Maths & Ideas (Session 3)

I'll bite egg sandwich. Yeah, I woke up a little late, so I'm... Yeah, me too. Boy, that sounds good. At some point, I'm going to be listening, and I think I'll make some friends. Excellent. Okay, welcome, everybody, to Maths and Ideas from Antiquity to the Renaissance with Resumate. I'm going to hand it straight off to him so we can get started. Thanks, everyone. So, sorry for last session I was in Florida, I couldn't really find intern, I was just running around all the time. So let's start, see if you have any questions, any thoughts from the previous session and

then I start continuing a little bit about gestures, then we introduce our first pair of rudimentary concepts, recursion and iteration. And then, if hopefully there is time, I will introduce Egyptian methods, which are considered to be the first forms of algorithms. We will do some exercises, and I think that would be the material for this session. So, but before that, any questions, thoughts, observations

there's one question I had sure so my notes from last time you so I had a a chance to look at video for the first session because I hadn't thought that that was all about the evolutionary background to the capability to do mathematics and the last session you were saying that gestures represent a sort of reversal from yes though and so in predatory activity this whole matrix

of vestibular and categorical um a lot of that is used for predatory sexual um acts and that just was a reversal and they are for the sake of the process rather than for an act. So it's a mental act comes first. So, yeah, I was just kind of, that seems like something very mysterious about how that would ever happen, you know, or like how it's possible to go from the evolutionary development to the cultural development, or like what happens. Uh-huh, uh-huh. You mean, You mean, correct me if I'm wrong, what you are asking is that, so we know all this stuff

about these proto-abstract models of actions, vestibular, I mean the whole idea of the predatory invariances and things and then there is a gap you know of course like a vast gap between those kinds of uh proto-abstract actions and mathematical actions and mathematical norms yes and i'm curious about how to fill in the gap or specifically you refer to it as a reversal too like that that it's sort of a turning around of the apparatus in a way. Yes, but the reversal wasn't really about this gap. The reversal is the idea that gestures, so basically the whole idea starts with this,

that reversing from what to what is basically reversing from what you might call already abstracted concepts to back to their underlying mobilities, you know, kind of like the gesture of mobilities. We talked about this idea that Archimedes really couldn't deploy any form of experimental heuristic gesture unless he had such and such concepts like a concept of density, concept of weight, concept of volume. As soon as he starts to destabilize the relation between these concepts by way of a specific gesture and all gestures have, are connected to the

problems of mobility and invariances of a space, then he could reorganize different kinds of relationships between the concept of density, volume and weight. So that reversal was basically from already instantiated concepts back to what you might call those gestural possibilities, like kind of what you might call, if you imagine for example a diagram being this static frozen snapshot of a specific form of a spatial organization,

mobility and so on and so forth, then the gesture can be understood as a method of dethawing this stasis of the diagram and seeing what other kinds of mobilities or what other kinds of processes, also in a very Schottelian sense processes of individuation, can have given rise to this specific diagrammatic picture. So this is the, what I meant by reversal was this, that simply gesture is not some sort of like this in the sense of an activist or the way that at least the proponents of

extended cognition defended. It's not something that you simply remobilize heuristics without any regard for already instantiated concepts. But it is the whole idea that you start with really a domain that you have access to certain abstract concepts, there are certain relationships between the abstract concepts, then you would want to deploy a thought experiment and this thought experiment precisely has something to do with that kind of intuitionistic constructivist approach to mathematics that you use here heuristics, experimental methods, you use your own body as a kind of a virtual site

for this experimentation, like Archimedes. And then using these kinds of experimentations you want to reorganize and re-study the relationships between these concepts that you have already developed. So this is the reversal. Now, I think the idea from that evolutionary gap from those proto-abstract, proto-mathematical actions to mathematical norms. I think it is something that needs to be understood along different trajectories, but I would

say that language is definitely one of the main factors that can be understood as some sort of transitioning glue that fills this gap, not just language but language being one of the main factors, precisely because, you know, what language ultimately does is at least two things. One, the stabilization of some perceptual mechanisms and that's Stanisla Dehaan's thesis that the limits of language, particularly natural language, are perceptual mechanisms.

What is the significance of a stabilization of perceptual mechanisms? Precisely because once civilizations or basically any form of communication and accumulation of knowledge couldn't happen, that is really a significant epistemic constraint, unless you can stabilize, maximally stabilize models of action or any kind of model precisely because then you can, once this stabilization happens, language can do that, once this stabilization happens then for one thing it can be transferred across different generations but also members

of particular community can access these shared information spaces and that's what concepts ultimately are. Concepts are what you might call to be tools for informational chaos control precisely because they are kind of like these units of memory. I mean, when Tom talks about this, that imagine this whole idea that from an information theoretical standpoint is that any form of information bank that you have, and memory is one of them,

every time that you actually access it, if this informational bank doesn't have some internal stability, and that's what syntactic semantic language allows for this stability. If you do not have this kind of stability, every time that you access a memory, part of this memory changes. That's the whole idea of situatedness of memory. So basically, you introduce more inestability into it. So there should be a way, from an information the perspective, there should be a way to stabilize information by demarcating their boundaries by allowing certain formal computational mechanisms to be in place.

That one, this information can be transferred with relative stability across different, you know, diachronically, but also can be shared synchronically among members of different community. So this is one. Another one is that if you think about perceptual mechanisms, they are parochially representational. What I mean by parochially representational, precisely because they are representational in the true sense, causal representation. But causal representation always happens to be in a one-to-one correspondence with what it tries to represent. And causal representations are de facto representations, meaning that even though there are representational

indices, nevertheless they do not yield any form of semantic or inferential import. So one of the ways, one of the solutions to this problem of representation was the evolution evolution of language precisely because language, the symbolic regime and the units of language do not stand in a one-to-one representational relation with any item or object in the environment. They only stand in inferential symbolic relationships with the members, with other units of the

same language or other languages if we are talking about meta-languages. So imagine that these one-to-one parochial representational correspondences that we see in kind of those perceptual proof of mathematical abstraction are being plugged into the inferential networks of language where what is privileged from now on is not the one-to-one correspondence with something objective, namely representation of an object, an actual object in the environment, but really how this representation can be diversified within the inferential network

of the language, namely how concepts stand in friendship with one another. And that's basically, I mean, that itself has, you know, what you might call, again, different imports. One it allows, again, significantly reduces the computational cost of information processing. Because if you try to simply start to picture reality or picture objective environment by these parochial representational one-to-one correspondences, then you need what you might

call an infinite memory but also an extremely expensive computational cost to the point that it becomes making a model of the environment becomes almost impossible so what you need instead is a finite a limited finite report or of symbols that can stand in a combinatorial relationship with one another and then you would be able to plug these parochial de facto causal representational correspondences into these inferential networks. So you see that basically this gap between perceptual mechanisms, the mathematical abstractions and conceptual abstractions, namely where you have something like mathematical norms and mathematical objects,

is in response to so many things, I think, to so many, in fact, issues in both what you might call natural evolution and cultural evolution, in the sense that from an information theoretic perspective, a stabilization of information, from computational costs, namely idea of how to construct models, how can an agent construct models that are costly. Then representational problems, the whole idea that I said that processing one-to-one correspondent one-to-one relationship representational relationships is

untenable for information processing but also for memory so then you need to overcome this and our number other number of issues but just to sum up I think language precisely because it covers so many of these issues and solved so many of these issues I think it would be useful to think about at least not the only factor but one of the main factors and one of the main what you might call elements for filling this gap. Without evolution I don't think that you could have this transition from simple perceptual mechanisms to these kinds of formalism that are you know familiar in mathematics

so you're saying that or one of the things you're saying is that that means that like the birth of the other or the you know the storehouse of language like traditional knowledge that the rise of that can be explained in part by computational cost by sort of the efficiency of it basically. Yes, yes, yes. I mean that's the whole thing that I think, you know, this is of course should be interpreted usually this whole idea that language is so central is interpreted in a very conservative way, you know, the kind of like a philosophic, linguistic way that language is central and you know, sapient, use symbolic regimes and that's what makes them special.

But I think there is a much more important and non-trivial interpretation of the centrality of language and that's the idea of language as this vast computational framework that covers issues regarding computational cost but also ultimately language is a computational framework, meaning is a computation from a semantic perspective. At the level of syntax you will still have formal syntactic computation and the constraint of those syntactic computation. Then you have the pragmatics, the interactive aspects of computation. So yes, I think this computational, the computational framework of language is definitely needed

for such a mechanism from causal representational mechanisms to something like the formal regime of mathematics being sentiated another just like last thing for me like another question but that leads me to that I'm interested in and curious about is the degree to which language is isomorphic with the reality that generated it like I'm curious about a couple of different versions of like the structural realist position you know sort of from like like Max Tegmark and people like that sure and yeah do you

have thoughts about that yes I mean I would say that's the function of language, the function of language, I mean, what you might call the intra-linguistic, the intra-linguistic behaviors of language, I don't think that they have any form of correspondence with reality. I mean, that's the whole idea of the inferentialism, or what Szilard calls non-relational theory of meaning. But when you look at the units of language, particularly natural language, by the units of language I simply mean what you might call to be the naive grammatical

aspects of any natural language that yes I think there are there are representational no of not reality really but experienced reality and this is again not I don't think that's a new thing it's something that Kant and Hegel somehow we're trying to discuss as well in the sense that for example like think about the spatial and temporal connectives of natural ordinary language like and, on, within, before and after. These are derived from perceptual mechanisms in any language. But of course being also, you know, molded by certain cultural contexts.

But then there is a, I mentioned that there is a really fantastic book on this front, particularly on the spatial structure of natural language and how it reflects, again, not reality like in the sense of like objective reality, but perceptual mechanisms of experience. And that book is called, I think, A Spatial Propositions, A Case of Study in French, I mentioned by Claude Vandaloy. But also there is another great book on this front. It's called, it's by Wolfgang Welchen, Wolfgang Welchen, this is probably the spelling is

wrong. It's something about language. to find it for you. So yes, I don't think that on any level we can say natural language, let's say just restrict the answer to the natural language, that natural language have some sort of representational already instantiated representational correspondence with reality. representational correspondence with reality. They have representational correspondence with perceptual mechanism, namely experienced or experienced reality and that's a completely

different thing, it's precisely, it has something to do, again, back to the idea of the causal structure of perceptual mechanisms, but at the level of the function and interlinguistic behaviors I don't think that language has any representational capacity thank you just before we move on from this topic do you think how do you think that written language complicates that picture like for one thing in terms of alphabetic combinatorics for producing words versus like one-to-one pictographic correspondences with concepts and then also you know the way those two

different things like are different perceptual like not perceptual configurations but configurations of the faculties that we have for perception so like visual recognition of letters being directly pinned to auditory perception of sounds or basic auditory elements as opposed to pure picture to concept mappings, like is that kind of variation still within the computational regime you're talking about or? I think so. I mean, that's the topic that Katarina Dottel-Novais brings up. Yep, I think this whole idea of a non-pictorial writing system, which was developed, you know, after simply spoken language was a further you know intensification or

further phase in development of these non-representational inferential linguistic mechanisms precisely because what it allows you to do it allows it writing These systems, I mean, even in the most rudimentary way, can be thought as a form of platform or scaffolding for processes of de-semantification, simply or in the sense that you are capable of, it allows you to develop formalistic technologies.

I mean that's the whole idea, that formalist languages couldn't have emerged and formalist languages, by that we also talk about logics, mathematics, computation, theoretical languages, English languages couldn't have emerged if there weren't some basic technology through which you could further dissociate language, particularly the written language, from those you might call a still rudimentary forms of writing in which there is a form of isomorphism

between the symbol and what is represented. And this is, you see that all, I mean, in the evolution of the writing system you see that there is a direct transition from the iconic regime to symbolic regimes. With iconic regimes of the proto-writing systems are being based on some sort of what Peirce calls indexicality, in the sense that you have some sort of perceptual mechanism about some perceptual experience of something, for example, happening and then you are trying to represent it by way of a picture and this picture stands in iconic relation which is

also parochial but also arbitrary because iconic relations are based on similarity which means that they are already limited in terms of their representational range or representational capacities because they are based on similarity, they are based on overall isomorphisms. So this is the iconic sign or iconic regimes and then you see that as they move towards symbolic regimes, in the symbolic regimes all of these forms of resemblance or similarities or isomorphisms are being discarded and often what you have are really the combinatorial

connections between symbols and these symbols you do not need to have many of them to represent what is out there all you need is just a handful of them but you have precisely because of the discreteness because of that they are fully dissociated or abstracted from any pictorial isomorphic relationship with the environment, you are capable of combining them effectively. Once you have this syntactic combinatorial repertoire, then you can build on the kind of inferential dynamics or semantic dynamics that they ascend with regard to one another. Okay, cool.

And is that related to the way that once you have, I mean even if it's not alphabetic, if it's just like symbolic and not iconic written language, that you're able to sort of like freely invent new written marks for new concepts or new words or whatever and vice versa. You can write down a new word or a new symbol, whatever it is. Yeah, absolutely. the whole thing that you see semantic emancipation of language couldn't have happened this whole idea that you can invent and form new concept couldn't have happened unless you had in fact a more rudimentary syntactic technology that could support it with the iconic with the iconic relationships and with

those parochial isomorphic pictorial relationships, you can never in fact make this leap toward this semantic emancipation of language formation, you know, this kind of like a versatile formation of new concepts, precisely because everything is ultimately constrained by pictorial iconic relationships. As I said, iconic relationships are mostly between these relationships built on resemblance and isomorphism, and once everything is based on these kinds of isomorphism, everything can also be quite arbitrary, because that's

one of the critiques against iconic form of regimes and particularly Pers's defense of iconicity or iconic relationships happen to be quite arbitrary. Anything can be similar to something else based on some relationships and that's not really a good way to infer what other possible relationships two things might have with one another. Okay, that makes sense. Thanks. Okay.

So we talked about gestures and And I mentioned that the most important, we can't define gestures and no one has ever tried to even define gestures, but at least there are some useful points to characterize gestures, particularly the kind of mathematical, scientific gestures that mathematics and scientists use in order to embark on what you might call mathematical construction. And mathematical construction, as I mentioned, is very different from mathematical proof.

So what are these characteristics? One, the gestures involve with mobilities. We know the role of mobilities in perception of a space, particularly by way of forming new spatial invariances. He talked about that the entire human perception is entangled with the problems of a spatial perception. Even temporality for us, we always abstract it by way of spatialized relationships.

So gestures involve invention or at least perception of new mobilities. Mobilities can in turn, different forms of mobilities can reorganize how you perceive a space, namely, and that is, you see that this is in direct relationship with the task of geometry, geometry being an art of organizing spatial relationships, namely reconstructing

and deconstructing spatial invariances to find new form of relationships between the components, abstract components that you are studying. A big part of what you're saying is that with spatial invariances, like the capacity to within our perceptual capacity to construct a spatial invariance it already has within it and then later on with its development the capacity to bind the temporal dimension as

well so it ends up becoming almost a spatio-temporal invariant yes i mean this is one of the things things that you see, like particularly philosophers in early 20th century coming from the Husserlian phenomenology, that there is a constant struggle to try to see if you can develop a theory for perception of time dissociated from perception of a space.

I'm not talking about like philosophy of time and non-spatialized representation of time in that metaphysical, you know, German idealistic sense, but simply phenomenology of time abstracted from phenomenology of space. no one when you look into you know after Hosserl and Hosserl himself no one has managed to really pull this up to the point that you know in later Hosserl you get that there is a fundamental entanglement between representations of temporality and representations of a space so a spatial temporality really is

what you might call a spatial interpretation or a spatialized representation of temporality. This is one, I mean, and that's, there is a reason for this. We talked about this that what you might call to be the transcendental structures of experience and these transcendental structures of experience in Kant are outer sense, inner sense, intuition, imagination and understanding. We are not talking about reason because reason is not a transcendental structure but understanding

has rational components but also has structural components because it's derived from, you know, the functions of imagination and the manifold of intuition. Then you see that all of these, what you might call to be the transcendental structures are required for understanding, for development of categories, pure concepts of understanding, They all have in fact a spatial components. You have in causality, in the idea of modalities, the way that Kant tries to formulate them are a spatial components and not even ultimately for example, the idea of temporality in Kant

for causality in Kant, yes, it has sequences of before and after built on this idea that if you start to see a pattern emerging in which certain causes are always preceding certain effects, you would derive a category from them or capable of seeing an invariance emerging, but this idea that this particular perception of this invariance, namely causal invariances in Kantian understanding of categories, again is based on detection of certain spatial

invariances between antecedents and the consequence, between causes and effects. So yeah, there's definitely, I think, this is again quite a Kantian thesis that behind all categories of understanding there is what you might call to be the tyranny of a space and a spatial perception that everything that is built in categories all forms of categories in one way or another are connected to a spatial invariances

Again, I highly recommend on this front read Adolf Gurenbaum, The Problems of Space and Time. I think I have a question. I'm not sure if it's intelligible or not, but I'm having difficulty understanding how we determine whether the model is primarily spatial or primarily a causal representation of temporality. It sounds like you're saying it's primarily spatial and then the causal somehow fits into that, but it's not as fundamental.

No, it is. You see, okay, let me, sorry for confusing you. So it is, you see, when you In Kant, the idea of temporality, I mean the representation of temporality, or representation of time, which is an objective representation, is based on causality, on category of causality. And category of causality itself, the way that Kant formulates it, is about a spatial transformation and spatial invariances between what you might call befores, elements coming before and elements coming after, the so-called B-Series, sequences of befores and afters,

in the sense that if there is... So you see, what is really befores and afters is the so-called B-Series, they are not tensed temporality. It's just some elements that it stands in, think of like this, you have a film and this film, you call it the observer and that's really what the notion of observer in physics is, it's not really some agentic observer, it's simply you might say like a sensitive film or sensitive camera that registers certain changes or certain registers. For

example, a light makes a change in this film, then another light changes something else in this film, then the relation between these changes, these are not primarily temporal, there is no temporality in this because temporality is about the synthetic facts that they stand in a specific temporal relationship, the first light, the register of the second light and the register of a third light. This one second third, again you shouldn't interpret it temporally because these are causally de facto changes. So how causality is based with, is connected to the spatial invariance is the understanding

that the first, the second, the third register, they have gone through some sort of spatial transformation. one, and certain components have been preserved through this transformation, that in fact you are capable of linking them to one another, and hence saying that whether this is really a cause, for example, the first light caused the second change, and the second change caused the third change, or there was no connection between these registrations at all. This is something called the tracing method, or the marking method, marking interpretation of category of causality

put forward by Hans Reichenbach. And he tries to argue specifically that at the level of B series, namely before and after or sequential series, ultimately the idea of sequence, what comes before and what comes after, which is necessary for the category of causality, is based on some sort of minimal spatial invariances, what is preserved from the earlier series and carried over to the later series or what is what you might call whether there is a spatial

transformation between the earlier series and later series or there is no spatial transformation at all. Nevertheless the whole idea that you can connect and infer some sort of causality from the earlier series and the later series is how they are being transformed to one another or spatially undergo some sort of transformation. It's basically the idea of change, and change is not here becoming, the idea of change between earlier series and later series is undergirded by concept or by some sort of rudimentary forms of spatial invariances and spatial transformations.

Because if you do not have some sort of capacity to differentiate the kind of spatial transformations that these series go through, then there is no way for you to in fact connect these series together. Hence coming with the synthetic fact that the later series is caused by the earlier series. Hence what comes after is an effect of an earlier cause. Thanks, that was really clear and answered my question. Let me write it for you. It's Hans actually called I think tracing method, Mark method. Mark

Any question? I think I'm still trying to register it. I'm still not sure then. I understand that the way in which you would measure, that you would even determine that a cause happened is because some variable changed. or some variable was no longer consistent, right? Yes, either transformation or preservation. Right. Which of course these are co-constitutive concepts. Right, right.

I guess let's move on because I haven't formulated a question really yet. Okay. So, as I said, gestures, I mean the main characteristics of gestures, what's always accompanied gestures are mobilities and spatial invariances. And we saw that people like Longo, Châtelet, Bernard Tazier tried to argue that it's because

the most basic form for us to render something intelligible is done by way of detection of spatial invariances. That leads to the conclusion, for example, for Chatelet, that what makes mathematics reasonably effective is its capacity to render physical dimensions intelligible by way of reorganizing different senses of space.

So there is a, again, direct connection between the problem of intelligibility, rendering articulation of intelligibility, and the problem of organizing of a space. We saw this in the Archimedean experiment. If you remember, in the previous session I ended up with something called the method of cutouts and briefly talked about the idea of the Cauchy residual or Cauchy integral

paths. So let's… Could you… I didn't get the name of that in the last session. Could you type that in? Oh, sure, sure. Let me turn this up. It's Cauchy. Agustin Duikashi. So let me give you some examples of these forms of gesture and how you are capable of, By rearranging or reinterpreting the spatial organization of your entire system, you are capable of deriving new intelligibilities and basically rearrange the formal components

of your abstract system of mathematical thinking. So a good example of this, as I mentioned, is…sorry, I'm trying to… Can you see the screen, the iPad? No. Okay. Let me…

Are you seeing it? Yes. Okay. Okay, can you see the whiteboard? Yeah. Okay. So we talked for example that the experiment with Archimedes was to determine whether the

The votive crown was forgery or it contained pure gold. And this is the thought experiment led to the principle of buoyancy and re-articulation of a new relationship between weight density and volume. Now from a formal perspective or from a more abstract perspective, what ultimately Archimedes did, we mentioned that if we think of Archimedean thought experiment, top experiment, is that

he put himself in place of a body of water, he removed by the force of thought this body of water, put it outside, abstracted away. This is volume V of water, this is your system s of some fluid and you have some cut out here whoops

A cutout. So what happens when you have, by some force of thought, emptied or made a cavity within a system of fluid? We can think of the system of fluid before this thought experiment as a system of fluid being in the state of equilibrium, state of E, state one. State two, the gesture, this thought experiment de-stabilized this system, literally de-stabilized the system by opening a cavity in the body of water where you have a disequilibrium.

Why? Because if you think again very abstractly in terms of this thought experiment, what Archimedes does by opening this cavity within the system of water, the system of fluid, the water wants to seep inside, inside this virtual cavity, this virtual cutout opened up by Archimedes. So you have created, by introducing a designated instability within the system of fluid, which was in the state of equilibrium, you have created a new set of disturbances in the system.

And these set of disturbances in the system can now make explicit, new implicit relationships between the concept of volume the concept of weight and the concept of density now we have a new problem here this is and we can think of this fluid going inside Ultimately, what we want to articulate and what we want to study is really these forces,

these physical forces that are trying to push themselves in and fill this gap, fill this cavity. So we want to know a method of how these forces can be articulated, can be made intelligible, what these forces are and how they are going to transform their state of system from this designated inestability back to its formal serenity or formal equilibrium. Now any idea how these forces can be made intelligible, how these arrows, these trajectories,

points of liaison between the body of fluid and the virtual cavity that try to seep in can be articulated, that can be made in fact explicit, how can we make them intelligible? What would be your thoughts regarding a method that allow us to in fact see these forces these forces but also we can articulate them we can formally understand it any idea well we'd want to be inside right because we want to see the shape of the surface

of that internal cavity and how it's changing over time. We can infer forces from that, but we can't see anything inside of an opaque fluid, so we'd have to be inside the cavity. Yes. Yes, absolutely. So, basically, the first problem is that in order for, because you can never see these forces, so you need, you want to develop some sort of new, again, a spatial invariance. a spatial construct, a reorganization of a space that allow you to articulate and in fact see these forces. And that would be how you are going to see these forces, is that you want to see the transformation of this virtual cavity as the forces seep inside.

as the forces seep inside. Namely, you want to develop a method that can study how these forces express themselves, sorry, I don't know what's happening here, how these forces can express themselves along the surface of this virtual cavity. Namely, how this virtual cavity is being transformed from its state 1 to state 0, where it's completely being filled by these forces. This is a problem that Cauchy has started to think about precisely through a study of

hydrostatic pressure is something called a Cauchy path integral what is why is that it's integral because you want a method for integration of these forces these unseen forces along this along the surface path of this cavity in order to develop this path integral, a path integral that not only can study the forces along the surface of the cavity but can also trace the transformation of this cavity from a state

one to a state zero where basically the system retains its equilibrium, regains its equilibrium is development of a new component, a new mathematical object that is at the same time a point and a surface. It's something called complex point or Cauchy residuum.

The problem of Cauchy Residuum is also interesting. It's something like this. oops you see the state of shrinking the surface path

to the state of zero where the cavity can be said to be completely filled with forces of the fluid can be understood as moving you have two points on a surface on a line as moving from point x to point xn but what happens and you can be expressed mathematically as f of x equals to 1 over x minus xn. So what happens in this formula when x, when you move, when you push the, or you know, get closer, the point x to the point xn? In traditional mathematics, this

leads to absurdity, right? It becomes like this. So what would be the solution to get around this problem? Any idea? Did you get this part? Why it leads to absurdity? Because as you try to shrink this path from x to xn, bringing x close to xn, within this formula, you basically encounter the problem of 1 over 0, which is undefined. It leads to absurdity.

So how can you shrink this path without leading to an absurdity, to a mathematical absurdity? You need an infinitesimal, right? Infinitesimal, but again that one over x minus xn is really the whole idea of a new infinitesimal, new interpretation of infinitesimal. Even inside your idea of infinitesimal, Cauchy encounters a new problem that in order for shrinking this even further in a geometrical sense, he encounters the problem of mathematical

absurdity, 1 over 0. So imagine you didn't have any infinitesimal because you are already inside infinitesimals. What would be a way of bringing X close to Xn, namely shrinking this path, you are trying to shrink this surface, this virtual surface and bring it to zero where basically the equilibrium of the system is brought back. So how can you bring x close to xn without resulting in a mathematical absurdity?

So Kashi's solution is something like this. Just go around it. He discovers a new point called Kashi Residuum that can be understood both as a surface and as a point.

And that's really the whole idea of a new infinitesimal called Cauchy residual, namely which enables him to not only articulate, render intelligible, and make explicit the unseen forces of the fluid that try to push into the cavity, but also a method that simultaneously

studies and analyzes the shrinking of this virtual cavity from its initial state, its what you might call largest surface, to its most infinitesimal surface. So Cauchy Residuum is ultimately what you might call finding a method of mobility. Here we are talking about shrinking of surfaces. a method of mobility that can be seen as a way of constructing any form of surface,

any form of what you might call geometric surface. Once you have this infinitesimal, you can dilate it and construct and study the transformation of any form of surface. So, and Kashi is famous that he in fact studies Archimedes, reinvents Archimedes' thought experiment but within a much more advanced formal system. He sees the body of this cutout in new geometric terms and he tries to see what arises, what kind of new geometry arises if he introduces new forms of instability to the system, articulating

forces of liquid that try to seep in. First, what he wants to do, organize a space in a way that these forces can be articulated. And this new organization of space is a concept of path integral. What is path integral? It's really the surface structure of this cavity upon which forces exert their influence. This is one. And the second thing is a new spatial object or spatial geometrical intuition of point,

namely Cauchy residual, that allow you to study the transformation of every closed surface into a point, but without leading into a mathematical absurdity, namely 1 over 0. So, this is, you know, a kind of what you might call advanced reorganization. You have rendered intelligible the behavior of fluid system by way of simply reorganizing its spatial relationships. its geometry and reorganizing its geometrical relationships, the relation between the surface

and the point, a method of re-articulating the concept of geometric surface, so on and so forth. Another example of these gestures and how, for example, they are connected to the idea of a spatial construct and reinterpretation or reorganization of a spatial relationship is something that we will study much more detail, is invented by Nicole Orym. something called latitude of forms I'm not going to detail what this is because we are

going to talk about it in later sessions but imagine like this that the problem of addition you have three trees, how can you make a new mathematical object from a simple addition? Because you know that in rudimentary mathematics addition is a form of iteration.

And iteration is a trivial form of operation. It does not, what you might call, it never results in novelty. How can you express an addition, something that is inherently iterative by nature, in a form that can in fact result in novelty, creation of a new object that can be said to be different from its elementary components? idea of how you can put these three trees together in which the result would be non-trivial,

namely the product will be a different object altogether. What about something like Boolean algebra, like the OR function? Can you elaborate a little bit? Well, so if you're articulating it with the OR operation, you end up with a result that has, let's see, I'm not really sure, but it's different in a non-trivial way because it is a combination. Yes, but does it still, you see what we want to do is still we want to make the addition.

1,3 plus 1,3 plus 1,3. In OR you don't essentially get the addition, it's just you change the operation. Hmm. Okay. Okay. So Orym's, we will talk about this again very in detailed way. Orym simply changed, starts to think about that the relation between these trees is really this. What is really the spatial relation between these trees is that they are, they have been thought along a straight Euclidean line. That's what their geometrical representation is. So in

order to create a non-trivial product, you need to change the spatial relationship or the spatial organization of how these trees added together. And what would that be? Sorry, seems that I have problem to this he tries to distribute them along an optical line

So, would the presence of multiple trees be, so to speak, the gestural provocation to produce that new concept to integrate them along a single optical plane? Yes, you see, and of course we will talk about what exactly this, distributing these trees, adding these trees together within a new geometric construct, which is this idea of classical perspective really, which of course the first person who really developed the first fundamental, conceptual fundamental of classical perspective was Aurim himself.

Yes, so you see you have reorganized the space and re-studied or reformulated an existing abstract method here of addition within a new what you might call sense of a space or organization of a space and this reinterpretation, this spatial reinterpretation of an existing abstract method or concept can lead to a completely new result in Shatleyan terms, a new individuated mathematical object or a mathematical novelty, something that you couldn't hitherto imagine

from the perspective of your, what you might call, a stabilized or existing spatial construct which was implicit to your concept of addition. Addition as iteration 1 plus 1 plus 1 is like this idea that you are moving or distributing three objects along a straight Euclidean line. But imagine if you were seeing this Euclidean line differently. Then the addition of these trees takes a different shape.

Let me be ridiculously literal. Noticing that those are not the same trees, the trees on the line have been scaled up along the angle, and so they've been multiplied by a I'm wondering how that operation is introduced. You see, we will talk about this, that's why I wanted to leave it for like three or four sessions from now. So I'm sure that you know that in Euclidean geometry, and in fact we talked about it I

I think last session, we have some principle called homotacy or homotity, self-similarity, namely a scale invariance, right? You can contract or dilate any geometric object without transforming the essential internal properties of these geometric shapes or geometric objects. This idea has something to do as we mentioned with the parallel postulate. So you can dilate or contract a triangle without changing its vital properties.

What makes it a triangle? the idea of self-similarity or this kind of a scale invariance only holds true within a Euclidean system you are still these are still the same objects because you see from a mathematical perspective particularly in geometry and these people or him of course is working within an age that Euclidean geometry is at its peak, is that for example the idea of a triangle or any form of geometric shape is not defined by its resemblance, similarity, but is defined

by its part-whole relationships. For example, the idea that it should have three sides and these sides should have these internal angles. As long as the part-whole relationships are being preserved through this dilation or contraction, you would say you still have your triangle but you have something also more. You have a different spatial organization. So to answer your point, yes, that only works within a Euclidean system, where you have a scale invariances, where you have self-similarity. Well, what I was going to say is that, so the invariances change in each particular

level of geometry. starts off with a congruence geometry where rotation, translation, and reflection are the only operations permitted that preserve the congruence. But there's a step, there's a move up, I guess it's still within Euclidean geometry, into similarity geometry, I guess is what you're talking about. And I don't quite remember what the name for that is. And then there's one goes to projective and other stuff. And affine geometry where you can perform shear and strain operations. But you're saying that both the rigid movement geometry and the similarity geometry are just different levels of Euclidean space.

Yes, yes. Again, when we are talking about the Euclid's elements, I think when we are talking about similarity between, for example, two scales of this triangle, it's not similarity in a kind of again naive sense that similar this for example the idea of a triangle simply being similar to another triangle it is similar to another triangle by virtue of preserving some essential part whole relationships for example how the

relationship between the side the three sides and the three internal angles if you have the same internal part-whole relationships, then you would say that your larger triangle is congruent or equivalent to a smaller triangle. This idea of equivalence is extremely important. We are not saying equal, we are saying equivalent. Whoops. Reza, so when you initially asked the question of how we get from iteration to a new concept,

But I was thinking in the sidebar, it's that precisely at the point where we can explicitly mark the operation of additions that we have one plus one plus one. And then we immediately have something that can be equals three, which is our new entity. And that's still following this sort of like. um, erity conditioning the explicitness of an operation that makes this new This new number novel mathematical. Do you think there's like an equivalence there in terms of the spatial invariance Seeing the three trees next to each other versus the ability to linearly specify the operation of addition? Yes, yes, yes, absolutely. Yes Yes, yes, and this is again comes back to this idea that you know 1 plus 1 plus 1 equals

to 3 is really a diagram and depending on how you see the spatial configuration of this diagram, what you might call the productive mechanisms, mechanisms of individuation that were responsible for giving rise to number three, the product, the final product, we can put together these numbers, these ones, differently and create, again, new forms of mathematical objects by basically finding different processes of individuation that

could yield the same number. OK. And so is that? Again, example like this. You remember that we talked about this. So sorry, I have it. Oops. A circle is a diagram, but processes of individuation responsible for giving rise to this product,

this final product can be different. The idea of a gesture is to really melt this diagram and find its possible process of individuation. We said that it can be one like this, a curve or a straight line that has topological characteristics that you can squeeze it and bend it differently, or simply iteration of a smaller scale or point by virtue of the principle of self-stimilarity or homotasy from a center to its outer edge. These are two processes of individuation

that can be understood to be responsible for giving rise to this diagram, the circle. Cool. And all of those, I mean, I think every example so far, so I'm guessing this is sort of the next conceptual move. All of these involve recursion or a shift from iteration to recursion, right? Yeah, as soon as you have 1 plus 1 plus 1, you have like the ability to parenthesize the first 1 plus 1 and say I'm repeating this same operation. Now I have 2 plus 1 necessarily because I can put that equals at any point. And sort of the same story with like this series of concentric circles is at any point I have the whole circle and I'm like repeating that operation to generate.

Yes, absolutely. Yes. What is a homeomorphic function? Homeomorphic function is when you are capable of continuously, without damaging the internal properties of an object to a different object. It's basically continuous differentiation, a smooth differentiation. So imagine you have a straight line that you could, if it had elastic properties, that you could continuously bend it so you can create a half circle out of it.

Okay, so I'm going to stop with these gestures for now. We are going to get to our recursion, iteration and recursion topic, and then we come back can study these gestures that involve with mobility and forms of reorganization of the space as we move forward and we study them under different pairs of concepts particularly we I mean as I said one of the most really best

brilliant examples of deployment of gestures was by ORIM or I mean was even more adept in these kinds of simply creating mathematical novelties by changing the spatial parameters of existing mathematical concepts or mathematical relationships. So we will get to these as we move forward. For now, let's look at the concepts of iteration and recursion in a very rudimentary way and give a few basic examples that can illustrate the basic mechanisms undergirding iteration

and recursion. So should we have some sort of bathroom drinking break and come back, or should we just go on? If we could do maybe a short five or ten minute break. Yeah, superb. Yeah, sure. Sure, sure. Everybody run out for cigarettes. We'll see you in five minutes. Cigarette is good. refer to bathroom okay I'll take my fried egg break okay see you in 10 minutes then okay Fraser are you still there?

no Okay. Thank you.

Thank you. Thank you.

Thank you. Thank you.

Thank you. Thank you.

Thank you. Thank you.

Thank you. Thank you.

Thank you. Thank you.

Thank you. Okay, to answer Theodore about the question,

you might call you see we talked about that one of the main principles that derived mathematical construction is the invention of invariances because if you have a new form of invariances or what you might call abstract regularities you You are capable of deriving new relationships between your variables. We saw that, for example, the simple iteration of numbers created one result, that was one process of individuation, and another form of invariance using Euclidean homotasy or

self-similarity, a scale invariance, contraction and dilation, it still produces the same results but through a different process of individuation. So the process of creating new mathematical invariances which are fundamentally spatial, they relay some new forms of mobilities, spatial mobilities can be deployed in order for the mathematician to arrive at new forms of relationships

between variables, between mathematical objects. Once you have constructed this new geometrical spatial invariance, then you are capable of utilizing it to put it into exercise and create in fact new mathematical objects or sometimes new mathematical domains. Sorry, I think I came in just a little bit towards the beginning of that.

Were you responding to the question that I put in the sidebar? Yes. Okay. Yeah, I guess I understand how creating new invariants would help create new mathematical objects and new ways of imagining the space. I'm just not exactly sure how that even takes place. But... As I said, you see, think about this. that's mathematical construction if if we accept this thesis that not mathematical proofs what mathematical constructions there is an intuitive in

the Kantian sense there is an intuitive components to them namely they are somehow at their most fundamental core connected with sense of a space and organization of a space namely the problem of geometry then if we rearrange these parameters responsible for organization of a space or sense of a space then we might be capable of developing as I said new forms of

transit not only transformation with how variables are transformed being transformed to one another but also we are capable of distinguishing or We're extracting new formal relationships between our variables. You can think of gesture being about individuating processes in the mathematical domain. And what are these individuating processes? processes, namely constructive principles, all of constructive principles of mathematics

and this had to wait for the advent of late 20th century mathematics to be confirmed. All of constructive principles of mathematics are at their base geometric, namely they have something to do with how a space is organized, whether in the most intuitive parochial sense, whether in the most formal abstract sense of a space. You see, there is a historical story here to be told. Socratic, the way that they understood mathematics was using a language of force, dynamics, mobility,

so on and so forth. You see in Heraclitus, you see in Parmenidians, you see in Eleotics, Zeno, so on and so forth. But the history of Western mathematics after pre-Socratic was fundamentally changed by Aristotle. Aristotle believed that mathematical forms are pure forms, a very kind of a naive interpretation of Plato, whereas physics represents impure forms. Because these impure forms, why they are impure? Precisely because they are entangled with the problem of physical mobility. So as the Western Canon of mathematics emerged out of this Aristotelian Metaphysics, where

you have a disjunction between mathematical purity and physical impurity, geometric problems were being understood as some sort of what you might call a mathematical cop-out. It was always thought to be impure and numbers could be thought as real mathematics precisely because geometrical shapes, for example in Euclide, they still represent some sort of physical impurity. There are mobilities, there are spatial configurations, so on and so forth.

This metaphysics of mathematics was held until what you might call 20th century in fact, where the thing became reverse. Somehow people started to question whether you can in fact develop mathematics or understand mathematical creativity and construction in terms of pure forms, pure formalisms. So there was somehow this historical revival of pre-Socratic interpretation of mathematics,

mathematics has something to do with the logic of dynamic elements, mobility, spatial transformations, so on and so forth. And then you see for example, particularly after the advent of category theory, this is being fully employed in the language of mathematics. In order for us to fundamentally broaden the scope of mathematical universe, you need to bring back this... You shouldn't see mathematics as being fully abstracted from physics, but in fact what

What really drives mathematical construction, mathematical creativity is this implicit entanglement between physical impurity, the problems of mobility, and the problems of mathematical forms. Spatial invariances that are usually being relayed in the language of physics through different forms of mobility and dynamic transformations and on the other hand formal mathematical objects that have maximal stability once you define a definition for a mathematical object is going to remain the same

Now how can you manipulate these estabilities within this formal system of mathematics? How can you destabilize them in order to create new estabilities, namely new mathematical definitions, new mathematical objects, using, by tapping into this implicit entanglement, hidden entanglement between the language of dynamicity, mobility and forces within physics and the language of forms of maximal stability within the domain of mathematics.

So just like for help in visualizing, I think Theo is looking for examples too, so like in physics for example, the spatial invariance that we're talking about, are we talking more about like conservation of momentum or the inequality that defines the difference between space like and time like dimensions like in terms of sea? I don't know, how broad a term is spatial invariance, like with respect to physics as an example field? Well, you can think about this, that ultimately within physics, at least let's try to think about this problem anachronistically.

In physics, you have geodetic principles. What are geodetic principles? Conservation of energy with regard to the problem of the shortest spatial path. Okay. Hamiltonian dynamics. Okay, so the conservation law is included in the larger set of the definition of Euclidean or relativistic space, in terms of these geodetics of space-time, the metric. Yes, yes. The whole idea is that you see that in fact historians of mathematics now believe that

every time that a mathematical revolution namely the invention of a new mathematical field took place was due to this going back and forth between some new physical discovery and reinterpretation of this mathematical physical discovery within physics how you can use some for example some you know what you might call physical principle that has something to do with mobility and reinterpreted between mathematics and then use again this new mathematical interpretation applied back to physics to find out some more stuff about possible physical

invariances okay and this is like the space of the gesture right as you get this sort of orthogonal disruption from a new physical discovery and then your construction of like a formal invariant in mathematics is like the integration of that new of that new previously orthogonal yes yes in mathematical terms like I so my natural just because of what I've been working on elsewhere is to describe that as diagonalization but diagonalization of the philosophical sets also you know starts in math is that is it right to describe that space of the gesture as like being related to the diagonal method or to diagonalization arguments I

mean probably would be a little bit of we needs to be taken with the grain of salt precisely because it leads to some confusion but at least what you might called to be the main idea yes it actually executes the main idea of diagonal method yes absolutely cool cool thank you so um actually maybe Maria's question might help bring out sort of what I'm getting at too it says so would Would it be correct to say that mathematics is creatively destabilizing through gesture

while at the same time gesture being delimited by physical limits? Yes, yes, absolutely. And this has of course some really also this question if you confirm it or you affirm it, also has some bad news for mathematics as well which of course some mathematicians are willing to to to endorse it it's precisely because comes back to this idea that hunter was talking about and you know we talked about this idea that you know it's not about the correspondence with reality

It's about the correspondence with experience reality, the transcendental structure of physical reality, the transcendental experience of reality. So, yes, gestures are destabilizing mathematics in order to create more new forms of stability, namely new definitions, new mathematical concepts and objects. and this process of destabilization can be escalated but where the principle of the destabilization are coming from are coming from and not physics as such physics as pertaining to the reality of the universe but for the most part in

fact they are coming from the intuitionistic the experiential idea of physics we saw it like in thought experiments that mathematically like for example Archimedes or Aurene conduct in order to arrive at new spatial construct is that they are simply employing some, what you might call some physical aspects of, or some heuristic aspects of their experience,

and how these heuristic aspects gain traction on physical reality. reality so if if this is the case then it means that if there are limitations to the transcendental structure of experience the intuition the intuitive component of mathematics then by virtue of that we can we can speculate that that there are mathematical domains or mathematical objects that can never be fully unlocked or can can be fully explored through the employment of

these kinds of transcendental structures the transcendental structures of sensibility at the level of neurobiological organization of human agency at the level of its productive imagination at the level of for example categories particularly representation of causality and time so on and so forth um i have like a follow-up just just basically are you um are you describing a like a hegelian dialectical kind of vector in this process of back and forth between perception and knowledge

conceptual knowledge yeah and then never never fully arriving at reality and not being kind of like the driving force yes yes it is but I mean the thing is that when we are talking about the idea of dialectics we also need to be careful of not trivialize the idea of dialectics yes if we mean by dialectic as some sort of what you might call going back and forth by way of some sort of negative force or force

of negation. Yes, yes, absolutely. It is. It is Hegelian in that sense. So, actually, there is this fascinating topic in today's computer science which covers the intersections of logics, mathematics and computation. I have talked about this in the previous class, I think Jake remembers this. called computational Triturianism or the holy trinity of mathematics logics and computation the idea is that if we take computation as interaction an interaction that is

basically mobilized by logical negativity and logical negativity is really the idea of involution switching of rules between for example the falsifier, the verifier, the player one, player two, between me and you, you know, kind of a very traditional dialectical sense. And we found all logical behaviors and mathematical structures on this rudimentary form of computation, fundamental form of computation which is interaction and negativity. The so-called, this is in computation and logic, it's called the duality.

And duality is in volution which is negation really. is for example minus and plus so you have minus one you have minus minus one you have this this is involution and also you have interaction if you have these which are this is a fundamental all form of fundamental forms of computation are driven by the concept of duality or the process of duality then And they have observed and they have seen that behind all mathematical structures and

all logical behaviors and connectives can in fact be laid out in terms of various computational interactions, namely negations and interactions. Yes, so there is this even more fundamental dialectical process at the base of logics, mathematics and computation that connects them together. Which is really the whole idea of computation in the most general sense of it, not computability, is computable what is computation were you just referring to a book there by

somebody did I which which which was that oh maybe maybe you weren't oh I I did I talk about Holy Trinity on the Holy Trinity of math logic computation so this is the guy that has a fantastic blog but this is again that's not him but he is the one responsible for making a term for this but in the technical sense it's referred to as this grandiose title Corey Howard Lambic isomorphism or correspondence what is so basically

Corey and Howard saw that there is a fundamental correspondence between computation and logic. Lambeck saw that the Corey Howard correspondence, namely the correspondence between logical behaviors and connectives and computational processes also holds within the mathematical domain as responsible for the instantiation of mathematical structures and that was due to Lambeck. Joachim Lambic

thank you the what about the epistemological status of the thesis that all principles of mathematical construction are ultimately spatial is is that something that like how to what degree is there like proof of that or is it yeah induction mostly because of the way that math works like it could could it be proven to not be true you see I think there are two levels to this idea

of dependency on spatial invariances. One, you might say that spatial invariances precisely in that intuitionistic sense, Kantian intuitionistic sense, which is derived from the transcendental structures of experience. Which I would say that yes, at that level you might say that it is purely inductive precisely because of this entanglement between mathematical deduction and physical induction, yes, which can be disproved or cannot be proved at all. Nevertheless, it's just a thesis. This is one level. This is another level that you might say that we are not talking about spatial invariances

with regard to the intuitionistic component, namely the transcendental structure of experience, of a space, perception of a space, but we are referring to the mathematical concept of a space. This is, I think, a different question. This is something that from, you see that from 1960s, 1950s, this mathematical sense of space, namely the idea of formal geometry, and by geometry I do not mean geometry in

the sense of Euclidean geometry but what you might call to be formal understanding of topos from topology to topos theory but also formal understanding of categories and categories are generalization of sets, that they all can be proved or be shown that operate within all mathematical specificities, all mathematical objects are operating and exist within general mathematical spaces.

But this is a wholly different concept of a space than the kind of physical concept or experiential concept of a space that we have been referring to. Right, like a topological space. Yes, it's actually a topos theoretic concept of a space. And so there, I mean modern mathematics is really derived by this, again, by this process of dialectic between generality and particularity. Particularity being the domain of objects, and generality being domain of concepts.

concepts all concepts of mathematics are topos theoretic that's that's a fundamental insight that Alexander growth and they came up with and so and And what's inductive is the relationship between the topos theoretic. At this level, we are not dealing with any form of induction, really. Right, right, right. But I mean, but the connection between this space and experience space.

Oh, I think at this level, what you might call, there is still connection between those intuitive or intuitionistic methods of mathematical creativity. But if you have noticed that I… Oh, yeah, I see what you're saying. Yes, we are thinking or making this relation, referring to this relation only in terms of mathematical construction. But mathematical construction is only one dimension of the mathematical universe. In fact, the more important dimension of mathematical universe is the mathematical proof. So you might say that this relation between, this relation of this, for example, this topos-theoretic

interpretation of mathematical concepts and that kind of experiential understanding, inductive experiential understanding of a space only holds true for methods of mathematical creativity. But then mathematical creativity are not sufficient for broadening the scope of mathematical universe. They are necessary but they are not sufficient, precisely because they need to be tested within the formal proof theoretic dimension of mathematics. That does not, at that level, you do not have these relationships anymore.

So the fact that all math can be defined in terms of category theory has nothing to do with gestures. Gestures in that sense of heuristic, yes, yes. Gotcha. Yes. You have abstract gestures. You have abstract, you have mathematical gestures that are fully decoupled from the physical aspects of gestures. Okay, so let me read some and then hopefully if there is a time we will start our examples

like some rudimentary math problems from the time of Egyptian civilization. Before I start this idea of iteration and recursion, we know that in common sense, ordinary language, iteration and recursion both mean repetition, or at least convey a sense of repetition but then what is the difference between iteration and recursion if we say that okay they both imply some sort of repetition but then what

would be the element that distinguishes the recursion from iteration any idea maybe that recursion is nested that like after it completes a task it goes back up to a higher level yes yes absolutely anymore Jake? Jake, you're muted. I did have a question like 60 seconds ago, and now it's possibly a flip from my head.

Oh, no, no, no, it was actually just for Topos Theory, do you have a good book about Growth and Deke's innovations in mathematics or is there like I know a layman's introduction to top us theory is probably I don't know where would you go for like chief chief logic and top us theory like basic elementary category theory okay but yeah the basic category theory is the best best one is really the whole conceptual mathematics by Louvre and Chanel. But when it comes to topos theory, that becomes, everything suddenly becomes really nasty,

complex. I think really the best, what you might call, layman friendly introduction on at least the conceptual fundaments of tapas theory is the first i think two chapters of tapas of music by green Could you spell that out? Sure. Just the names. Topos of music by... Oh, not typos of music, topos of music.

music and the little airbook was what was the name of that one which one bear book oh it's conceptual mathematics by William lover and Chanel but you just type lover that's in a PDF that you've given us and yes yes yes oh that's in there cool oh well we're on the subject um we have readings up until next week i think or or maybe this week um are we gonna have more readings and actually no reading until we get to

the euclid uh elements all right okay this session and probably half of next session we are just talking about recursion and iteration and looking at the Egyptian methods, like the first forms of algorithms. And then when we get to the Euclid elements, yes, that's I remember that I suggested, Danielle Macbeth, right? I think so. You referenced two female authors, Macbeth and another one. oh yes yes yes yes i remember that yes but macbeth one is i mean you can just read the macbeth essay

all right so jack any idea of adding to hunter's um definition about nestedness and relation what makes them what makes recursion different from iteration is the idea of nestedness any anything that you can add to that well definitely before when I was thinking about sort of the explicit when you're posing of the operators it's like conditioning the new the production of the new mathematical structures that like I definitely just saw in my head at each point where the operation occurs you now have the opportunity to invent two to invent three if you just had an

endless stream of one plus one plus ones that nesting in parentheses each point at which that can happen is the point at which you can construct a new number to represent it so i guess in terms of the relation between recursion and iteration or the move from one to the next it has something to do with this ability to map to a new series or to map involutions or to map nestings to another series. That's the only way you define nestings, right? It has itself like a second iteration, which are like the steps of the recursion, like recursion at 0.n or 0.n plus 1. Uh-huh, uh-huh. But any idea, any of you, why this nesting is important?

I mean, let's think about it not in terms of mathematics proper, but why is it this idea of nesting is computationally important? Well, it's because you can specify a very small algorithm for a very big, potentially very versatile process. Right, like if you had to specify, if every algorithm had to be specified as its full length, as like a listing of every single step, then no algorithm would be shorter than, no program would be, no program could be shorter than the algorithm carried out. True, okay. So, because of the pressure, it's possible to have small.

Yes, it's a compression problem, yes. Yeah. Good coding, yes, and good coding comes back to this idea of, again, compression. But what else? Self-similarity? Yes, self-similarity is a result of it. And cost, yes, of course, when we are talking about compression, we also mean memory and cost. It's basically also the instantiation of long-distance rules. Because if you have nestedness, you create new rules between constituents of your recursive

regime. So what is exactly the technical definition, does anyone know about the technical definition of recursion? Because we know that recursion is a word that is being thrown around quite liberally, but Does anyone know what might be the technical definition of recursion in which we can distinguish recursion from iteration along these lines that you have been referring to, you and Hunter? I'm a really fast Googler.

OK. Recursion is a specific type of hierarchical process that embeds constituents within constituents of the same kind. So embedding of one constituents within the constituents of the same type is what recursion ultimately is. And then you see that this embedding process, by definition, results in nested hierarchical

structures and the kind of compression that you were discussing. Theoretically, would that be like sets that contain themselves? The construction of sets that contain themselves? Yes, but we will look into it. How is this possible? I mean at least what would be the most rudimentary form of embedding of constituents within constituents of the same type. So the first thing to mention that is very important is that iteration differs from recursion.

First thing is that iterations, first of all, they are both forms of coding operations. You can organize data by way of these two forms of operation, iteration and recursion. Now one of the ways that iteration differs from recursion is that iterations are memory-less generative processes which allow for identical repetitions. See, they are memoryless. I will make an example to make this clear. Recursions on the other hand require memory of operation, previous operation, and can precisely because of this retaining the memory of the previous

operation, they can generate hierarchical relationships or dependency relations between constituents. So this is really important that recursions precisely because they retain memory of previous action, they can generate new dependency relations between constituents of the same type. For example, think of this, you know, the instruction, chop the garlic until it turns into a paste. You do not need to have the memory of your previous action.

It's a memoryless operation. This is an iteration precisely because you do not go back and build on top of the memory of your previous action. Now, whereas cut the pizza into six equal slices is a recursion because you cannot create the final product, namely pizza cut into six equal slices, unless you preserve the memory of your previous action. First two slices, then three slices, then six slices. Just the

whole idea that every cut needs to be built on top of the previous cuts. The Wikipedia page for recursion suggests the making of sourdough bread because you You need a batch of sourdough bread from the previous time you made sourdough bread to start the next process. So you require memory of the substance that you used to make sourdough bread the previous time to make it again. Yes, I mean cake making is the best example of recursive operations. You still have stuff that are completely undifferentiated, but nevertheless this whole idea that you it's very different from simple cooking and mixing stuff together precisely because you need the memory of your previous executions.

Yes, absolutely. Iterations cannot contain recursions but recursion can contain iterations, yes. So back to the idea of chop the garlic until it turns into a paste, which is an iteration. Since the operation requires no memory for reaching the terminating condition because there are no dependency relations between iterations of the operation, the very fact that you can liberally iterate the process without preserving the memory of the previous execution means that you do not create or generate new dependency relations between

your repetitive actions. So iterations doesn't generate dependency relations, one. Whereas when it comes to cut the pizza into six equal slices is recursion because it requires memory of each instance of the operation that has taken place. So in this case, each instance can be said to be dependent on the previous operation. First cut to 2, then 4 on the basis of 2, then 8 on the basis of 4, for example. Now, when you look into, as I said, the concept of recursion is being thrown around quite

in a really like a wide connotation in computer science, in math, in logics, in language. For example, you often come across this definition that recursion is defined as an operation that calls itself. Recursion as an operation that calls itself. But the problem is that these kinds of formulations don't actually define what recursion is. They merely describe how recursive procedures behave. So what we need to understand what recursion is, is really this understanding of the embedding,

of a specific type of embedding, embedding of constituents within the same kinds of constituents that generate new dependency relations and these dependency relations are hierarchical within a nested structure between the elements or constituents of your system. And once you have dependency relations you can extract long distance rules between how these constituents can be combined or grouped together for example

let me bring the ipad again Thank you.

Thank you. Thank you.

Is Raza still there? I think we lost him. Jake, I think you're muted too. Jake, I think you're muted. Oh, sorry. We've lost a bunch of people, haven't we? Did all of the... I think a couple people were dropping out just because they had to leave earlier.

Like, to live back to this. Right, right. Well, let's just... Sorry, everybody. Let's just stay in here for a few minutes. Just anything you wouldn't say in public because I'm not going to stop the broadcast. Hi, everybody at home. And we'll just give this five or ten minutes. I'll email Reza and see what's going on. Has there been a formal reading list created yet for day by day? Or is this just... Actually, in the classroom, I went through the last two chats and compiled a list of, like, names and books mentioned and things like that. I'm going to add the ones from today when we're done,

and then I will try to go through and, like, anything. And, like, I'll email Reza about, like, PDFs for some of them because, like, the McNanny book, whatever it was called, Philosophy and Geometry is, like, all of that guy's books are a bunch of money. But some of the stuff is online, and I'll, like, go through and try to find links to the PDFs. okay cool thanks yeah I found out PDFs for most of this stuff yeah there's playing like money's on our right I assume a bunch of his stuff is at least what what about realizing reason did anybody find out but Danielle Macbeth physical realizing reason Danielle Macbeth was in the reason I tried it

an ARC I couldn't find it. Yeah I just did. I wondered if anybody had found it. Yeah I'll keep an eye on it. I brought it to my own student. What do people think about the idea of kind of bracketing some of questions until the end so Reza can get through iteration recursion and Egyptian mathematics otherwise we will be perpetually falling further and further behind I mean sorry apparently as I turned the iPad on it kicked me out of computer and connection everything all together okay back so instead of me trying to turn the iPad on so I will just type it here so you

can see iteration something like this that it creates cycles whereas if we embed constituents, for example let's imagine that these alphabets were constituents of the same type, let's call it type beta. If they were all constituents of the same type, the idea of embedding them within the constituents of their own group would be something like

this and This way, you can execute this process of embedding, embedding procedure with multiple constituents of the same kind as well, and generate even more complex hierarchical branches.

Now, using this embedding, Embedding, this embedding that pertains to constituents of the same type, generates dependency relations in a self-contained hierarchical way. Also hierarchies or dependency relations it generates are self-similar. that what you are ultimately doing is that you are embedding constituents within constituents of the same type, hence the total self-similarity of your structure. Now the advantage of the recursive construction is that the rules of new generated hierarchies

don't need to be determined in advance and this kind of permits for a what you might call a flexible constructability, a flexible construction. So another important point for understanding recursion is that it allows that you do not need to fix or determine rules in advance for how this procedure should, or this hierarchy should be constructed. Now, again, from a kind of a cognitive perspective, recursive methods are cognitive friendly.

They can create abstract prototypes that can be subsequently permuted and diversified. They allow combinations and abstract groupings, for example, production of complex concepts out of more rudimentary concepts and models that require qualitative distinction between hierarchies and their roots. Now precisely because of this cognitive friendliness of recursive methods, they are prevalent in language, in computation, in mathematics, in simple abstract activities, so on and so forth.

But also there are prevalence precisely because at least in part the prevalency is due to this evolutionary tendency toward a kind of representational encoding that allows organism to contextualize complex situations and parse hierarchical information or the relations between wholes and their constituents without deviating too much into details. This whole idea that I mentioned that they compress information and they also create self-contained hierarchical structures.

From a computational perspective, recursive procedures prevent combinatorial explosion when you are encoding information, when you are building a model, a representational model. And that's exactly ultimately what recursive methods do. They represent a complex environment as a self-similar structure or a global schema in which some details or relations between elements can be implicitly represented without the need for developing separate abstract representation of each distinct hierarchy

or set of dependency relations. So in other words, rather than differentiating the specificities pertaining to each set of dependency relations, namely details concerning constituents, the self-similar hierarchical structure produced through the recursive procedure allows the selection of one set of dependency relations whose attributes and behaviors can then be applied after the appropriate permutation

to the rest of the environment, to the rest of the totality of the structure. So then it allows you, precisely because of this idea that it compresses information and And then you can extract a specific set of dependency relations and use these rules, these long distance rules, reapplied back to the totality of the structure. So this idea that the definition of recursion as an operation that calls for itself is not really the definition of recursion but it's a result of recursive procedures. A rule, a set of dependency relations is being extracted as a rule and then reapplied

back into the totality of the structure. Go on, Jake. Oh, sorry, I was just thinking about reversibility and irreversibility. like the hierarchies created like is recursion kind of is this is a technical definition is it agnostic about whether you are able to go back up the hierarchy as opposed to there being like one-way cascades of information or dependency like when change on a top level cascades down through recursion to change at the lower level like picking up dependency

from the changes in between each time. Does the kind of one directionality of that, is that generally characteristic of recursion? Yes. Yeah, okay. Yeah, forward chaining. This idea of forward chaining. And then you see in the, I will talk about, it's actually connected, you can see it very much in the very idea of computation, you know the idea of sequent calculus particularly the recursive interpretation of modus ponens. Yeah I read some of the- I will talk about this, yeah sure. Yes I mean so you see that it's basically

Actually, another much sophisticated version of this interpretation is in the idea of cut elimination in natural deduction, particularly sequent calculus, or abstract rewriting in computer science. We're with abstract rewriting. That's like not just refactoring but like constructing something that has to be compiled or interpreted into the original language? Yes, abstract rewriting, not that definition of abstract rewriting. Abstract rewriting in the sense that once you have, for example, constructed a proof of a type, then you can completely get rid of all the details about the construction

on how this proof was constructed and then use this proof to bootstrap new further constructions out of it. Okay, I got you. Like normalization process. So... So in this sense, you know, the recursive encoding generates a self-similar hierarchical representation through which attributes and behaviors of various dependency relations between constituents, whether they are of the same type or not, can be efficiently guessed and anticipated.

And that's the whole idea of self-similarity. similarity can also from an evolutionary cognitive perspective you can see as a technology that allow you for anticipation of a pattern and hence you see it in the idea of inductive reasoning. But of course this is also a flaw of it precisely because if recursive methods are preferred from an evolutionary cognitive perspective as a result of computational limitations of organisms through which you can always anticipate certain patterns. But these patterns that are specifically generated or you can only guess them precisely because

you are using a certain embedding procedure, how you encode information. So there is this, what you might call to be the codependency between the procedure through which you have made your model and how this model is used to anticipate behaviors in the environment. So if your model, the procedure by which you made your model is faulty, then your anticipation is also faulty. So again, recursive encoding is vastly preferred because it enables long distance induction

in the statistical sense of induction, you know, this whole idea of anticipation or predictability. And which of course is again being reinforced by the idea of self-similarity or the kind of rigid regularities that recursive methods generate. If an organism can represent its environment as a self-similar hierarchical structure, namely build a recursive, recursively build a model of the environment, then it can efficiently perform inductive moves, anticipate how certain behaviors emerge according to this recursively

instantiated model. I mean, very rudimentarily put, it can navigate an unknown situation without having all the information in advance. It can reduce uncertainties, compress information, overcome certain issues related to computational cost, to the cost of memory. It can predict how hidden variables work or how a complex situation unfolds in real time. From a certain perspective, we are evolutionary trained to anticipate self-similarities in our environment and apply models of recursive encoding to various phenomena because these

abstractions allow us to respond to situations quickly and reliably. So while these kinds of inductive capacities are extremely valuable for the acquisition of early cognitive capabilities, their extended use as modes of thinking rather than modes of model building, I would say is actually detrimental if not disastrous. That's because they are developed to resolve problems associated with evolutionary biological

constraints, the computational cost of the agent, the memory, the combinatorial explosion, so on and so forth. They do not provide us with means of systematic information processing. that the recursive procedures they generate long distance rules. Long distance rules as Jake was suggesting and as I mentioned they are limited by the idea of forward chaining. So it basically completely discard the details that has come before, render them implicit so you can only work with the explicit rules that you have extracted through your recursive

hierarchical structure and then reapplying back to move forward. So basically you are discarding huge amounts of, while it allows you to parse information reliably and create an anticipatory model from a statistical sense, but it also throws up a lot of useful information but also precisely because recursive methods again from a cognitive perspective they are directly connected to bias in entrenchment why is that you can see it if an error has been introduced at one level of your recursive procedure precisely

because the whole idea of recursive operation is an operation that preserves the memory of the previous action, the error will be transferred from an earlier stage to a later stage. You can see then, application of recursive methods from a cognitive perspective can entrench previous biases as statistical biases but also cognitive biases we will see it i mean you can you can see it in real algorithms i mean probably not today but next session we are looking at the egyptian algorithms if you make one mistake in one line of your recursive operation how rudimentary it might be

it will be transferred and carried over and completely gives a real deviation from the correct answer. So as I mentioned, I mean, recursive methods are useful for model building, for creating methods that are mathematically and computationally reliable, but they are not really good for systematically looking at details, namely systematic thinking.

For example, recursive encoding forces us to think via strong structural rules. due to the self similar you know hierarchical representation it creates we start thinking only via weakening and contraction does anyone know what weakening and contractions are in logic. So it's basically weakening and contraction is the idea that the structural rules of logic

are three, exchange idempotency of entailment and monotonicity of entailment. Idempotency and monotonicity of entailment, or weakening and contraction, concern the idea that if you add new premises to your antecedents, the conclusion or the consequent won't be changed. So they are basically the idea that you can easily, within a certain range, if you duplicate

add new premises or freely copy your existing premises that doesn't really result in change in the consequence. So it's basically based on this idea that precisely because of this strong structural rule that you have in classical logic between your consequence and your antecedents, P's then Q, you can easily transform and manipulate your antecedents and your consequence always will be true if your initial antecedent was true. It's precisely because of this long, strong structural rules that are preserved between

consequence and antecedents. So I kind of dropped the ball and completely forgot that we're not ending at 5, and we're like 35 minutes over right now. OK, so let's just finish this very quickly. Actually, let's just talk five minutes about this. As I mentioned, another great example of recursive method is in computation, the very computational interpretation of modus ponens. So what is modus ponens? Modus ponens is simply kind of a Latin word for the way that affirms by

affirming. For example, if there is cloud, if it rains it's cloudy. Today it rains, therefore Today it rains, therefore it is cloudy. So a formalist interpretation and a computer use of modus ponens from A and A implies B we deduce B. So it's like this so A B

Thank you. and a so if a is true but also a implies b then b if today is rainy and also every time that is

rainy there is cloud in the sky then what today is cloudy so So modus ponens from A and A implies B we deduce B, that is, it's, that's also this in computational interpretation of modus ponens also is called operational semantic, consists only in controlling in a mechanic way that define the sequence which codes for the second A. Following this, then, like if then the structure, then the machine writes, actually copies B.

This sequence matching, rather than the idea of pattern matching, this sequence matching is basically what ultimately a digital computer can do and it's a completely a recursive method. There is no general pattern, just sequences. Modulo, a very simple syntactic unification procedure is being produced by way of recursive method by way of how you recursively handle the law of modus ponens in computer science. And of course, one of the, you know, what you might call paradigm shifts in computer

science emerged precisely because once you suspend the structural rules of classical logic and you disturb modus ponens, then you can't simply use this recursive procedure in computation. Hence your paradigm of computation also shifts precisely because, and that was due to Girard, because you can never fully replicate, copy or add new premises. So the idea of the recursion, we need to, I mean, as I mentioned, we need to understand

it, that while it creates an almost mechanic way of creating modules, and precisely because is reliable, is efficient and it creates new forms of dependency relations that you can extract and turn them into algorithmic rules or computational rules, it is preferred. But precisely also because it is built on some structural rules but also throws out details about how actions are being performed, it has its own limitations. And these limitations are specifically highlighted

when either the recursive procedure is applied incorrectly, or an error creeps on your recursive process. And then we will see in the next session that what happens precisely and first how this recursive method is being instantiated in mathematical thinking by way of introduction of the first algorithms and basic operations but also the kind of consequences that might arise when this recursive methods, rudimentary recursive methods are applied incorrectly to mathematical procedures. Questions?

I've got like three, but we've got, I don't want to like make this run on so long that people have to leave and miss things that at least give me one of the questions I got one which is the issue of path dependency and of sort of historical evolutionary bias is getting entrenched upstream so is this kind is one of the places where the space of the gesture comes in where so even if they're not like errors that get fixed in but there are things that used to like cognitive biases that were useful like you know over recognition of patterns apophenia I think that were useful in a certain context later on encounter whatever it is that makes them suboptimal that wasn't like part of the

environmental experience of the operational environment before and then you need you discover this sort of space of plasticity yeah yeah yes and Basically, we might call the gestures capable of destabilizing the kind of rigid, recursively instantiated patterns. And is this kind of what we mean by generative entrenchment? So looking from the other side, like the ability of path-dependent biases to open up these spaces of gesture, or like condition, production of novelty? But basically you might think about gesture that instead of being forwardly chained, it's a backward chaining process.

Hence the idea that I said is reverse. You destabilize a current recursively instantiated hierarchy or structure in order to access the kind of information that was responsible or the kind of performances or actions that were responsible for the individual for individuating this hierarchical structure namely this pattern okay so kind of like like breaking the black box in order to observe like new symptoms that will tell you what's for not symptoms but outputs that will tell you what's producing

what's inside the black box in the first place? Yeah, kind of like that. But also in the sense that if you think about that, let's think about this, that precisely because of this, the cognitive dimension of recursive method, you can see that recursive methods are context-independent. They are context-insensitive. Precisely because of this context-independency, they can also be misapplied or they also have limited range of application but gesture is precisely by deconstructing the recursive method rather than abiding by it by deconstructing

in the precise sense of deconstruction by deconstructing the recursive method is capable of seeing possible contexts, basically we contextualize your recursion, your recursive, your individuating processes. Okay. You can see that basically gestures here, you see that for example some recursive method that you have been using in algebra, once you deconstruct it by way of a mathematical gesture, then you can find actually use a novel or find a novel geometric context for

this algebraic method that can not only changes your existing algebraic method, but also create a new geometrical interpretation of this algebraic method. So this, like, if we were to transport this very summarily over to philosophy, this would be something that Deleuze really excels at, right? Is sort of deconstructing a particular set of philosophic ideas and thereby creating new spaces in other branches of questioning in philosophy to apply? Yes. Yes. Okay, cool. Thank you.

I want to ask, because I, this has, I sort of notice, especially when it comes to, I guess, ethical questions just from day-to-day life, like, that, like, if I, like, I'll end up thinking about a problem that I haven't reflected on for a while. and then I it comes to me like I used to think about it but then I see that like the my my operation of how I determine ethics has sort of changed and then you know that sort of makes me deconstruct the black box that constructed that old that that the old perception and um

it it changed like to me it's like like what you said how it changes um it um it changes the old mode of perception while also making a new one on top of it as well yes it's it's what you might We will talk about this, but what you might call to be another hidden principle that drives the domain or the universe of mathematics and this dialectics between stability and instability, equilibrium and disequilibrium. Not physical essentially, but conceptual in the sense that you have a stabilized concept,

gestures destabilize them, and this process of destabilization initiates a dynasty of new problems that can lead to new, higher levels of stability. And then again, the process will, gestural creativity will be applied again to these new forms of stability, these higher, more elevated forms of stability, and then ad infinitum. how you basically broaden the scope of your mathematical intuition or geometrical intuitions. So would that deconstruction or that destabilization itself be a recursive process? Yes, but that would be more as a

yes, that would be a recursive process, yes, but not in the in the probably not in that technical sense that we have been talking about in the, you know, the idea of introducing constituents or embedding of constituents of one type within the constituents of the same type. But it's more in the sense of just like what you might call the ordinary linguistic sense of recursion you know it's just a form of bootstrapping really thank you okay last call I think because we're at 420 so we're quite a bit over anybody

Anybody else, one more? Just a very simple question. Do you have any kind of recommended readings that put together Archimedes, Cauchy, Path Indugrills and so forth and lay out what's going on in a way accessible to a layman? I'm typing it here. Sorry for the caps. Oh, great. OK.

And just another comment. I mean, we're dealing with kind of boundary concepts that are on the boundary between philosophy, mathematics, and experience or whatever, and tracking between them. And I'm wondering if we could, I know that I noticed a tendency to want to translate every one of these concepts, let's say recursively, back into philosophy and trying to figure out how to frame it philosophically. And I think I would love to hear many more particular mathematical examples to see how a particular concept plays out within mathematics spatially

and so forth. So that, you know, and then maybe save the tracking back for philosophy after we're totally grounded. Yes, yes, that's actually what I'm planning to do next session. First, I mean, give the most rudimentary example of recursion. And that, what would be the first rudimentary example of recursion in mathematics? The relation between multiplication and addition and division and subtraction, but by way of a completely different method that we have been taught in elementary school, by a method called unit fractions, the Egyptian method of algorithm. Right. Okay, cool.

Unless anybody has questions about next time or anything like that, I think that we're going to finish up here. So again, Reza, we're good for next week. Yes. Yes, absolutely. 1 p.m. EST. Thanks, everyone. See you next week. And I will put the books definitely in the Google Drive. Oh. Yeah, thank you for the extra time. That was really awesome. I appreciate it. Take care. Have a great day. Bye bye. All right. Take care, everybody. Bye bye. See you. OK, this time I'm going to stop the broadcast. Sweet.

Thank you.