself. Hence, survival becomes basically out of question. It's just simply there is no complexity if autophagia is dominant. So this is an evolutionary pressure, evolutionary pressure that needs to be solved through development or evolution of a system that allows the organism to make this differentiation at a causal level.
And we are not even talking about the conceptual level or abstract level. simply talking about purely a structural morphogenetic level of organism. So we see that in order for this differentiation to be happen which also is known as figure ground differentiation, simply making a contrast at the most basic level between the organism and the ambient environment, the ambient space in which the organism is embedded. At this level, the most minimum thing that can, that needs to be done is some sort of sensitivity to a stimuli.
In fact, this sensitivity to a stimuli, this capacity to receive and react to a stimuli takes place in the first billion years of evolution through sensitivity to light. Light is a dominant factor on the terrestrial horizon. So evolution, we know that evolution tends to work with the most dominant, basically,
parameters light, gravity, so on and so forth. So the first instances of sensitivity to a stimuli are required for the figure-ground differentiation and hence development of what we might call a sense, a rudimentary sense of the space happens to sensitivity to light. This sensitivity to light that you see throughout the course of evolution, it simply becomes a form of entrenched zone for evolution of ocular system, the eye.
As I said, another prevalent and dominant parameter in a terrestrial evolution of organisms is gravity. There is no organism in this planet that does not have a sense of gravity. what would be then what kind of precisely what kind of system can address the question of gravity and what would be the role of then gravity in order for the organism to differentiate itself
from its food from its environment namely to develop a rudimentary sense of a space In terms of light, it is very intuitive to think about how light or response, reactive response to lightest stimuli can allow an organism to make a contrast between itself and the ambient environment. But then what would be the role of gravity and how the nervous system can address the question of gravity uses the gravity in order to differentiate the organism in order to differentiate itself from the surrounding the space does anyone know
what kind of what kind of part of the nervous system address the question of gravity and how it precisely is instrumental for any organism to make this kind of develop this rudimental sense of a space to differentiate itself from the surrounding ambient space. Yes, vestibular system, yes. Inner ear, the structure of inner ear or the vestibular system is precisely the result of the first in fact it's really interesting that so yes sensitivity to light is
something that is quite ancient in evolutionary timeframe so as the evolution of the vestibular system these are really the two poles that every nervous system has them in one way or another and the vestibular system is precisely a organ for detection of grav- inertia gravitational field. But then how does gravity, how detection of gravity, that was the second part of the question, how does the detection of gravity help an organism to differentiate itself from the surrounding space? We still don't know about this.
But then what would be the rule of inertia in this? Oh, we don't know anything about law yet because we are, let's abstract away anything that is conceptual about the nature of the evolution. We are just simply talking at the very causal and structural level of organisms. It's simply that inertia is a field of continuity. that is an inertial trajectory will remain on the same path if it is left alone, if you do not interfere with the process.
Yes, so there is this, by inertia gravitational field, the paths of mobility that it creates for any organism that simply is mobile is that it creates a what you might call in a I will define this in a very physical sense a geodetic trajectory a geodetic path. What is it exactly a geodetic path? A geodetic path is the shortest path, usually a curvature, that corresponds with something
in physics that is called Lagrangian optimality. Lagrangian optimality means that it consumes, roughly speaking, it consumes the least amount of energy. projectile that travels across a geodetic path a curvature is moving in that direction precisely because it requires the least amount of energy I'm sorry could you possibly type those two words in the text box geodetic like Yes, and Lagrangian.
Thank you. Yes, Lagrangian is a principle of optimality for any physical object in the universe. You can think of celestial motions, you can think about throwing a projectile, throwing a stone. It always travels along a geodetic path, the shortest path and its correlation with energy conservation. This is simply something that every physical object is bound to. It is the pressure. is something that you cannot find anything that is exempt from this
principle so these two reaction to gravity and reaction to light on the earth's biosphere are two most dominant instruments that can be integrated with one another and create a rudimentary sense of a space simply a capacity and so far as it as I said the sense of the space is not given for it in the organs and the sense of space this idea that an organism can simply differentiate itself from its background
is not something that is given to an organism it's something that is being developed and hence needs to be understood in terms of an evolutionary afforded capacity this sense of a space is what you might call the first gesture the first evolutionary affordance of mathematical thinking. And this is fully embedded in the evolution of the nervous system. Now let's talk about how is it really, how can we talk about this rudimentary sense of the space, which as I said, it only happens at the level of causal, at the level of the structure of the nervous system,
might be connected with basic mathematical intuitions. In order to start to elaborate this connection, I think we need to approach this on two opposite directions. One, what is exactly, at its most basic level, mathematical intuition? So the question from later to earlier, from already complexly instantiated cognitive framework that affords mathematical thinking toward this rudimentary nervous, evolution of nervous system.
And the other direction would be talking about precisely what kind of affordances, this fusion of, for example, visual signal and inertial gravitational signal affords the organism in terms of certain cognitive affordances that later on, as they become more complexified, afford something far more complex, something like what you might call ancient pre-linguistic mathematical intuitions. First, let's start from the evolution of the nervous system and move forward and then we
approach the opposite direction. So as I said, if the sense of the space is not given, the first evolutionary pressure for the organism would be the development of something like nervous system, where the understanding basic function of evolution is solving the problem of a space. Why? Because precisely it permits, it affords the organisms to continue its functional requirements
like predatory and sexual activities in the most optimal way. And hence also it undergirds the complexification of life. At this level, figure-ground differentiation, organism differentiating itself from the ambient a space is a necessary element. It is not sufficient, it is still not sufficient for carrying out these activities, predatory and sexual activities in the most optimal way. Even though it is necessary, this figure-ground differentiation, it is not by any means sufficient.
It does not, if an organism simply can differentiate itself, contrast itself with its surrounding environment, this doesn't warrant that the organism can indeed optimally carry out survival functions. And this is the question of again further complexification of the nervous system. And at this level, at this level of more complex phase of nervous system, we are dealing with a new problem. It's a problem of invariances, a spatial invariances.
What are a spatial invariances? Does anyone know what we mean by a spatial invariances? why is that they are again necessary for conducting optimally basic survival functions? Is it like a certain like continuous synthetic manifold that within the differentiated space in the organism? Like inertia, but like it's a further development where it's not just inertia. We have like a particular object in like a differentiated space where things persist.
Yes. Yes, absolutely. Yes. It's precisely this idea. What is we were talking about? I mentioned I associated the word continuity with inertial gravitational field or geodetic path. So these are just generic what you might call physical pressures that are simply prevalent and they don't have any particular context. precisely the whole idea of optimality of optimal conduction of biological functions requires contextuality, requires physical contextuality in the sense that, for example,
a predator who chases a specific prey needs to have a, can't simply use this generic form of invariance, namely inertia, gravitational field, or geodetic path, to chase the prey in the most optimal way, to ensure that it can catch the prey, given these structural parameters of its body or the body or the structural parameters of the prey that it's chasing. So the question of a spatial invariance is simply what you might call a proto-abstract
way of seeing these geodetic structure of a space that has been afforded by in response to stimuli through usually response to light stimuli and detection of inertial gravitational field. Invariance at this level are not simply environmental pressures but some things, some what you might call causal connections that are being constructed by the nervous system itself rather than simply
being in the environment because the inertial gravitational field is just in the environment. And precisely because it's so prevalent that it doesn't warrant optimality of exercising biological functions. So nervous system needs to construct a specific forms of continuity, a specific forms of invariances that are while being derived from these generic principles like gravity, they have something more, they have some spatial context, they have some specific parameters.
So and what are exactly invariances? Invariances, roughly speaking, we can think about causal connection, that's what we mean at this level of invariance, not conceptual invariance. Invariances are causal connections between different occurrences that can be interpreted by the nervous system as patterns or regularities. Using these regularities, a prey can escape the predator in an optimal way. The predator, again, using these constructed invariances, these causal connections that
have been compressed into patterns or regularities by the nervous system can predict for example the motion or you know the flight of a prey. So, but nevertheless the question is that how these invariances are being constructed by the nervous system. Again we need to look into a new problem that happens in evolution and we are past the Cambrian explosion. We are already, we have already entered a phase in which the complexification of nervous system
has already been instantiated. We are in the phase of complexification of life on earth. At this level, simply response to stimuli doesn't suffice to develop more optimal navigation of environment. We need to have something in addition and above to response reactive passive response to stimuli. Any idea of what these new components required for construction of spatial invariances might be?
New components that as I said are not simply responses to a stimuli. Some sense of autonomy. Well, yeah that would be helpful if you elaborate what you mean by autonomy at this level because obviously you know we are not talking about anything like agency at this level. We do not have, there is no such a, we don't even have abstraction, we don't have even like basic ideas of self at this level
yeah I was gonna say away that an organism would optimize its own capacity to survive, but maybe that's a step backwards now, that it would self-differentiate itself. Yes, it does self-differentiate at a higher level, but what would be the components that required for the transition of the organism from simply reactivity to a stimuli to move to these higher levels of differentiation through the use of constructed invariance of the space. I noticed that memory was just mentioned. Yes, memory. Memory is really one
of the things that allow for the construction of invariances. And without memory there is no such a thing as a spatial invariance at this higher level. Hence you, there is no in fact condition of possibility for the complexification of the nervous system. But at this level what we mean by memory, we do not mean something like episodic conceptual memory of human beings. It's simply something what we might call constructive memory. Yes, habituation. What is exactly the function of constructive memory? Predictive memory is, plays a very fundamental function. It creates internal model for the organism to anticipate something.
It's basically, it's just a predictive device, anticipatory device. But what would, what is required exactly for, you know, anticipatory behaviors? So in a very like a Kantian, schema and at least we are dealing with we are talking about memory we are talking about three distinct things one is impression impression of organism of of an object in real time an object that instantaneously is interacting with with
the organism this this is what can't associate with the function of outer sense sense impressions sense impressions are simply impingement of of objects on the nervous system. As most basic level, this is precisely what reactivity to a stimuli is. Impression. So the first, no, impression not in the sense of predictability, but impression as I just said, receptivity of senses to a stimuli. That's it. That's what Kant means by sense impressions.
So sense impressions, imagine that we didn't have the concept of pen, we didn't have any of these further complex abstractions. When I'm interacting with this thing, it's not an object. Because the objectivity of this pen is afforded to me by how I use, apply different invariances, a spatial and then later on conceptual invariance or generalities to this object and then I construct this object. But at the level of sense impression there is no such a thing as this being anything. It's just a stuff. It's just the stuff, the color of it, the shape of it, these are all stuff.
We shouldn't even assume that there is an ordered sense of how this is stuff being processed. So memory and other functional memory is simply also ordering sense impressions. But nevertheless you can think of the first basically instance of memory is recording in real time, not in a kind of an idea that recording it so as to be capable of recalling it perfectly in a later time. this level by sense impression you mean it in a very kind of a instantaneous recording of information about the nature of the interaction between the
organism and an item in the environment so there's a sense impression then at the second level what memory does is something that can't called reproduction reproduction today's lexicon we might say that memories for organisms are not when we recall a memory we just it's not like our hard drive that we're simply recalling it back and the data that has been restored is you know is exactly the same data that was there when we basically put in the system in the first place.
No, reproduction, that's the whole idea of reproduction, changes the nature of the initially stored sense impression. And hence Kant's thesis that every impression is ultimately a reproduction. We do not have access to our sense impressions. Why is that? Precisely because in evolutionary sense, organism is situated in the environment. Every time that it is trying to, every time that its nervous system tries to access the initial sense impressions the parameters of the ordering of how these sense impressions
were ordered initially are being modified in accordance with how the organism is situated in this environment in real time. So this idea of situatedness the organism being situated within active parameters of the environment needs to be understood as something that actively modifies original stored data, original sense impressions. So this is the second element. We add the first one impressions, the second element reproductions. These are all there for the third element which is the function of constructive memory,
anticipation. So what is exactly anticipation? Anticipation, what you might call a representational model of an anticipated action. Imagine that for example a predator that has encountered, engaged with some other organism. The first instance, real-time, the information regarding some of the characteristics of the organisms, you can think of them as sense impressions. But precisely because if, for example, if this organism that the predator shows interest in, this encounter is being repeated a couple of times, every
time this initial information regarding the characters of the prey are being updated by the situation in which the predator encounters the prey by the parameters of the situation, in terms of spatial orientations, in terms of different visual field, in terms of temperature, so on and so forth, depending on the structural complexity of the predator. As these informations are being modified, so you see that as a time and of course the
encounter is being repeated, then you see that from our outside perspective, not from the perspective of the predator, we see that we can think about these different variations of the same encounter. All of these encounters have some specific parameters that have in common, but they are also are different. So this is the function of the constructive memory to single out what is common, what What is invariant, what is unchanging between these different instances of reproduction, between these different instances of encountering a specific organism?
This is the function of constructive memory, singling out invariances between different variations of reproductions, memories, initial memories that have been modified in accordance with the situation. Why is that these invariances are being singled out? Precisely because they are being singled out in order to create a predictive model, a forecasting model, an anticipatory model that can forecast, used as a generalized framework in order to forecast the encounter with this particular organism, this prey, regardless of the context
in which the predator is situated, simply context's independency of interaction with an item, interaction with prey. Precisely because if you have an invariant model that represents the characteristics of the prey and how you are interacting with it, the memory can use this, the nervous can can, the system can use this model, this representational internal model in order to forecast how you can optimally chase a prey and what would be the consequences of your action in the environment. For example, as you run or chase the prey, what would be the reaction of the
prey, what would be the possibilities, statistical possibilities of the prey moving in that direction assuming this orientation, whether it's going to attack you, so on and so forth. So there are all these, you know, as I said, these are all these parameters that are being compressed in these causal invariances, these representational models. And the function of the constructive memory is, as I said, is the construction of precisely these invariances necessary for forecasting, necessary for anticipated action in the environment, the consequences of an anticipated action in the environment.
In fact if you do not have this function of the memory, if you do not have constructive memory there wouldn't be anything remotely similar to what we may call action actions are the results of constructive memory any questions on this this. Do you need any clarification on this front? That how actions are correlated necessarily with the function of constructive memory without which there wouldn't be any action, any form of action, even the most basic level.
So this is the first role of memory in constructing causal invariances that can be utilized as internal representations for the agent, the purpose of which is the creation or anticipation of the consequences of action in the environment. The second role of memory is something that Freud had already talked about in his early
books. Memory does not really, hasn't been there to remember really. It's not, it's a very constructive memory or, you know, organic memory is very different from the function of a hard drive memory. Memory rather than remembering, it forgets. But why is that forgetting is necessary for the function of memory? What kind of, what role does it play in optimality of action for organism? What does forgetting do? No, because rewriting, re-recording is
more in the sense that, as I said, protection, compactness, reduces processing requirements, yes. Reduction of variances, yes, yes. Reduction of variances, yes. So forgetting is a form of reducing variations, compressing further the invariance. because the initial invariance that we have is still not optimal. It needs to be compressed further. You see the role of why is that memory forgets and what is really the function of
this forgetting? In order to answer this question we need to first understand what is exactly the nature of action for the organism, nature of anticipated action for the organism. Well, of course, organism, we define organism at this rudimentary level in terms of simply most important biological functions, sexual and predatory activities, namely reward-driven activities. So when it comes to reward-driven activities, evolution wants to ensure that
the organism activities conserve energy, that they also ensure that these activities can yield maximum reward, at least in the context of the rudimentary organism's activities. So the function of forgetting is simply a way of filtering out any mode of anticipated action that instead of being successful, instead of behaving as it was anticipated, it behaved differently. Hence, it did not yield the maximum reward.
Hence, it did not, for example, hence a predator chasing a prey using certain invariances, constructed by the nervous system using these invariances didn't really resulted in predator reaching prey within certain energetic and spatial parameters that's where forgetting for functional forgetting comes comes to the picture. It filters out bad representational models. It's in order to further refine the
model of anticipated action. Any action that the predator has done to this point and didn't result in success or in maximum reward would be forgotten by memory, by the constructed memory. But also more than that, forgetting is also a way of filtering out additional details about the environment. It completely gets rid of the kind of details that are present in the environment, but nevertheless are not necessary for the conduct of a specific action by the organism.
For example, a predator who wants to chase a prey, some of these details, signals in the environment are not necessary. It does not, it actually, if it starts to process these signals, the chase would be unsuccessful, either because it takes too much computational resources, it consumes too much amount of energy, or simply they interfere with the model of optimal anticipated action. So forgetting in a very rudimentary sense plays these roles.
It further refines spatial invariances or causal invariances constructed by the nervous system. Now we have already seen that the construction of spatial causal invariances requires filtering out huge amount of information to create some representational internal model of a specific action. This is what you might call, even though I hate to say it, but you might think about it as really a proto-semantic germ, proto-linguistic evolutionary germ of some basic condition
required for the generation of the very capacity for abstraction. Because what is exactly abstraction at its most intuitive level? It requires compression of information, it requires singling out invariances, hence generalities, it requires subtraction of extraneous or excessive information, so on and so forth. So memory, what you might call is a system that turns simple sense of a space, in the
sense of an organism differentiating itself from the environment, it transforms it into a more complex sense of a space a constructed sense of a space a constructed sense of a space in the sense that this sense of a space from now on is coupled with at least two things action in the environment and And the other one, generalities, generalities that are not simply in the environment, but are being constructed by the nervous system, more specifically constructed memory.
And our story essentially starts from this moment, from this more complex sense of space, more complex sense of space that we try to connect it with the most basic mathematical intuitions of space and hence mathematical intuition as such. Any discussion by any of you? insight or observation before we move forward so forgetting comes after memory
so in order to forget we would have had to have already established like forgetting is almost a type of memory you're saying it's not like not forget it's not not remembering but yes it's absolutely the function of the memory in fact the function constructive memory which is the most basic form of memory Looks like we've got a few questions in the sidebar here.
Yes. No, again, at this level we haven't even moved from sexual predatory activities, graduation from sexual predatory activities. No, we are, basically we have just moved from doing these activities less optimally to doing them more optimally. And the other question, as that's one of the Kantian intuition of geometry. Yes, we will talk about Kantian intuition of geometry and also the idea of refining the idea of intuition particularly the idea of philosophical intuition and how it's linked with for
example geometry then you know for example idea that Euclide axioms are not logical axioms in fact so many people argue that they are not axioms in fact but nevertheless they belong to a class of axioms that are considered to be intuitive and what do we mean by intuitive and how they are connected to the Kantian idea of intuition I will talk about this yes in the next sessions. It's actually something that this idea of erotic math, what you might call it, it's called proto-math. There is this French, so basically there is this group in France,
France I think it's they founded this group I think in early 2000s or late 19 it is called the GEOCO like GEO and CO is simply an acronym for geometry and cognition and one of the people who is part of in this of this group is his His name is Bernard Tessier and Bernard Tessier wrote this essay, famous essay is called Proto-Mathematics and he was one of the first people who started to look into the neuroscientific basis of mathematical thinking particularly within this broad overarching evolutionary
basis of nervous system and the kind of cognitive faculties that it afforded organisms with. This is a space through memory for forgetfulness is developed in order to act within an environment yes and to it so it is really it's just for action it's just for optimal action now optimal action requires two things one anticipation the idea of rip how
information yielded by sense impressions are being modified by reproductions of them and how the nervous systems start to single out what is common within different modes of reproduction of an initial memory in order to create a cohesive model of anticipated action. So the sense of a space through memory forgetfulness is developed in order to one, there is no one or two, in order for exercising or conducting optimal action in the environment. But conducting optimal action in the environment requires two things.
uh creation of an internal anticipatory representational model for the organism one two uh compressing or refining uh this model this invariant model that's that forgetting function of the memory the idea that if the action does not yield reward needs to be forgotten or some of the information regarding the situation in which for example a
predatory is embedded needs to be filtered out precisely because they lead to problems. They interfere with the optimality of action either through growing computational cost for the organism or simply as I said they literally interfere with the representational anticipatory model. And a couple of times I mentioned the idea of computational cost. So does anyone know what computational cost is? Why is it important? Basically nervous system absolutely, like anything, any evolutionary process, particularly
nervous system is always under another evolutionary pressure. But it's not really evolutionary pressure, it's in fact a physical pressure. It's called computational cost. Does anyone know what computational cost is? So computational cost is basically, it starts with this very rudimentary idea that you see, every observer requires a model of the environment. Now as the size of this model grows, so does the computational cost. Basically the bigger the size of this internal model, the more energy it takes for this observer
or the organism to process information. So there is this direct connection between the size of the model, size of the representational model and the growing cost of observation so if you are an organism the growing cost of observation for you means that your metabolism will increase you consume more food you basically require more you know a stabilization processes in order to not having meltdown so on and so forth. This is really the basic idea of computational cost. But the thing is that interesting is that as the complexity of the representational model
goes up, the cost of computation of these models also grow. the pressure caused by this computational cost on the observer or on the agent becomes more of a significant problem. And depending on what is the nature of this observer or agent, it can cause basically serious problems to the point that the agent won't be able in fact to observe the environment using a certain internal representational model precisely because it's too costly in
computational terms. It requires too much energy, it requires too much metabolism, so on and so forth. Yes, exponential, yes. And a really great person who has talked about this, I mean two great computer scientists So I've talked about computational cost, particularly kind of in connection with evolution. Is one, his name is Paul Vittany.
I don't know if I know the spelling. You need to check. It's Paul Vittany. And the other one is James Crutchfield. Crutchfield is. So this was, as I said, this was the conditions required for the construction of invariances that are no longer simply given to the organisms,
but are being constructed by the nervous system of the organism in order for the organism carrying its actions in the most optimal sense, optimal in the sense that not only it yields optimal reward, optimal yield maximum reward, but also be action, be context sensitive, can be done regardless of the specificities or specific parameters of the environment in which the organism is active. So and I said that precisely because the nature of these imbariances are always connected
to refinement of information to compression of information to basically forming models that deal with generalities rather than details that they can see in these invariances, constructed invariances can be seen as what you might call proto-semantic, namely causal abstractions
or conditions of possibilities of abstraction. Without these invariances, without these forms of generalities or compressions of information, you wouldn't be able to imagine any other complex form of conceptual or even representational mode of abstraction to emerge. And again as I said, at this level of constructed invariances, we are not talking about invariants variances of time or invariances of some basically complex parameter.
At this rudimentary level of evolution, constructed invariances are directly connected with sense of space in the sense that they are simply complex spatial invariances. These spatial invariances, of course, are being constructed by really complex modes of information processing, fusion of different signals, manipulation and modification of of memory, etc. etc. Let's get back to our, again, predator-prey example.
Let's think about the construction of invariance, a complex construction of invariance, a spatial invariants that is more than simple reactivity to a stimuli. Those of you who have watched these nature documentaries, you have seen a phenomenon predators as I mentioned tend to always chase the prey along the shortest path and this shortest path is not essentially shortest path in the sense of distance but shortest
paths in the sense of energy consumption, in the sense of conservation of energy, which I defined it in terms of geodetic principles or geodetic paths. So it's not shortest path in terms of distance, but shortest path that requires the least amount of energy. For example, when an octopus is being chased by a predator, it always tends to escape along the shortest path, finding a secure spot behind a rock. You see this
in, for example, other aquatic creatures. You see this in just simple, a lion chasing, for example, I don't know, like a deer or something, always has a specific trajectory. this trajectory might change but nevertheless the divergence from the main trajectory is always minimal precisely because diverging from the main trajectory requires switching from one mode of invariance to another mode of invariance hence it takes more energy,
more information processing energy for the nervous system. So chase lines, so-called chase lines, trajectory that the predator chooses in order to chase the prey, always have a very specific structure. It's not that we should completely forget that at this level organisms in fact think about what they are doing. They are just doing it. part of the nervous system, there is an initial gesture, the predator gesticulate as if it's putting its body in motion. That's usually an orientational move.
Orientation in the sense that basically this orientation is being formed in accordance with that's what I called the function of the memory the anticipated model of action model of anticipatory action so if you had this previous encounters with this kind of prey under these parameters and your nervous system has this internal model representational model the first thing is that the body your muscular thresholds, your sensory motor alertness are all being converged on a specific spatial orientation that corresponds with that internal model constructed by your nervous system to
anticipate a certain form of action. So the first one is the orientation. Then the next one is activation of this orientation. And the activation of this orientation is simply the predator putting its body in motion. Well, of course, you know, I'm filtering out so many details about how this system of activation is taking place. But nevertheless, once the orientation is being activated by the sensory motor organ, the task of the nervous system is to make sure that the movement of the predator or
the organism along this orientation, within this orientation, remains the same as much as possible. it needs to stabilize the movement of the organism within a specific orientational trajectory precisely because of all the problems that I mentioned. The problem of switching between different internal models, the problem of basically non-optimal action so on and so forth. This is still a stabilization is required for the exercise of optimal action, yielding the maximum reward. Now let's give an example of how does this really take place in nervous
system, this process of orientation and stabilization that then can allow the predator to gain maximum reward. As I said, the two most basic parts of the nervous system are organs for detection of light or simply reactivity to light and stimuli and the other one is reactivity to inner-sure gravitational field. Inner ear and ocular system. So I mentioned that the The inner ear is an organ for detection of the inertia gravitational field which can
be interpreted as a continuous trajectory. That's the whole definition of inertia gravitational field. A prevalent invariance that is simply out there and doesn't need to be constructed. We want to move from these to a constructed spatial invariance to allow the organism to travel along the shortest path. Shortest path, again, not in terms of distance, but in terms of the least amount of energy. So the first thing that happens, so if you have the visual signal and you have the inertial gravitational signal, when you're looking at different models of the nervous system
throughout the evolution, you see that all of these signals, different types of signals, are being fused within the nervous system, within the central part of the nervous system. For us it's the brain. So one of the main tasks of the brain is to fuse sensory signals regardless of their origin and their type. For example, you know, if we did not have a mode, an organ for fusion of different varieties
of sensory signals, we absolutely could not by any means identify, I mean I'm talking about us humans, we could never identify or associate a sound with a specific face. This This is like a good example that there is, so this sensory fusion processing unit in our brain is essentially something like a computational machine, but a computational machine in which some of these data come very late.
You should never assume that all of these sensory signals are coming at once and simply brain welds them together. No, some of them are actually coming extremely, extremely late. So the computer needs to wait for specific signals to arrive so it can integrate them with existing signals. We know that the speed of light is much, much faster than the speed So, then again that's what I was saying about the idea of association of voice or audio recognition with a specific face. Precisely because the face recognition is a visual signal processing task and the lip
syncing, this whole idea that you hear someone's sound and you start to associate it with that particular phase precisely because the sound signal, the sound signal reaches the brain millions times later than the visual signal if the brain did not have a form of diachronic fusion of signal processing that is being done through waiting for a specific signal to arrive and then integrate them with the existing signal, we absolutely could not even imagine that we can associate sound to objects, to visual signals.
A mechanism in the brain is called temporal window. A temporal window is something that according again to different parameters waits for a specific amount of time for the audio signals to arrive and then integrates the visual signal. So it basically stores the visual signals without even processing them and once audio signals come, it starts to basically process the visual signal and integrate them with the audio signal and that's how basically we think that when we see someone talking, We think that basically we literally syncing our brain, syncing the voice with that specific
face but precisely because of these two extremely different speeds of acoustic and visual signals, There is no reason for us to assume that there is any connection between sound and visual signal, or simply it's the same person that is talking. So there are all these really complex mechanisms in the brain that allow for fusion of different sensory signals across these different time scales. And when it comes to the visual signals and the vestibular signal, the inertial gravitational
field, the same thing happens. They have to come at different speeds, and according precisely the time stamped arrival, the central nervous system starts to fuse these two information together. The signal received by the vestibular system and signals received by the ocular system or the eye. Now what is really the function of the eye or the ocular system? Ocular system does two things. is simply what you might call retinal integration. You might think of this as taking snapshots
of an item in the environment, of an object in terms of shapes and colors, etc. So this is the more general function of it. Another function of it is something called saccadic jerk. Saccadic jerk is one of the fastest conceivable movement that any organism on the planet can exercise. The saccadic jerk is when your pupil moves very, really fast rate from left to right. That's a very predatory protosemantic gesture. It's specifically a predatory gesture, it's precisely for predators to be capable of orienting
the position of the prey without consuming energy, without basically putting their body into motion. It simply anticipates the trajectory of the prey, the saccadic eye movement. And then you see that when something suddenly happens, like for example behind us, some of like a noise, even without us moving our head, the first thing that happens is that our eyes at a really fast speed our pupils shift to the direction of the sound, and then the neck starts to shift. So the psychotic movement anticipates the trajectory of the prey.
The ocular, the retinal picturing simply takes a snapshot of the environment. Now if you did not have the vestibular signal to be fused with these two forms of visual registers, you absolutely shouldn't anticipate that what you see of the environment would be a stable image. I mean, if you did not have the, you know, I will say that how basically this happens, if you did not have the gravitational signal, the inertial gravitational invariance that are being registered by the vestibular system and only you had I, a complex ocular system,
You wouldn't see, for example, a picture of a wall in front of you. You would see cracks, simply cracks, as you move your head around. This idea that we see a seamless visual picture in front of us is precisely because our visual signal is being fused and stabilized by the inertial gravitational continuity that is being provided by the vestibular system. At the level of the ocular system without any vestibular signal processing, the picture
of the world that you see would be full of cracks, would be glitchy. There is no association between one snapshot and the next retinal snapshot. Basically you can't even have a minimal sense of trajectory for mobile object, so on and so forth. Now, the reason that this visual picture is stabilized is precisely because it's being fused with the information provided by the vestibular system, inner ear.
So, once they become fused in the central nervous system, what happens is that the vestibular system precisely because it has it works with a form of signal that is absolutely prevalent and is highly stable. This stability, this inertial gravitational invariance is being used by the nervous system to use it as some sort of what you might call an informational glue to simply glue these
retinal snapshots registered by your eye together. It's simply stitching them together and it stabilizes them. People who usually have no damage to their ocular system but their vestibular system has been extensively damaged, they see basically the world full of glitches, they lose their sense of orientation. Basically all sorts of visual anomalies are being registered for patients who have lost their or their vestibular system has been damaged.
Of course there are other parts of body and nervous system that also register the inertia gravitational field. But the most significant one is really the vestibular system. So even if your inner ear is fully damaged, still you have a minimum stability in your visual processing, how reality, the visual picture of reality, seemed to you as something purely seamless and stable. But as, imagine if all of these inertial gravitational sensors were being deactivated in your body for one reason or another, the whole picture of reality falls apart for you.
There is no way that you can see a wall as a wall. You can't, as you move your head in fact, you won't be able to see this as the same object. For example, if there is an iPad in front of me and I'm moving my head, when I'm moving my head this object does not appear as the same object for me. So there are these anomalies. These anomalies happen, as I said, when there is no robust fusion of signal processing between data provided by the vestibular system and data provided by the ocular system, namely
trajectory or orientation provided by the saccadic jerk and retinal snapshot provided by retinal. Now once, imagine that now once the nervous system is capable of sufficiently fuse these two forms of signal together, a couple of things happen for the predator. The first thing is that the predator ocular system starts to form a form of a spatial
invariance in which not only you have a trajectory, a specific trajectory, and this specific trajectory is always projective. You can use this in order to guess the position of prey that is running. So you have a projective trajectory, but you are capable of constructing using this invariance in order to anticipate not only the position of the prey from point A
to point B, but you can use this model of anticipation, this representational model, this spatial invariance in order to optimally mobilize your body toward the point B and the point A at the same time, in the sense that you are capable of having this really complex forms of invariance, spatial invariance, that can simply deal with a prey that is in motion rather than being at rest. So the first complex spatial invariances are invariances of mobility, invariances of motion. Not only the motion of the prey, but also mobilization or motion of your body.
And these invariances of motion or invariances of mobility could never be constructed if there was no structure in the nervous system that could fuse the signals registered by the visual ocular system and the signals registered by the vestibular system with regard to inertial gravitational field. Precisely because the fusion of these two signals is precisely what allows for the stabilization of an object, the image of an object. And this image consists of orientation, trajectory, direction, so on and so forth.
A stabilization of an object that is in motion. And precisely because the kind of organisms that at this period of evolution we are talking about have gate systems, namely are capable of mobilizing themselves, some at extremely fastest speed, the predatory activities require also to cope with them, to design modes of interaction, modes of spatial invariances that can solve the problem of a prey or a predator in motion, of an object in motion. For that, you need to have something that a representational model, or a spatial representational
model of something that is in motion but also at the same time without these representations become destabilized by the problems of mobility, by problems of an object that changes its position from one point to another point in completely unpredictable fashion. So these new forms of spatial invariances, invariances utilized to tackle with objects in motion are different from the previous forms of spatial invariances because they're
not just about rudimentary spatial invariances like orientation, trajectory, shape and color, What are modes of representation that require a stabilization of different types of information in order for the organism to change the parameters of its action as it's interacting with this
object in motion. There are simply requirements for any interaction with mobile object precisely because interaction with a mobile object requires constant updating of information. So if information is going to be constantly updated, then again, you run into the problem of computational cost, processing different types of information, changing positions, so on and so forth. So you essentially require a spatial invariance. I'm in class. It should be over soon. Oh. What time? I was just taking, putting that comment in actually. We got about 15 minutes, I figure, since we started like 10 minutes late, something like that.
Okay. Okay. So, yes, let's finish this. And then the next session we talk about the other direction of the math to this evolution as a kind of like a cognitive scaffolding of mathematical thinking and how it's connected to the mathematical intuition. So I was saying that this stabilization is really stabilization of motion or invariances of motion is what I propose to be the germ, the initial germ of mathematical intuitions. Now next session we look into more into these stabilization or invariances of motion.
a more technical, less evolutionary scenario. So I think it would be, I mean you can think about this a little bit more after this class that how can we see mathematical intuitions, mathematical objects and concepts as different variations, different more complex variations of invariances of motion which are again a spatial invariances but are complex spatial invariances. And of course how, you know, they are basically also connected with the problem of mathematical
abstraction and usually mathematical abstraction they all have a mobile components to them not simply in sense of concrete motion but certain abstract mobility okay let's go for the question and answers and then we can also extend time for a few minutes if you know the questions are being extended we are not a strict time limit. Can I hop in and ask a question here? Sure. Yeah, before we get too far into the sort of more social aspects and societal aspects of math,
I really like how you started the foundation within individual organisms and how the disruption of invariants can occur within the individual itself and how it sort of becomes like a site of agency in the sort of distortion of invariance and I kind of I kind of see some of this and the work that you did with Hecker how you're sort of challenging sensory stimulation to sort of challenge like external reality within the subject and I was just wondering if perhaps is there is there gonna be any sort of social implications like that that might be valuable to touch on now I know later on we're probably not going to be focusing on individuals and their you know their
sense systems and how those become like the sites of disruption yes I think well I mean specifically when it comes for example to sound I mean it's a it's not really a new topic that's us acoustic signals and especially are when we are listening to music, things like melody and composition and these kinds of stuff are always not really essentially acoustic signals or have anything to do with acoustic signals. It's the way that acoustic signals are always being processed in conjunction with spatial data like orientations, directions of sound. So in a sense that there are these huge amounts of the study being done under a field that's
called a spectral, basically it's a part of studying of receptive fields. The acoustic receptive fields share, are being shared also by visual receptive fields which means that some of our acoustic auditory intuitions, some of our auditory intuitions are in fact being informed by our visual intuitions. And some of our visual intuitions are also being informed and influenced by our auditory intuitions which then means that it's a huge field potential for further manipulation precisely because if you disturb some of these visual intuitions that have affected through the
course of evolution but also culturally our auditory intuitions then does it really lead to a different experience of sound? And then the question brings back to what you were saying, that how much can we highlight these, what you might call, different disturbances or different process of dehabituation of hearing on the level of simply, you know, experiential phenomenological encounter and how much can we register it on a semantic level. And the semantic level, I think, is the difficult one. That's precisely requires also understanding how some of these in fact quite rudimentary
perceptual mechanisms have impinged on the evolution of natural language and how we, our semantic capacities are formed in accordance with these quite, you know, specific forms of signal processing. There is this fantastic book by Claude Vandeloy called Spatial Propositions. It is not about acoustics, but it nevertheless talks about how perception of the space is tied to the evolution and a spatial proposition, the natural language. and what would be the anomalies caused if we create, manipulate language or if we had different kinds of perceptual mechanisms.
I don't know whether this is, and it probably does not have an E at the end. So yes, no, I think it's actually a really interesting question, but I think, as you say, the semantic level, is really important and I really need to think about it to see if it can be done effectively rather than just, you know, kind of feign or pretend that you are basically are actually modifying or destabilizing some of these, the semantic entrenchment of these processes of a stabilization and intuition.
Could I follow up with a question? Sure. So, I put it in the sidebar too. It's a question regarding the organism optimizing action and maximizing reward regardless of its context or its ability to do that in outside of its immediate sense context. I'm having a bit of difficulty seeing how we bridge the gap from like the immediacy of sense and the locality of it and the mapping of that and the memory and forgetability and how
that connects to mapping something that doesn't have context. I'm not sure if it makes sense. You can ask me questions too. Yes. I think the whole idea is that context independency doesn't mean essentially that it does not have any context. But it's simply something that has what you might call, it does not have a restricted contextual parameters or strict contextual parameters. Yes, of course it still has some constraints about context but these kinds of constraints can be malleable can be
plastic and essentially a spatial invariance is different spatial invariances as particularly the complex ones yes they have context and they are being derived from these contexts and the process of invariance always has some sort of connection with these different contexts what is that That's precisely because as different contexts by nervous system, sense impressions, which are context sensitive, being integrated with one another, the constraint of the context, the strict range of the parameters responsible for the identification of context, become broader and broader, rather than, but broader not in the sense that you lose your context sensitivity.
Yes, absolutely. I mean, navigation is a contextual problem. You can't use, when we are talking about that invariances are genericities or invariances are generalities that you apply in different contexts, this does not mean that they absolutely have, you can simply use them in any kind of context. No, they are basically what you might call that they can be re-employed within certain a range of concepts rather than just in a specific real-time instance within a particular restricted context. No, they can be reapplied, re-employed within a certain range of situations.
That's what invariances are. So you see that plasticity as coming from the factor of memory that forgets the part of memory that forgets yes but also the other parts of it that's it starts to fuse different reproduction can't in reproductions together different initial memories that have context sensitivity no sense impressions that are being identified under really strict parameters and how as the organisms reproduce these sense impressions it creates different deviations different versions of them and then how the memory puts these
different versions together and single out what they have in common usually regularities that what regularities are thank you sure I just want to tell you, if it's okay, about this project that I recently found. It's called SEPTER. And the idea is to build a mashable, deprogrammed, it's building a semantic mashable, fully decentralized
internet that completely operates through you know what they call it they call it escape but the whole in the the the projection of anticipated actions it's it's precisely that exact same thing and I just wanted to share that with you excellent so let me actually I have to copy paste these after the class thank you very much I will look at it it's incredible I think that it's actually this is going to be the pathway to break through the internet interesting also one of the things that I forgot I mean so basically this whole
idea of the what I said that a spatial a complex a spatial invariance of motion that has a sense of direction a sense of trajectory a sense of orientation and is also a visual line that is that you might say to be generic in the sense that it can be reapplied in different contexts it's precisely the kind of a spatial invariance, neurobiological spatial invariance, that if you did not have
that, you did not have the concept or intuition of Euclidean line. And we will talk about this, that Euclidean line, the so-called witless line, witless abstract line is precisely something what you might call to be a normative counterpart of this rudimentary causal proto-semantic abstraction, proto-semantic spatial invariance. Of course when we are talking about this connection between evolution and math, we need to be very careful what separates this complex spatial invariance afforded by the
nervous system to the Euclidean concept of a continuous line is a ion of natural and cultural evolution the nature of one is normative the nature of the other one is a causal but what all I am saying is that we could not have possibly have the normative version of the continuous line the Euclidean concept of continuous line if our nervous system could not afford us with that causal a spatial invariance that has trajectory that has orientation but it has context
independence in the sense that it can be really employed in different contexts the predator chase line you cannot you couldn't have the euclidean continuous line if you could not have the predator chase line in your nervous system more questions observations There's this one quote from, it's from Hegel's philosophy of nature. But Pinkard, Terry Pinkard quotes it in the Hegel's naturalism.
And it's a poster right here. It's so nice. It's such a nice quote. So it talks about how he says idealism begins with the organism being stimulated by external potencies rather than being affected by causes. And that is because it emphasizes the spatial invariance, because rather than, you know, I mean, you know, there is obviously something stimulating it. But the important thing is that we're constructing an invariance from the stimulation. And that's the most important aspect of it. Yes. I mean, that's essentially Kant's thesis that, you know, when he tries to talk about objects or object,
object is purely a constructible thing. you know there's this it's something that's you know the whole idea of different triple synthesis that happens at the level of you know apprehension at the level of imagination at the level of understanding these are essentially what you might call three stages of construction for you in fact to have anything resembling to any object or any concept of an object. The objects are pure constructions at different levels of invariances from the basic sense impressions to intuitions to imagination and understanding which at
that level we do not have absolutely causal invariances. There are what Kant calls pure concepts of understanding categories that's part of his Eschema Theism thesis we will we will talk about some of these as we move forward yes well one of the things I was free associating to as you were talking I kept coming back to cinema and the things that I like when you were talking talking for example about how the audio and the visual effects what audio you hear.
I remember a famous statement by Chris Marker who says, the audio track and the visual track are parallel paths that only occasionally intersect and cross. And they are in his films as opposed to traditional documentary where the image and the visual track just simply repeats itself. as opposed to presenting two different things that occasionally converge. And then, of course, when you talked about stitching together the images, the retinal scans, which are stills, I thought again of Godard's famous line about cinema's truth, 24 frames a second. And I guess it made me think, God, I should really be reading more of these people who like to lose on cinema or people who are someone has a book called the neural image i
wonder if that in fact i haven't read it but if if that in fact uh uh kind of riffs or uh speculates around things we're talking about now the vestibular system and stabilizing images and so forth that's the book uh i highly recommend it uh on on this whole scenario is by a very famous French neuroscientist who was a head of a research group at NASA studying astronauts' loss of sense of gravitational pull in space and how it affects their representational models and visual processing.
The book is called, which of course he had published both in very technical research and also in a very layman friendly book called The Brain's Sense of Movement. His name is Alan Berthold. Let me... The Brain's Sense of Movement. it's an extremely great book unfortunately it's out of print so if you can find it or you can borrow it from the library but yes that's but also I mean you know you mentioned cinema that's one of the things that usually is
there's this conjunction that's people who talk about the idea of so-called called Asperious Present or Asperious Now. It's simply what you might call a retention-protention account model of memory, or a durational mode of memory in which basically what are Asperious Now or Asperious Present? It's a very specific, extremely compact, quasi-form of time consciousness in which what you experience
as the present turns into an imminent past. What you experience as an imminent past turns into a distant past. And what you are experiencing as imminent future is becoming present. Your imminent future also, your distant past also becoming your imminent future. So it's like this kind of successive ordering of different time stamps that are becoming one another. This is called, so our notion of now is precisely this. this idea of transitory frame and the thing is that I think you know Deleuze
uses this but of course the origin of this is coming from in phenomenology and in phenomenology takes it from William James work on time consciousness and he's the one who actually puts forward the idea of a species now was experience present that our experiences are delivered within these units a species now where times has a kind of a synthetic durational property and without this minimal duration not in the Berksonian duration but simply transition fast transition between these timestamps you wouldn't have the other forms of
basically you you wouldn't have in fact even experience what you might call even Kantian experience you couldn't experience anything so these are the units of experiential encounters a species now or experience present so So it's a kind of, and later on, of course, once these species nows are transformed under, subsumed under categories of causality and, you know, fused with inner sense, they produce something what you might call a durational representation of time.
But yes, it's very interesting that kind of we have these kinds of almost neuro-phenomenological of some of our conceptualizations and some of our ways that we theorize about some of the stuff like cinema and as I said, even mathematical abstraction. What was the first name of the Crutchfield person you mentioned? It's James Crutchfield.
James Crutchfield all right thank you absolutely is there a last question or any other like short clarifications like that people want to do just we don't take up time and we got like a couple of okay cool so Reza do you still want to do 2.30 in the future as a set time or change that again? No, if you prefer it earlier, every one of you. I can move like 30 minutes earlier, but if you want to have it right at this time, I'm still fine.
Sidebar, is there anybody who would rather it be moved earlier, needs it to be? Yes, so no, okay. Yes, I will send you a reading list. As I said, today's session was just very, very introductory. It was all these kinds of, starting to think about these deep links of, or evolutionary, neuroevolutionary affordances that needed to be in place in order to, we that needed to be in place for us to have mathematical abstraction into details the next sessions we're gonna do 2 30 starting next week reading list is
gonna go out everybody got the syllabus if anybody didn't see the syllabus was quoted in the email with this link and then I'm assuming that there are at least a few people I mean I know there are at least a few people who don't have classroom access so if you could email me afterwards I'm also the mass email with instruction but if anybody still has trouble just make sure to email me because that's where the chat sidebar will be and that's where you how various stuff will get posted. Thoughts, questions, concerns? And also feel free to ask questions on the classroom page, and I will definitely answer them. And any kind of question, discussion, reading
recommendation you want, please post it on the classroom page. yeah we'll definitely put the syllabus in the classroom as well okay excellent excellent thank you so much this is okay thanks thanks everyone have a great weekend thank you bye bye bye bye bye bye Jake, are you there? Yeah, I'm sorry. Did you stop broadcast?