All right, welcome to the first session, the second module of complexity and computation. We are still in the first module, sorry. It's the last session of the first module, yeah. Oh, we're one behind, right? Well, kind of, yeah. I mean, I promised to do some examples and stuff. All right, so welcome. Yeah, okay, fine. Welcome to the fifth session of the first module. All right, Reza? Okay. Okay. So, before moving forward, any questions, comments, discussion from last sessions?
Last session we talked about Kolmogorov complexity, right? Yeah, Kolmogorov complexity, yes. And we touched a little bit of sort of Boltzmann and Gibbs and the sort of thermodynamic treatment of entropy. I think that was the last session, right? Yes, yes. And also the end of the one before it, yes. But it seems like in thermodynamics, we have sort of both this presentation of state space and presentation of dynamics, or forces that are sort of acting, or what are the physics of complexity
or kind of well-described thermodynamics. But in information theory, like Shannon and Komogorov, They're both sort of describing dynamics or a process in particular, or they're measuring how an object is generated. But actually, there's no forces or physics. Yeah, so it seems like almost an incomplete correspondence of thermodynamics. So it's pure metaphor, but it leaves out something really meaningful, actually, which are sort of constraints.
In thermodynamics, we have, you know, under fixed pressure and temperature. And then in information theory, we just very loosely make reference to some unobserved, or in Shannon's, We don't observe the generating process. We just measure it empirically and then say, yes, yes. These things, then yeah. Yes, that's true. Yes, yes, very interesting. Yes. No, I mean, the thing is that you are completely right. I mean, the only measure that kind of has kind of like loyal to this original Boltzmannian thermodynamic, basically, content of the system
is really the measure of thermodynamic depth. Whereas Shannon and Kolmogorov, it's basically, as you say, it's a highly constrained, as I said, it's basically the same formula of Boltzmann, but without the physics behind it. as you say, thermodynamics basically reappropriated as the informational content. And that's simply at the cost of basically massively truncating the underlying physical forces, physical phenomena behind it. And that's partly due to one of the things that
there is a response for this is basically when you move from the microdynamics or microphysics of thermodynamics to microphysics of thermodynamics, then you can do, in fact, talk about entropy as information content. And that's basically how, basically, they justify this transition. But nevertheless, as you say, at the cost of massive filtering out of this kind of interesting physical phenomena. That's why I said Boltzmann, even though it's still a purely physical measure of complexity, this idea of entropy and thermodynamics, is still very interesting to study it,
without this kind of linking it back to Shannon and Kolmogorov. Because once you talk about actual thermodynamics in a Boltzmannian sense, then you can also talk about temporality and the arrow of time and entropy gradients and so on and so forth. Whereas in Shannon entropy and Kolmogorov, likewise, you don't have really the factor of time. And that's what we talked about in terms of modern measures of complexity, Kolmogorov and Shannon are atemporal, whereas, for example, Bennett's and Crutchfield's have the factor
of temporality and cost. I think cost is really a missing bit here. And in thermodynamics, you also have this other force, enthalpy. So you actually have, in Hamiltonian mechanics, you have this idea that it takes something to move and that something is being consumed in these dynamics, even under the constraints, like the very strict constraints. But in information theory, things just change atemporally and without any kind of ending. And we just sort of observe these transformations of process and then measure that as complexity. So it's actually describing something else. The system is really unbound in the observation.
In information theory versus in thermodynamics, it actually seems pretty concrete what we're talking about. Yes, yes. Do you have a good reference material for this Hamiltonian dynamics behind, and it's linked with both one in thermodynamics. Well, yeah. I mean, it's the very definition of Bolson mechanics is under these constraints, right? But is there any good book discussing the relations between the two? Yeah, I found some papers actually
sort of outlining the sort of systemic description in information theory versus, in particular, even describing the difference between Shannon and Kolgorov in terms of the role of the objects versus the process and complexity. Yeah, I'll throw those on the drive. SPEAKER 2, super, super. Excellent. SPEAKER 1, and I guess it's like the topic of my very late assignment. Just kind of like. SPEAKER 2, OK, OK, excellent. We're really looking forward to it. Okay, questions. Aaron, okay. Sorry, I just yesterday I saw your axiomatic stuff and I wanted to answer but then I thought if I answer it then I will spend my entire
day working on the answer and not preparing the slides for today. Okay, no, I thought we were going to try to prepare something. Yes, yes, yes, yes. There's iron behind, so let's stop. Go on. Ask questions. Oh, good. Can I get in? No, I was just hoping, I thought we were going to open today's discussion with that, I guess. So I was hoping, like, we would have time to prepare something, but I guess we can push that back another week, or I'm still confused as to what that assignment was. OK, the assignment was a simple question. Considering an axiomatic system with the idea
that axiomatic system and axioms respectively have different meanings and basically have different connotations as what an axiom is. Now, considering this, is an axiomatic system factoring in and taking into account in a specific axiomatic system and the notion of axiom that we are talking about. Is an axiomatic system considered to be a deep object or not? And discuss then. Why? Because as I said, for example, in
You see, this discussion is a bit tricky. And that's also, basically, I'm trying to crowd sourcing your brains to also answer the question myself. Because for me, it's also kind of a difficult question. Because not only that different axioms and different axiomatic systems can be treated differently in terms of this logical depth, but also There is a kind of a jam-spaced nature to this question. Because, for example, a book, as I said, you know, basic example, a book in number theory, you have basically premises.
Then you can derive all of your hypotheses and theorems from these premises to very astringent kind of operations. And you can actually, for example, in some cases, for example, in Halbert in axiomatics, you can easily use basic operations of iteration and recursion to generate theorems from your premises. So when you have that, then it seems that in the context of that book, the book on number theory, for example, written in the Hilbertian school, in form of Hilbertian school, cannot be considered
as a deep object. Because there is no, basically, time, a slow time scale that is involved in generating your theorems from your premises. Whereas you think about, for example, another axiomatic system, for example, Euclidean axiomatics, where axioms are basically intuitive in the Kantian sense. Then there is a link outside of this book. So basically, you can't contend your process of generation of theorems from axioms to simply what you have and those simple operation of iteration and recursion.
Then you step beyond the book. Basically, your lived experience in the world. And that's basically the idea of the Euclidean and the geometrical term, that you can simply stick to these kinds of purely formal rules and basic operations of production of theorems from axioms. You need to encounter, basically, is a kind of Kantian sense. You need to navigate the world to understand, really, in the Kantian sense, apprehend, basically, what, for example, a line is. You know, it's basically the whole idea of the long go
and Alan Bertho and all of these people coming from the Poincaré School of Geometry, that, for example, the concept of the line, the intuition of the line cannot be formalized really. It's basically a product of million years of evolution, in the sense that, for this whole idea that Berthoud talks about in his book, basically how do you get the intuition of the line? What are its evolutionary basis as a lived experience? This idea, for example, the vestibular system, the inner ear, is basically a gadget for detection
of gravitational force, vertical gravitational force on the planet. Then the saccadic eye movement, basically on the retinal saccadic eye movement, is detection of orientation, visual orientation. So for example, a predator who detects a prey in the environment, it orients its body in the environment, so it involves the body, the body movement, locomotory command. And the saccadic eye detects the orientation of the prey in the environment. As you put your body in the environment, oriented toward the prey, the vestibular canal, the
whole idea is that the gravitational force and its pressure on the vestibular system needs to be understood as an inertial system. So once you move your body in a direction, in a specific direction, you basically diverge from this inertial field. And this basically breaking of the inertial continuity is the first formation of the line, the basic core of the intuition of the line. It's a kind of you create a continuity. though you do not know as an organism doesn't know about any of this stuff, doesn't have any kind of intentionality. But nevertheless, these are the kind of elementary components
out of which our conceptual intuitions of lines have emerged. So that's why, basically, you have the axiomatic system of the Euclidean system. You have generational theorems from axioms. For example, in a Euclidean book about geometry, but nevertheless, you can't simply talk about the relation between axioms and theorems in terms of simple generation rules. You need to step beyond the book in order to explain and understand and see the origin of the axiomatic systems, namely the intuitively lived experience of the organism within the
environment that makes these intuitive axioms possible in the first place. So in that regard, it can be, for example, said that Euclidean axiomatic and Euclidean, for example, axiomatic system, is a logically deep object, because it involves massive time scales of evolution. I guess this is similar to what Bennett refers to as, I think it's like the non-arbitrary causal history of pi or something like that. The pi being a similar one of these. Yeah, yeah, yeah. Just for me, also, it helps in Kantian terms. Or I guess also it's a, Nietzsche asked the same, sort of asking this question, is this a deep object? I'm trying to sort of rephrase it as asking the Kantian question,
like, how must a world be constructed so as for, in this case, rather than morality, for geometry to exist, right? Sort of what would have had to happen so that we as creatures can intuitively perceive geometry and then having to, like, go back through sort of the whole causal history of what... Isn't that what you're doing? Yes, but the thing is that you see, I think these discussions are quite tricky, and I'm still trying to grasp this. Basically, I think it's fundamentally people like and this kind of Kantian term have tried to talk about this. this is that basically once we talk about this evolutionary time scales and try to basically
describe the phenomenon that are at stake that we have reached to this level, this is a kind of, first of all, it's teleological, you know, weak teleological project. I mean, it's not bad, it's not a kind of like a faulty teleological project, but we need understand it's a teleological project. And we need to be very careful in treating it so we do not have kind of like a theological teleological project, let's say here. But nevertheless, when it comes to the teleology, basically we are talking about functions, teleological functions. And these teleological functions
are always basically analogically presented. They are analogically presented with regard to our linguistic conceptual resources. So basically, in fact, every talk of functions in any system, whether it's basically simple heart valve to these kinds of evolutionary function, these are analogically presented. And we need to be careful when it comes about analogy. Analogy is a good. It can be a virtuous circle. because that's all there is basically in describing evolutionary forces and evolutionary functions. But you need to be careful that you do not overextend analogical models, basically.
And that happens a lot of time in neuroscience. It's basically people inflate their analogies. They talk about intentions as if, for example, an organism has intention. but just it had intention by analogy to our basically linguistic sense of what's at stake. And if you do that, if you basically are careless with your analogies, you basically, you can And start, as I said, inflationary attitude usually leads to deflationary attitude. Basically that's what so many people in neuroscience and people like the kind of very rudimentary
and skeptical account of people like Scott Baker. It's like this. Once you do this, once you see, for example, that an organism has an intention, an intention is simply being simulated by information on, sorry, modular information processing units inside, you know, for example, the nervous system, then you start to deflate, basically. You start to reduce without any constraint whatsoever, human intentionality to that of a basically rudimentary nervous system, or basically the modular processes. But that's the whole point. It's an analogy. Analogy needs to be taken into account within the primacy of linguistic resources
that make this analogy possible in the first place. And when it comes to, for example, the construction of AGI, human-level AI, I think it's fruitful to talk about these things in analogical terms. And in fact, again, in a careful, cautionary, controlled manner. We do not overextend analogies. Simply, in Kantian terms, we can think of the construction of AGI from this kind of basic mechanistic automaton, which is kind of isomorphic to, for example, a rudimentary organism. Analogically, start to posit this and explain its functionalities and behaviors,
and then construct different analogically posited additions, different, basically supplemented, makes this AI more complex in each stage. And each stage can be understood as an analogically posited stage. So you basically, in Kantian sense, this idea of a stage construction, multi-stage construction of AI can be understood as a kind of analogical Buddhist wrapping, that you start from a very constrained analogy to the point that where basically you do not need analogy because the AGI is simply isomorphically
mapped to your linguistic practices, precisely because it is capable of doing the same conceptual work. So I think this idea of analogy and, you know, the talk about evolutionary functions are really important. And people who do AI work and people who do evolutionary biology need to take this into account. Otherwise, it's really, really easy to make those kinds of inflationary leaps and in the same way also make the deflationary movements. Once you do this, you become overexcited about your project or very skeptical.
For example, you say, for example, this kind of basic statistical inference model of AI can simply be understood as a kind of human level AI. Or you say that, for example, as the kind of the skeptical projects in normal science, say that, for example, human intentionality in the linguistic sense that we understand can easily be reduced to these modular information processing units. basically you can justify it. Yeah, so I guess like two questions from these then. Yeah, I guess I'll go to the second one first.
So would I be wrong then in understanding sort of this idea of logical depth of these other kind of mathematical formulas like formal representations of this idea of complexity and depth? as like a step in, as an important step for artificial intelligence in kind of giving a basic formulaic understanding of what it's like to have a complex causal history like this. So to be a reasoner is to have a history. Yes. Will we go back through and draw the inferential connections in that history, not just as a sequence, but as like a recombinatory inferential? Yes, yes, absolutely. It's basically once you have it.
And this also brings another discussion to the full, the idea that in classical project of AI, it's basically based on kind of a explicit behavioralism. And basically, you have an unconstrained account of functions. You simply, basically your functions, you can easily generate these functions through basically pure abstraction, formal abstraction, precisely because there is no causal history involved. And this was one of the many faults of the classical projects of AI. When you look into, for example, new advances in AI,
kind of like the work of Itamar Aurel and other people, is that they talk about this causal history. And this is, again, can be thought about in Kantian terms. So many people understand Kant as a functionalist, precisely because of that you can talk about cognitive abilities in terms of functions. But the thing is that also some people, because they think that functionalism is only about this input-output inferential process, that you can purely abstract it, formally abstract it, they say that Kant is absolutely not a functionalist. But the thing is that Kant is a functionalist,
but a functionalist whose account of functions, whose account of cognitive abilities, are linked and constrained by causal history, by link to mechanisms. And this is explicit, as I said, in his account of basically threefold synthesis, specifically the synthesis of, basically, intuition. It's the idea of the inner sense and outer sense, basically intuition of a space and intuition of time. And these two basically are part of the lived experience of the human, of the cognitive
agent, which can only be thought in terms of its causal history. And that's basically what sensibility is, receptivity to how objects affect you, namely have a sufficient structure to be differentially responsive to these stimuli, you know, of items in the world. So there is a causal connection here, and respectively a causal history that constrains these accounts of functionalist, accounts of cognition in Kant. Yeah, I think my second question is then on sort of moral metaphysics and Kant and where this kind of gets both more difficult and then more problematic. I think you were talking
about when people start giving sort of just so evolutionary stories and sort of equating. I don't know, you either have skeptics like Backer or sort of the kind of positive psychologists in the New York Times who say that we're naturally wired for morality and kind of make all of these bad conflations and I think Kant is good here because he's sort of explicit about this being sort of analogous in your terminology or sort of the going back to this idea of the technique of nature from the critique of judgment where he says we kind of then pose this teleological view on natural phenomena sort of necessarily in order to bring out sort of a conceptual understanding of its history,
of it having a causal history. And that makes a lot of sense. And I guess you're sort of giving the explanation of that from the point of view of sensibility, but when we get to sort of language and moral metaphysics is where I have the question. And I guess now I'm forgetting the name of the book, but it was the book on sellers, like the joining the images. Fusing the images. Yeah. Or sort of the implicit project of, was it Jay Rosenberg or was it? Jay Rosenberg. Yeah. Someone like that is to sort of rehabilitate that sort of Kantian moral metaphysics, right? Yes. Yes. But actually, Rosenberg is really, I think he is, basically, he's usually considered
to be a right-wing Szilardzian. But the thing is that he's not really a right-wing Szilardzian, like in Millikan or Zeitz's sense, basically, that they actually justify these kinds of moves. For them, there is no difference between sentience and sapience for right-wing Szilardzians. For Rosenberg there is, in fact, because he is a holistic Szilardzian and he is extremely Kantian. So he actually, there is a book I mentioned, Thinking Self. He talks about these kinds of things there and also his introduction to Kant, which I said, best book I think ever written on Kant, most comprehensible at least, accessing Kant.
He talks about these things, actually. But yes, there is, as you say, there is this kind of still these kinds of problem in Rosenberg. I think this is why I think ultimately when it comes to this idea of normative ethics and kind of basically the modal vocabularies and stuff and its relation with empirical sciences and empirical observations. Brandon's work is absolutely the best one, because it's basically the whole idea that you can deploy an empirical-based vocabulary of any special science
if you do not have a modal vocabulary, basically all of your empirical observations are can only be generated by use of modal vocabularies otherwise it couldn't be done so it's it basically it kind of brings the primacy of language and the centrality of language which we will talk about all of this you know in the third module what I mean this really I think the centrality of language is really important so many people have talk about centrality of language, but the account of language they usually provide are extremely impoverished. Basically, language is reduced to some sort of like a symbolic medium, you know, repertoire of codes, or you know, like Habermasian system, or kind
of like a bad Hegelian, basically a medium of social recognition. Well actually I think if we are going to provide an account of centrality of language in sapiens cognition and also talk about all of these relations, basically how we talk about causation, about temporality, talk about, for example, empirical observation only within our language and basically they are only analogically presented in relation to our linguistic resources, is that we need to provide an account of language that allows for at least three levels, syntactic,
semantic and pragmatics. And we need also to be able to link a natural, basically a naturalistic account of linguistic evolution with an interlinguistic logical conceptual function. Basically the semantic core, semantic complexity of language. And I think a way of doing this is using kind of like modern tools of substructure logics, like Jean-Yves Girard linear logic, or computational games, paradigms of computation as interaction that's put forward by people like Abramsky
and Andreas Blass. We need to talk about these things. So we need to be able to talk about different levels of computational complexity within language, different levels of basically symbolic functioning, or sign design, sign behavior. And then we are capable of talking about basically what is it exactly in language that makes cognition, human cognition, possible in the first place. Yeah, and then with the goal of answering the question, I guess I'll put it in the Nietzschean
terms then, like how did nature create an animal that can make promises? Yes, or basically how nature allows for an account of rational behavior that is autonomous, autonomous from its causal history simply. This is Salar's, you know, ending question at the end of his essay, This I or He or It the Thing which Thinks. You know, is there, you know, the ultimate task in this kind of project would be thinking about an account of, you know, causation, an account of nature that allows for the intelligibility
and autonomy of reason. With reason simply being another register of what we call sapiens. Yeah, that's all. And Jim O'Shea was the other thinker I was thinking of. Yes. I mean, he's very good. But as I said, you should definitely, any of you who are interested in Kant and Szilard and Brando and this kind of stuff and Hegel, definitely Rosenberg is one of the best and kind of very obscure figure, is a really, really great thinker. Was. Questions, discussion before we start?
Adam is thinking. Yeah, I mean, I was trying to grapple with this topic around the writing. And my background is the software background. And I was relating these things to software, but while also keeping in mind your description of... Firstly, software is largely deductive, in terms of what it's constructed on.
It's a sort of symbolic foundation or substrate. that sort of not that undermines the idea of a complex system right and so and then sort of there's not an obvious thing to say okay software is a complex system and then the sort of critique right I was wondering if that was useful territory or it was too vague right you could sort of explore ways that software engineering in practice faces this problem of complex systems and that could be useful ground to write on. I think it is.
I think software and programs, program or basically, you know, I think a more general term in theoretical computer science, a program, you know, and softwares are kind of, you know, different, you know, constraint types of programs. A program is absolutely a complex system, I think. Because once you have programs, you basically do not essentially need to think about this kind of recursive loops and pattern matching and stuff. You can have functional programming, basically. The territory of functional programming requires a different understanding of program and how axioms and theorems are basically connected together,
which is basically this idea that we will talk about this in terms of computational games, logical games. And I think they are extremely, you know, they have, they might have, you know, like some of the, you know, lack some of the feature of complexity that we talked about, but they nevertheless are complex, and basically they are resource sensitive. They require massive amounts of scheduling with physical systems, with actual computers. They are basically interactive in the sense that axioms interact. Rules are not a priori generated, but are generated as basically you move in the programming stage.
I think programs are complex systems. I will try to talk about this in the next few sessions. And especially, we talked about concurrent processing and stuff. I think anything that involves concurrent processing, any kind of interaction, basically, because interaction is essentially a resource sensitive mode of computation, needs to be understood as a complex system. And the problem of, I think, programming, not only in the sense of how you schedule and synchronize them with actual physical systems, but even within their own symbolic medium,
is a problem of complexity. OK, cool. So it sounds like fertile territory. So yeah. I mean, it was basically saying to me that, Especially because software doesn't really assist in the abstract. It exists to solve some problem in the world. And so you very rapidly get tangled in the complexities and the dependencies of the world as well. So yeah. OK, cool. I'll take that away. JOHN MUELLER- Sure. And I think the reason that I talked about Bennett's
and Crutchfield, especially, for example, Bennett, it's basically his measure of complexity is basically a measure of computational complexity. And we will talk about computational complexity. And I think it's very good that basically we distinguish computational complexity from that kind of causal physical complexity. There are correlations between them, but we don't essentially need to restrict computational complexity to that kind of causal physical complexity. In fact, language, natural language, but also theoretical formal languages
are completely complex artifacts in the sense of computational complexity. And we will talk about this at the level of syntax, semantics, pragmatics. These are different complex. And without these kinds of complexity, no matter of causal history does not amount to something like human level of cognition. Yeah. OK, cool. OK. So what I'm going to talk about today, I promised to talk about a model of, talk about, for example, an example of complexity
phenomena at the end of the complexity module. So today, I'm going to talk about, basically, deploy this example. But the thing is that I started to think about different concrete examples, like ecology, economy, and stuff, use them as examples. But as I started to think about it, it seems that a lot of what is usually in social sciences and environmental sciences, even in military studies, passed off as complexity are examples of complexity
in the way that they model them. They are extremely impoverished at the level of mathematical conceptual modeling, logical and mathematical modeling. They are using extremely clunky math to basically model a complex phenomenon. So this brought me to this idea that maybe I should talk about a little bit this idea that if we are going to talk about complexity, we are going to appropriately model complexity, we need to all also be able to update our mathematical toolbox. We can use a 19th century or early
20th century mathematical field or mathematical armamentarium and use it to model a complex phenomenon that, for example, has hierarchical systems, time scaling, modularity, so on and so forth. So this brought me to the idea that, basically, the idea of today's session that first I will talk about a little bit of this importance of developing a proper mathematical toolbox for modeling.
And then I will talk about our example. And our example won't be economy or ecology. It will be connected to our course of our three modules. It will be about the brain, neuroscientific modeling, modeling of the brain. And the mathematical toolbox that I'm going to talk about is category theory. But also, you can extend it to algebraic geometry and topos logics and topos theory. First, one thing that needs to be said about this idea
of modeling, modeling physical systems and complex physical systems, is that people usually use mathematical models to engage with these kind of systems. But the thing is that mathematicians or physicists or biologists who use mathematical tools, mathematical derived from a specific mathematical field, are often blind to the fact that application of a specific mathematics to a physical system can only be justified on the basis of a logical conceptual
model that is already in place. So for example, applying a differential equation to a system requires a background, basically, of a logical model that makes this application justifiable in the first place. Now when it comes to complex systems, there are really strong mathematical tools for this application. But we yet do not have a good logical model through the lens of which we can appropriately apply a mathematical instrument, a mathematical model,
to a physical system. Without this logical modeling, we are basically at risk of always, as I said, applying very clunky mathematics and irrelevant mathematics to a system. So I want to constrain the argument to specifically hierarchical systems, poly-stochastic systems, multi-level systems, and suggests that category theory is in fact a useful logical conceptual model for understanding these systems on the one hand, and by developing
a categorical theoretical logical model of complex hierarchical systems, then we can also think about applying the right mathematical tools to describe and model hierarchical systems, complex hierarchical systems. So first, the first thing that I want to talk about is a little bit introductory account of the relation of category theory to hierarchical systems, and then talk about understanding of and describing and modeling of human mind based on category theoretical terms.
First, I need to know how many of you are familiar with category theory? At least just the basic stuff. Yeah, not at all, sorry. Not at all, OK. OK. Yeah. Yeah, only loosely. OK, OK. We talked about category theory a little bit about the basics in our, basically, previous New Center class, the New Rationalism one. So just to get the grasp, I mean, the best book,
best introductory book that you can get into category theory is Conceptual Mathematics, written by William Loubier. there. It's really kind of like it starts from the most basic stuff and goes to our very difficult stuff. This is one, there is another one written by Giuseppe Longo for Computer Science. That's also a very good book. And you can find basically so many helpful blog posts online doing the basics of category theory
and really explain it in kind of layman friendly terms. OK, I will talk about the kind of very, very rudimentary basics of category theory. But I will leave some of this stuff out. And then you can go and explore them by yourself. Like the concept of core limits, which is something that I will talk about a little bit today. OK. So before sharing the slides, let me just talk about a little bit of this whole hierarchical
systems again and the question of basically the ontology of hierarchical systems. You guys know anything about ontology and information systems? No? OK, I will talk about that too. So basically, very rudimentary ontology is kind of like a very broad concept. It can be applied very differently.
The word ontology in an information system primarily deals with levels of description. Basically in order for us to talk about, for example, a physical system, we need to have different levels of descriptive tools. And these descriptions need to describe accurately and faithfully the reality of that existing level in the system. So basically, ontology is also connected to the traditional understanding of ontology and philosophy. So for example, semantic web, the idea of semantic web is an ontological concept.
Basically it's what is usually talked about in terms of fundamental ontologies. Fundamental ontologies are basically, some are fundamental and some basically are called Meso-scale ontologies, is that there are descriptive tools that try to deploy semantic connection and map semantic connection between concepts and vocabularies and basically logical axioms that can, as I said, that can describe faithfully a specific level of a system. Now, in this sense, for example, when
we are talking about, for example, a system, like say an airport. An airport is kind of a social system, but also is a mechanical system. It has so many different components. the interaction of humans, norms of behaving at the airport, surveillance and cameras. So ontology information system, if there is an ontology of airports, it tries to provide us with descriptive logical tools that are capable of basically not only differentiating
these levels of a structure and functioning at the airport, but also apply faithfully appropriate descriptive tools to a specific level. For example, how cameras at the airport interact with one another, how basically people interact with one another. So it uses, for example, a descriptive tool. For example, at the airport, we need to have a set of descriptive resources for describing normative interactions. We need a set of descriptive tools for describing interactions between machines,
so on and so forth. So this is a very kind of a rudimentary account of what in information science is called ontology. A very good person who has done some really interesting work in the idea of semantic web and ontologies and basically this idea of differentiation of levels in complex systems. And also he's a kind of a very ardent Brandomian. is Daniel Pirello. I will put his name here. You can see his papers on his website. There are some very interesting on this idea of ..
So, let's start with the concept of ontology with relation to complex hierarchical systems. Ontology has acquired over time several meanings and it has also been approached in many different ways. However, these are all connected to the concept of an objective existence and categories of items. And these categories of items are basically the idea that I said, you know, sets of descriptive tools in information science.
How you classify your logical conceptual resources in order to accurately describe a specific level in hierarchical system. Here we can consider the word existence as a basic concept which cannot be defined in either simple or atomic terms, with the latter in the sense of Wittgenstein. Moreover, generating meaningful classification of items that belong to the objective reality is a major task of ontology. And basically, this is one of the basic ideas behind the semantic web of foundational ontologies in information science.
But the most interesting question by far is how human consciousness emerged subsequent only to the emergence of homo sapiens, its syntactic, semantic, pragmatic language, and appropriately organized primitive society of humans. So basically, the most interesting question when it comes to the question of ontology is this ontology of human mind. Basically, what exactly, what kind of classification do we need to construct in order for us to be able accurately describe a complex hierarchical phenomenon called the human mind?
The ontological theory of levels considers a hierarchy of items, structures on different levels of reality or existence, with the higher levels emerging from the lower, but usually not reducible to the latter. Different degrees of structural and functional asymmetries. When it comes to the ontological theory, at least four realms or levels of reality taken into account. People usually talk about these in different terms or they further differentiate them. Material inanimate, physicochemical, material living, biological, psychological, and social. You know, you can think about this in terms of the airport system and the ontology of
complexes and luck in airport. The social is normative interactive, psychological is the fear of going through the check-out, check-in, all of these biological is basically germs and stuff. The material, inanimate physical chemical are basically other dust and all sorts of stuff. you can have proper descriptive tools, proper classifications to make a semantic web of relations in these complex systems with understanding that semantic web cannot be constructed unless you have a theory of levels.
The ontological theory of levels particularly stresses a need for understanding causal and spatiotemporal phenomena formulated within a descriptive category called context for theoretical levels of reality. There is the need in this context to develop a synthetic methodology in order to compensate for the critical ontic data analysis. Although it should be noted that analysis and synthesis are not exact inverse of each other in this sense. Let me just share the screen with you.
At the same time, we can address in categorical form the internal dynamics, the temporal rhythm or cycles, and the subsequent unfolding of reality. And so on can be simply referred to as—sorry, the general corresponding concepts such as processes, groups, essence, stereotypes, and so on can be simply referred to as items which allow for the existence of many forms or types of causal connection in a complex system. The implicit meaning is that the irreducible multiplicity of such connections converges or it is ontologically integrated within a unified synthesis.
So ontology information science and also respectively the idea of the semantic web or semantic mapping of causal connections. Needs to be understood in terms of a unified synthesis, overarching synthesis, within different levels, between different descriptive tools, so on and so forth. The objects, now before moving to this idea of synthesis and analysis and ontology, is that we need to talk about ontology as a difference between object-based approach and process-based dynamic approach. In classifications, such as those developed over time in biology for organisms or in chemistry
for chemical elements, the objects are basic items being classified even if the ultimate goal may be, for example, either evolutionary or mechanistic studies. An ontology based strictly on object classification may have little to offer from the point of view of its cognitive content. Now, whereas the existence of different ontological levels of reality is well established, one One cannot also discard the study of emergence and co-emergence processes as a path to improving
our understanding of the relationships among the ontological levels, and also as an important means of ontological classification. Furthermore, the emergence of ontological meta-levels cannot be conceived in the absence of the simpler levels, much the same way as the chemical properties of elements and molecules cannot be properly understood without those of their constituent electrons. So basically the whole idea is that ontology is a process-based approach precisely because the relation between levels are process-based. And that's how we talked about in complex hierarchical systems,
the idea of there is links, structural and functional links, emergent links also, between these different levels. And these links are processual, basically, items. And so when we are talking about, you know, ontology in this sense in information science, and so far as there are links between different levels, we need to have a kind of classification, basic ontological classification in place that allows for us to be able to describe and take into account these processual links between various levels of a complex hierarchical system.
It is often thought that the object-oriented approach can be readily converted into a process-based one. It would seem, however, that the answer to this question depends critically on the ontological level selected. For example, at the quantum level, object and process become intermingled. Either comparing or moving between level, for example, through emergent processes requires ultimately a process-based approach, especially in categorical ontology, where relations and inter-process connections are essential to developing any valid theory.
Ontologically, the quantum level is a fundamentally important starting point which needs to be taken into account by any theory of levels that aims at completeness. But such completeness may not be, of course, attainable simply because an extension of Godel's theorem may hold here as well. The fundamental quantum level is generally accepted to be dynamically or intrinsically non-commutative. I will talk about these later in details. In the sense of the non-commutative quantum logic and also in the sense of non-commuting quantum operators for essential quantum observables such as position and momentum.
Therefore, any comprehensive theory of levels in the sense of incorporating the quantum level is thus mutatis mutantis non-abelian. I will describe these terms very shortly. So in a sense, the shift to process-based ontology requires a paradigm shift to a non-abelian categorical ontology. However, the implementation complex functionality in a machine recalls also the design and construction of a correspondingly complex structure or structures. A similar argument holds for variable machines, variable automata, and variable dynamic systems.
Therefore, if one represents an organism as a variable dynamic system, one affortuary requires a super complex structure to enable or entail super complex dynamics. And indeed this is the case for organisms with their extremely intricate structures at both molecular and supramolecular levels. Now, the essential properties that define the super and ultra complex systems, with super you can think of them as organism and ultra complex systems you can think of as basically something like society. derived from the interactions, relations, and dynamic transformation that are ubiquitous at such levels of reality, which need to be distinguished
from the levels of organization internal to any biological organism or biosystem. Therefore, a complete approach to ontology should obviously include relations and interconnections between items. This is usually a talk in information engineering and semantic web domain as semantic, idea of semantic mapping interconnections between items that are distributed at different levels of complex hierarchical system. So, therefore, a complete approach to ontology should obviously include relations and interconnections
between items with the emphasis on dynamic processes, complexity, and functionality of systems involved. This leads one to consider general relations such as morphisms. Morphisms are basically arrows. You can think of them as arrows, pointing arrows, such as morphisms on different levels, and thus to the categorical viewpoint of ontology. The process-based approach to a universal ontology is accordingly essential to an understanding of the ontology of reality levels, hierarchies, complexity, and dissipation systems, life, consciousness, and the universe.
On the other hand, the opposite approach, based on objects, is perhaps useful only at the initial cognitive stages in experimental science, such as a simpler classification system used for efficiently organizing data and providing a simple data structure. Now, we can note here also the distinct meaning of object in psychology, which is much different from one considered in this discussion. For example, an external process can be reflected in one's mind as an object of a study. This duality or complementarity between object and subject, objective and subjective, is a familiar topic in philosophy, beginning with Descartes and continuing with Kant, and so on.
A somewhat similar but not precisely analogous distinction is fundamental in the standard quantum theory, the distinction between the observed measure system, which is the quantum subject of the measurement, and the measuring instrument, which is a classical object carries out the measurement. So what I'm now going to discuss is a very brief account of the methodologies necessary for studying difficult as well as the controversial ontological problem of space and time at different levels of objective reality, defined as complex, super-complex, and ultra-complex synomic systems. These are biological organisms, societies,
and more generally systems that are not recursively computable. More on this point in the next sessions about this, that they are not recursively computable. Rigorous definitions of the logical and mathematical concepts that are employed in this discussion, as well as a step-by-step construction of our conceptual framework, of course, doesn't claim to have any mathematical rigor. It's just providing a basic conceptual mathematical The continuation of the very existence of human society may now depend on an improved
understanding of highly complex systems and the mind, and how the global human society interacts with the rest of the biosphere and its natural environment. It is most likely that such tools might have value not only to the sciences of complexity and ontology, but more generally also to the philosophers who are interested in keeping the rigorous side of the fence. Following Kant's critique of pure reason and Wittgenstein's critique of language misuse in philosophy, one needs also to critically examine the possibility of using general and universal mathematical language and tools in formal approaches to ontology and complex systems, the idea that people who usually model complex systems using extremely clunky
and questionable mathematical toolboxes. Throughout this session, I'm going to use the attribute categorical only for philosophical and linguistic arguments. On the other hand, I use the term categorical only in conjunction with the applications of concepts and results from the more restrictive, but still general mathematical theory of categories, functors and natural transformation. We can abbreviate this from now on as TCFMT. Theory of categories, functors and natural transformation.
This is basically the idea of category theory that we'll talk about. Now what is category theory? Very briefly, category theory is a general mathematical theory of structures, of systems, of structures, and of systems of structures. Category theory is both an interesting object of philosophical study and a potentially powerful formal tool for philosophical investigation of concepts such as space, system, and even truth. You know, it has basically come to occupy a central position in contemporary mathematics and theoretical computer science. It has also applied to chemistry, mathematics, neuroscience, which neuroscience part of it
we are going to talk about a little bit today. Traditionally, modern philosophy also aims at unity that might be obtained, as suggested by Herbert Spencer in 1862, through a synthesis of synthesis. This could perhaps be iterated many times. So first, before moving to this idea, the idea is that category theory has this basically powerful, it's basically power comes from its classificatory potencies.
It's basically categorizing stuff accurately, not only in terms of objects, but also in terms of processual relations between objects. Processual in terms of it allows for indexing transformations, alterations, shifts, basically, so on and so forth. Another thing is that category theory allows for construction of, or basically translation between local and global constructs, local and global spaces.
You can think of this intuitively in terms of local as a region on the map and the global map. more technically in terms of local mathematical objects and global mathematical objects. So because of this transition that enables between local and global, it simultaneously has an analytic and a synthetic balance. It can, moving toward the local constructs, You know, moving toward the local construct, you can focus analytically. Moving toward the global construct, you move synthetically. So basically everything, because of this kind of sophisticated synthetical horizon that
it provides, it allows for the refinement of the classical relation between analysis and synthesis. Or namely, as I said in her Spenserian terms, it allows for synthesis of synthesis. Synthesis are synthesis. It's this idea of allows for basically procedural, incremental iteration of synthesis at the basic level, constructing it to more sophisticated, more global,
basically, synthetic spaces. and through this kind of massively synthetic web, it's capable of analytically focused or synthetically basically described, classified. Now, Traditionally, the first person who brings this antimony, the opposition between analysis and synthesis in the philosophical discourse is Descartes.
Descartes is a champion of analyticity. He, with a great deal of success, he championed the analytical approach in which all available evidence is, in principle, examined critically and skeptically, first both by the proposer of novel metaphysical claims and his or her readers. Descartes equated the synthetic approach with the Euclidean geometric axiomatic approach, and so doing relegated synthesis to a secondary, perhaps less significant role than that of critical analysis of scientific data inputs, such as the laws, principles, axioms,
and theories of all specific sciences. On the other hand, Spinoza, Kant, and Spencer's styles might be considered to be synthetic. Whereas, for example, you see that a person like Russell in philosophy is more of the analytic in the accurate terminological sense. Let's say it's called analysis A and synthesis S. Descartes did not see that synthesis and analysis are inverse of one another.
And also not merely as bottom-up or top-down processes, basically the relation between analysis and synthesis. Interestingly, unlike Descartes' discourse of the philosophical method, his three ties of philosophical principles come closer to a synthetic approach in having definitions and deductive attempts, logical inferences, not unlike his synthetic predecessors, albeit with completely different claims and perhaps a wider horizon. However, we can see that if one, as proposed by Descartes, begins the presentation or method with an analysis, A, followed by synthesis, S, and then reverse the presentation in the follow-up treatment by beginning with the synthesis,
S star, followed by analysis, AS star, of the predictions made by S star consistent, or analogous with A, then AS wouldn't be equal to S star AS star, because we assume that A is isomorphic or equal to AS star. And that S is not equal to S star. Moreover, if one did not make any additional assumption about analysis and synthesis, then analysis to synthesis, moving from analysis to synthesis,
wouldn't be equal to the movement from synthesis to analysis. Or AS wouldn't be equal to SA. That is, analysis and synthesis do not commute. So the idea is that in analytical approach, which championed by Descartes, is that analytical methods, all you need is analytical methods first, and then you can construct synthesis out of analytical methods. But the thing is that analysis and synthesis do not commute. No matter how much analytical method you have, how much analysis you have, you cannot basically
produce a synthesis, a synthetic vision. Such a theory, when expressed mathematically, would then be called non-abelian, this non-communativity between analysis and synthesis. This is a very rudimentary but helpful meaning of the term non-abelian in philosophical epistemological context, namely the non-communativity between analysis and synthesis, between local and global constructs. Local and global constructs that are perceived analytically versus synthetically, because usually local constructs are analytical objects whereas global constructs are usually called
synthetic objects. If you have local analytical object, you cannot construct synthetic objects. Whereas if you have a synthetic construct, where basically you have synthesis everywhere, You can create basically piecewise procedure. You can create from a local object a global construct where the global construct look completely different and cannot be isomorphically understood or cannot be isomorphically mapped to a local construct. So basically you have this asymmetry between local construct and the global construct.
between the overarching synthesis and local analyticity. This is basically the basic idea behind the non-communativity or non-abelian, basically, systems. Philosophical categories, according to Kant, are quantity, quality, relation, and modality. and the most complex and far-reaching questions concern the relation modality related categories. On the other hand, mathematical categories are currently considered as the most general and universal structures in mathematics, consisting of related abstract objects connected by arrows called morphisms.
The abstract object in category theory may or may not have a specified structure, but must all be of the same type or kind in any given category. The arrows or morphisms can represent relations, mappings, functions, operators, transformations, homeomorphisms, and so on, thus between objects or categories, thus allowing great flexibility in applications, including those outside mathematics, as in logics. computer science, life sciences, psychology and sociology. Mathematical category also has a form of internal symmetry, as specified precisely as the commutivity of chains of morphism compositions
that are unidirectional only, as naturality of diagrams of morphisms. Finally, any object A of of an abstract category has an associated unique identity, usually denoted by A1. Sorry, 1A, basically identity of category A, identity of object A. object A. Therefore, one can replace all objects in abstract categories by the identity morphisms.
When all arrows are invertible, the special category thus obtained is called a groupoid, which plays a fundamental role in the field of mathematics called algebraic topology. The categorical viewpoint, as emphasized by William Laubere and Charles Erisman, and most mathematicians working in category theory, is that the key concept and mathematical structure is that of morphisms that can be seen, for example, as abstract relations, mappings, functions, connections, interactions, transformations, and so on. We can notice here how the philosophical category of relation is closely aligned to the basic concept of morphism or arrow.
In an abstract category, the implicit in it is that arrows are what counts. Basically in category theory, you have three basic elements, objects, morphisms, objects, and categories. Now, you can't even completely discard the objects and categories and simply say in category theory that everything is a morphism. Basically you can construct with morphism whatever thing you want, objects, complex categories, so on and so forth, transformations, translations, local global transitions. One can express all properties, attributes, and structures by means of arrows that in
the most general case can represent either philosophical relations or modalities. The question then remaining if philosophical categorical properties need be subjected to the categorical restriction of commutativity. As there is no a prior reason in either nature or pure reasoning, including any form of Kantian transcendental logic, that either relational or modal categories should in general have any symmetry properties, one cannot impose onto philosophy, and especially in ontology, all the strictures of category theory, and especially commutativity. Interestingly, the same critique and comment applies to logics as well.
Only the simplest forms of logics, the Boolean and intuitionistic, Hating-Brauer logic algebras, are commutative, whereas the algebras of many valued logics, such as Schubitz's logic, are non-commutative or non-abelian. I will talk about these ideas about non-Abelian character of general philosophical and logical theories including general ontology approaches in more details later on both this session and the next session. a structure of category ontology and theory of levels.
The categorical techniques provide powerful means for describing levels in both a linear and interwoven fashion. And in that, the central concept of our discussion so far, namely emergence, complexity, open non-equilibrium, and irreversible systems. Moreover, an effective approach to philosophical ontology is concerned with universal items assembled in categories of objects and relations involving in general transformation and or processes. Therefore, categorical ontology is fundamentally dependent upon space and time consideration.
One needs to consider first a dynamic classification of systems into different levels of reality, beginning with physical levels and continuing in an increasing order of complexity, for example, chemical-molecular levels and then higher to our biological, psychological, societal, environmental levels. Indeed, it is the principal tenet in the theory of levels that there is a two-way interaction between social and mental systems that impinges upon the material realm for which the latter is the bearer of both. Accordingly, any effective categorical ontology approach requires or generates, in a constructive sense, a structure or pattern rather than a discrete set of items.
And basically, category theory can index this pattern, this structure. General system analysis seems to require formulating ontology by means of categorical concepts. Moreover, category theory appears as a natural framework for any general theory of transformations or dynamic processes. Just as group theory, for example, in mathematics, provides the appropriate framework or classical dynamics and quantum systems with a finite number of degrees of freedom. Through category theory, we can model what is universal in some domain or in general. And that only for simple system, this involves commutative modeling diagrams and structures.
However, I should note that this ontological use of the word universal is quite distinct from the mathematical use of universal property, which means that a property of a construction on particular objects is defined by its relation to all other objects, i.e. it is a global attribute. Usually through constructing a morphism, since this is the only way in an abstract category for objects to be related. With this idea of ontological meaning, the most universal feature of reality is that it
changes. It is subject to countless transformations, movements, and alterations. In this select case of universal temporality, it seems that two different meanings can be brought to bear on one another through appropriate formalization. In addition, concrete categories may also allow for the representation of ontological universal items as in certain application to categories of neural networks. I will talk about these later on. For general categories, however, each object is a kind of a black box. In category theory, you can think of each object as a black box whose only exposure is through input and output, i.e. the object is given by its connectivity through various morphisms
to other objects. For example, the dual of the category of sets still has objects, but these have no structure from the categorical viewpoints. Generally, abstract mathematical structures are developed to define relationships, to deduce and calculate to exploit and define analogies. Since analogies are between relations between things rather than between things themselves, a description of a new structure is in some sense a development of part of a new language. The notion of a structure is also related to the notion of analogy. It is in one of the triumphs of the mathematical theory of
categories in the 20th century to make progress toward unifying mathematics through the finding of analogies between various behaviors of structures, i.e. not between objects, but the relations between objects themselves across different areas of mathematics. So why category theory is important and why it's an appropriate tool to talk about logical modeling of complex systems.
It's basically, it captures interaction, it captures relationality, it captures transformation. It describes structures not by what they are, but by how they stand in a processual relation to other structures. idea that basically it poses analogies between behaviors of the structures, between not objects, not the structures themselves, but between the relations between the structures.
Now, ontological classification based on items involves the organization of concepts and indeed theories of knowledge, and that's the basic task of ontology information science. Ontological classification based on items involves the organization of concepts and indeed theories of knowledge into a hierarchy of categories of items at different levels of objective reality as reconstructed by scientific modeling through either a bottom-up, inductive, synthetic, or abstraction process, or through a top-down, deductive process, which proceeds from abstract concepts to a realization in a specific context of the real world.
Most modalities, top-down, bottom-up, inductive, synthetic, abstract, or deductive modality can be developed in a categorical framework. I'm going to briefly talk a little bit about this bottom-up modality in categorical ontology. And basically, that's also this bottom-up. We talk about this bottom-up approach in our neuroscientific example of how you can apply
like category theory to logically model how a complex system like brain functions. But before that, I would like to very, very briefly introduce one of the main concepts of category theory called natural transformation. Basically, if you have a natural transformation, you can construct a really complex, constructive schema of natural transformation, like limits and co-limits. So what is a natural transformation? One of the major goals of category theory is to see how the properties of a particular
mathematical structure, say S, are reflected in the property of the category cat S. Let me bring the diagram for you. One of the major goals of category theory is to see how the properties of a particular mathematical structure, say S, are reflected in the properties of the category S of all such structures and of morphisms between them. Therefore, the first step in category theory is that a definition of a structure should come with a definition of a morphism of such structures. As I briefly mentioned, a structure comes with their relations, and that's one of the
powers of category theory. Usually, but not always, such a definition is obvious. The next step is to compare structures. This might be obtained by means of a functor, A, that maps the category S to category T. Finally, we want to compare such functors, A, B. What are functors, basically, in category T? are maps, basically categories of morphisms that preserve structures and identities across these translations or across these transformations, across different categories.
This might be obtained by means of a functor A that maps a category S to a category T. Finally, we want to compare such functors, A and B, that maps category S to category T. This is done by means of a natural transformation eta, denoted by an arrow, a natural transformation between A and B. Here eta, natural transformation, assigns To each object X of category S a morphism, natural transformation X maps AX to BX, satisfying
a commutativity condition for any morphism A that maps object X to object Y. In fact, we can say that natural transformation eta assigns to each morphism A of category S, a commutative square of morphism in category T. Now, in order to kind of like make this a little bit easier to follow, I've made this diagram. This diagram usually presented as this square diagram. But in order for us to see the links better, how it basically in natural transformation works is I've made this drawing this diagram. Sorry, my diagram software was expired so I had to draw this by hand. And also I missed for the red box,
the red box is a category D basically. So the black box is C, the red box is D. D is is missing this diagram. So the formal definition is that if F maps C to D and G maps C to D are both from category C to category D, A mapping can be constructed between F and G, called natural transformation and denoted by symbol eta. F yields a natural transformation to G.
And this is basically the diagrammatic map of its natural transformation. Now, eta, natural transformation, F to G, is a family of morphisms from F to G, satisfying basically in two conditions. For every x belonging to object C, there is a morphism that maps Fx to Gx associated to X called the component of eta at X, or natural transformation at X. And for every M mapping
between X and Y belonging to the home set. What are home sets basically in category theory? They are basically the collection of morphisms. There is a composition between natural transformation and formation at M being isomorphic to the composition of GM and natural transformation at X. Now, I understand that this is kind of hard to kind of grasp in one session, so So that's why I asked you to look into Lavers' book, Conceptual Mathematics, because he really
makes these sophisticated diagrams really comprehensible by using really intuitive, familiar examples. And then he construct them from these very, very really rudimentary examples to kind of the stuff that are very, very complex. And natural transformation is considered to be an elementary concept and category theory, along with morphisms, objects, and categories. But things like limits and co-limits are much more complex.
And basically, they can be constructed by natural transformations. And this was basically the diagram of a natural transformation. The notion of natural transformation is at the heart of category theory. Basically, it defines, as MacLean puts it, to define natural transformation, one needs
a definition of functor. And to define, and functor, as I said, is basically a map. It's a map between categories, a structure-preserving map. To define natural transformation, one needs a definition of functor. And to define the latter one, one needs a definition of category. Also, if you have noticed in our diagram, two arrows become three objects in the meta category, where three categories of functors and natural transformation. Here and here. Now it's possible to formalize the hierarchy of multiple-level relations and structures
that are present in biological, environmental, and social systems in terms of the mathematical theory of categories, functors and natural transformations. On the first level of such a hierarchy are the links between the system components represented as morphisms of a structured category which are subject to several axioms, restrictions of category theory, such as commutativity, associativity conditions for morphisms, functors, and natural transformations. Now, then on the second level of the hierarchy, one considers functors, or links between such first level categories that compare categories without looking inside their objects' system
components. On the third level, one compares or links functors using natural transformations. In basically three categories or metacategory of functors and natural transformation. At the first level, natural transformation not only compares functors but also looks inside the first level objects, namely system components, thus closing the structure and establishing the universal link between items as an integration of both first and second level links between items. And we will talk about this, how this works in, for example, some hierarchical system like brain. From the point of view of mathematical modeling, the mathematical theory of categories models
the dynamic nature of reality by representing temporal changes through either variable categories or through toposes. Certain variable categories can also be generated as a topos. For example, the category of sets can be considered as a topos whose only generator is just a single point. a variable category of varying sets, or thus have just a generator set. However, qualitative distinction does exist between organisms, considered as complex systems, and simple inanimate dynamical systems, in terms of the modeling process and the type of predictive mathematical models or representations that they can have. A relevant example of applications to the natural sciences, for example, neuroscience
that we will talk about would be the higher dimensional algebra representation of processes, of cognitive processes, of still more linked sub-processes. Additional examples of the usefulness of such a categorical constructive approach to generating higher level mathematical structures would be that of subgroups of groups of items, the two-group ways, double-group ways of items, so on and so forth. So one of the things with category theory is that, because it allows you to model, because basically category theory, as I said, you can construct
different mathematical objects belonging to different mathematical tools with distinct structures. You can simply construct them from basic categories, basic relations between categories, morphisms, fungators, natural transformations, so on and so forth. So when we, when basically, that's why it's appropriate to apply to basically the question of ontological semantic mapping of complex hierarchical systems. In the sense that, for example, to mathematically model a bottom level, for example, object
or a structure, you do not need extremely sophisticated mathematical models. In fact, a very inappropriately sophisticated mathematical model, as we talked about, can introduce a lot of basically distortions into your model of that particular level. So you need to have, basically you need to be capable of applying the right mathematical toolbox or right mathematical model to the appropriate level of reality of the complex system. And category theory, considered to be a logical model, allows you to basically
come up or solve the problem of this, the right application of the appropriate mathematical model to its respect corresponding appropriate, basically, level of the system. Precisely because of this idea that you can construct from very rudimentary elements, morphisms, functors, and natural transformations, not only simple mathematical tools and mathematical objects and mathematical structures, but extremely complex mathematical objects.
So any questions before I move forward? Questions? No, I'm good. Okay. Okay. What is, OK, just missed the, you're running low on time, huh, for some reason. It's OK. I will, if you guys are comfortable, I can go for an hour again.
OK. Okay. Now, a little bit about this symmetry, commutativity, and abelian structures. The hierarchy constructed, that we constructed up to level three, can be further extended to higher end level, always in a consistent natural manner, is using commutative diagrams, basically using the category theory of procedures. Now let's examine a few simple examples or specific instances of commutative properties.
Well, somewhat basically commutativity is. We talked about it a little bit, but a little bit more. The type of global natural hierarchy of items inspired by the mathematical theory of categories, functors and natural transformation as a kind of internal symmetry because at all levels the link compositions are natural. That is, if... Can you guys see the screen? Yes. OK. That is, F maps x to y and g maps y to z. Then there is a natural transformation, h, that maps x to z.
The composition of morphism g with f is given by another unique morphism. Sorry, I said natural transformation. I meant it. Just there is another mapping. Actually, that arrow needs to be corrected, double arrow. Then the composition of morphism G with F is given by another unique morphism, H equals to G composition F. This general property involving the equality of such link composition chains or diagrams comprising any number of sequential links between the same beginning and ending objects is called commutativity, and is often expressed as a
naturality condition for diagrams. This key mathematical property also includes the mirror-like symmetry, x star y equals to y star x, when x and y are operators and the simple star represents the operator multiplication. Then the equality of x star y with y star x defines the statement that x and y operators commute. In physical terms, this translates into a sharing of the same set of eigenvalues by two commuting operators, thus leading to equivalent numerical results up to multiplication constants.
In addition, the observations X and Y corresponding respectively to these two operators would yield the same result if X is performed before Y in time or if Y is performed first followed by X. This property, when present, is very convenient for both mathematical and physical applications. However, not all quantum operators commute and not all categorical diagrams or mathematical structures are or need to be commutative. Non-commutativity, non-abelianness may therefore appear as a result of breaking the internal symmetry represented by commutativity. As a physical analogy, this might be considered a kind of symmetry breaking, which we talked
about in our previous sessions, which is thought to be responsible, for example, for the expansion of the universe as well as many other physical phenomena like phase transitions, superconductivity, et cetera. On the one hand, when commutativity is global in a structure, as in abelian or commutative group, commutative groupoid, commutative ring, et cetera, such a structure that is commutative throughout is usually called abelian. However, in the case of category theory, this concept of abelian structure has been extended to a special class of categories that have meta-properties formally similar to those of the category of commutative groups.
Among such mathematical structures, abelian categories have particularly interesting applications, for example, to rings and modules in which commutative diagrams are essential. diagrams are also being widely used in algebraic topology. As one can see, both the earlier and more recent literature on category theory, abelian categories have been studied in great detail. On the other hand, the more general case is a non-commutative one when it comes to hierarchical systems. Several interesting non-commutative or non-abelian examples are are provided, a good example of thinking about these non-abelian, non-commutative diagrams,
are provided by certain asymmetric drawings by, you know, Morris Escher, such as his perpetual water mill or his illusory castle with monks climbing from one level to the next at the same height. It's not basically a very intuitive image of a non-abelian structure, or non-commutative structure. Any comprehensive categorical ontology is a fortiority non-abelian on account of both the quantum level, which is generally accepted as being non-commutative, and the top ontological level of the human mind, which also operates in a non-commutative manner, albeit with different
multi-valued logic. So in short, the operational logics at both the top and the fundamental levels of studying modeling hierarchical systems can be said to be non-commutative. These are invisible actors who behind the visible scene make both the action and play possible, both the quantum fundamental level and also the human mind that basically makes these observations possible. Now I suggest the work of some of this stuff that started application of categories theory to biological sciences and started with a seminar work of Robert Rosen.
I suggest people who want to get more into these kinds of mathematical modeling of biological sciences, especially this application of category theory to biology, look into the work of Robert Rosen. Now, in his 60s books, where he has started to talk about category theory and mathematical modeling of biology and how we should appropriately model biological systems, He's shown, basically, Robert Rosen has shown that complex dynamic systems, such as biological
organisms, cannot be adequately modeled through a commutative modeling diagram in the sense of digital computer simulation, whereas the simple physical engineering dynamical systems can be thus numerically simulated. This modeling commutative diagram for a simple dynamical system included both the encoding of the real system N in M as well as the decoding of M back into N in this diagram. Where small delta is the real system dynamics and LF is an algorithm implementing the numerical computation of the mathematical model M on digital computer.
Now in Rosen's description, however, there is an absence of logical model, which is, we talked about this most important thing before application of mathematical model to a complex system. This is especially important because mathematical model cannot be applied correctly unless there is already a robust logical model in place. But more importantly, without the proper logical conceptual model, we risk aligning or overextending distinction between different levels and their properties, not to mention the application of inappropriate mathematical tools. For example, there is a fundamental logical difference between physical systems and biological
systems or organisms. Whereas the former are readily represented by homogeneous logic classes, living organisms exhibit considerable variability and can only be represented by heterogeneous logic classes. One can readily represent homogeneous logic classes or endow them with uniform mathematical structures, but heterogeneous ones, as in the case of, for example, organisms, are far more elusive and may admit a multiplicity of mathematical representation or possess variable structures. Now, with these kinds of introductions, this idea of, you know, this link between logical model, mathematical model, and the actual hyper-complex hierarchical systems and Y-category theory
as important as a logical model that allows for this differentiating of mathematical structures and the appropriate application of them to actual levels of a system, we can go to our example and probably, I'm not sure if we have that much time, but because it's still massive amount left. But nevertheless, I tried to wrap it up. With this introductory account, we can now look at some of the conceptual ideas in utilizing category theory in the philosophy of neuroscience, primarily focusing on the human brain as a particularly hierarchical
complex system, or what is usually called in neurobiology, memory evolutive neural system, we have generative entrenchment, a scaling causal history, a structural memory, a structural functional asymmetry, nonlinear interactions, modularity, etc. Memory evolutive systems, memory evolutive neural systems, let's abbreviate this, men's, a common frame accounting for the functioning of the neural, of the mental, and cognitive system at different levels of description across different time scales.
It is not intended as a model of the invariant structure of the neurocognitive system, but as a dynamic model sizing of the system in the making, with the variation over time of its configuration and of its information processing. It describes how various brain areas interact as hybrid systems and generate an algebra of mental objects. This is from Shenzhou. Through iterative binding of more and more complex synchronous assemblies of neurons, It is, in its frame mental objects, are treated as conceptual higher level neurons called
category neurons. We can abbreviate this as cat neurons, category neurons, or cat-neur, on which to compute how cognitive processes of increasing complexity can emerge. Now, the bio-inspired development of men's memory evolutive neural systems has followed the two directions, analyzing living organisms as computational systems agents and implementing natural computation strategies. How, basically, different levels of computational complexities arise and how, basically, different computational strategies emerge through these kinds of hierarchical systems called brain.
A third step would be to develop an adequate kind of properly unconventional computation to simulate basically memory evolutive systems. Is an application of the memory evolutive system, MESS, which gives a model based on category theory for complex, multi-scale, multi-agent, self-organized systems such as biological, social, or cognitive systems. I will talk about references material for this, and people want to get into this more properly with more technical details and stuff. First, we need to ask why category theory can or should be applied to something like
a field like neuroscience, and especially the study of the brain and memory evolutive systems. Category theory, as I talked about, has a unique status at the border between mathematics, logic, and metamathematics. As I mentioned, the basic elements of category theory are particularly powerful abstraction to capture change, transformation, the sense of emergence of new constructs, and complex networks relations between interactive components. Different kinds of graphs or diagrams are extensively used to represent networks of any nature. Here, graph or diagram always denotes the data of a set of objects and a set of oriented
edges, called arrows or links, between them. Denoted by A ascending in relation or link to B. Several arrows may go from A to B, and closed links are accepted. A path of a graph is a sequence of successive arrows or morphisms. A category is such a graph equipped with an internal composition associated to a pair A and B of successive arrows, A mapping capital A to B and a small b mapping capital B to C.
an arrow from A to C, denoted AB, called its composite. This composition is associative, meaning A parentheses BC parentheses equals to A parentheses A, B parentheses C. Each object A has an identity. Now, each graph generates the category of its paths. The objects are the same. The links are the paths of the graph or diagram. And the composition is the convolution of paths. In it, two paths cannot have the same composites. On the opposite, in a category which is not the category of paths of a graph,
two different paths with the same extremities may have the same composites. So this was the extremely important definition in category theory. Basically, it's the primary definition of it, really. This property, one of the main reasons for which categories and simple graphs are used in the study of complex systems. We need to distinguish which paths play the same functional role, and the composition will be defined so that they have the same composites. We then say that they are functionally equivalent or isomorphic, functionally isomorphic. In particular, it opens the way to important universal construction, such as the co-limit
operation, which will model the binding of pattern P of linked objects. A pattern... Sorry for these messy diagrams I have drawn. The concept of core limit, as I said, it's a very complex concept in category theory, and you basically need to do some study on yourself. I think I will send some links. There are some really useful, intuitive blog posts on the concept of core limit. But also there are very interesting technical stuff about core limit and how it is constructed in a kind of a Kentian program of neuroscience.
I will try to find them and send them to you. A pattern or diagram P in a category is a family of objects P-I with some distinguished links F, P-I to P-J. A collective link from P to an object N is a family S-I of links S-I from the different different PI to N such that Fs, actually I, yes, let me see. Yeah, I think, let me look at this definition.
I think, no, no, no, this is correct, sorry. I thought that I have miswritten some of the notations. OK. Again, a pattern or diagram P in a category is a family of objects P, I with some distinguished links F, P, I to P, J. A collective link from P to an object N is a family of links S, I from basically P, I to N, such that Fsg equals Si for each distinguished link Fpi to PJ of P. The pattern admits a co-limit or inductive limit,
M, if there is a collective link Li from P to M, which factorizes any other collective link, so that the collective links Si from P to any N are in one-to-one bijective correspondence with the links S. M with the links S that maps M to N, basically binding them formally. For each I, we have the equation S I equals to L I S. This is basically a diagram of co-limit in category theory. Collective links and a fortiorly co-limits
make an essential use of the composition of the category via the above equations, as seen in the above diagram, and could not be defined in simple graphs. Category theory extensively uses diagrams, in particular commutative diagrams, in which two paths with the same extremities have the same composites. Proofs are often made more intuitive by reasoning on figures rather than writing the corresponding long sequence of equations. It is what's called diagram chasing process in category theory, ubiquitous in basically any study of categories. The memory evolutive systems give a model based
on the dynamic category theory, incorporating time and duration. For example, multi-scale systems with the following characteristics. One, the structure of the system is changing. Its components and their links are varying over time. As I said, varying structures when it comes to organisms. And basically, you need to have proper mathematical model to be modeled as varying structures. In memory evolutive systems, MES, the system is not represented by a unique category, but by what can be called an evolutive system. It consists of a family of categories KT representing the successive configurations of the system
at each time t, and partial transition functors from kt to kt prime, accounting for the change from t to t prime. A functor, as discussed earlier, is a map between categories, basically a collection of morphisms which preserves their composition and identities. Two, the system is hierarchical with a tangled hierarchy of components varying over time. The components C of a certain level bind at least one pattern P of interacting components of the lower levels so that C and P acting collectively have the same functional role.
Modeling this hierarchy raises the binding problem. How do simple objects bind together to form a whole that is greater than the sum of its parts? And how can such a whole interact? In the categorical setting, the hole C is represented by the core limit of the pattern P of interacting simple objects. And that's why it said core limits cannot be decomposed to simple structures. And the interactions between the holes are described in terms of not can be decomposed, that cannot be basically constructed from simple constructs. In the categorical setting, the whole C is represented by the core limit of the pattern P
of interacting simple objects. And the interactions between the holes are described in terms of simple links and complex links. Three, there is emergence of complex multi-form components the development of flexible central memory, hence the emergence problem, how to measure the real complexity of an object and what is the condition making possible the emergence over time of increasingly complex structure and processes. This condition is usually characterized as a multiplicity problem, a kind of flexible redundancy which ensures the existence of components which binds various lower-level patterns which are not isomorphic nor even well-connected. Such components are said to be multiform.
It is important to note that the multiplicity problem is necessary for the emergence of increasingly complex objects and processes with multiform presentations constructed by iterated complexification processes. The complexification of a category K described the universally constructed category K prime deduced from K after changes of the following kinds. Addition of given new objects, formation or preservation if it exists of a new object which becomes the core limit of some given pattern, suppression or decomposition of some objects. You can think about this in brain in terms of basically that, as I said, different information
processing units are not isomorphic. So then how is that we can have emergence in brain and this kind of process world translation between different levels that allow for structures at higher levels that are actually isomorphic, even though they are constructed from these non-isomorphic basic objects, information processing modules. Four, the system has a multi-agent self-organization. Its global dynamic is modulated by cooperation and competition of a network of internal functional subsystems called the core regulators. So coregulators is a network of internal functional subsystems acting as agents with the help
of a long-term memory. Each coregulator operates locally with its own rhythm, procedures, and complexity. But the commands of the different coregulators can be conflicting and must be harmonized. While the local dynamics are amenable to conventional computation, the problem is different for the global one. Now MEN's memory evolutive neural systems is a memory evolutive system that level zero of which represents a physical neural system, neurons and synopsis, while its higher level components are conceptual objects called category neurons, which represent elemental objects as the binding of synchronous hyperassemblies of neurons.
Now basically a picture of these can be thought about this. You have basically co-limit constructs. And then these co-limits constructions allow you to, basically from neurons and synopsis that are not isomorphic in structural terms, to construct structures at different levels of the brain that are in fact structurally isomorphic and then you can basically hence
allowing for emergence of basically global behaviors or global structures. Sorry. Properties of the neural system. Despite the huge progress of brain research in the last couple of decades, there is still no advanced understanding of the brain's larger scale organizational principles
allowing for the emergence of higher order cognitive processes. Interesting mathematical models of local nature have been developed for particular processes in specialized brain areas. And that's what I'm, you know, when you look at, for example, neuroscience and mathematical modeling, these mathematicals basically have unique categories, unique classificatory, basically, a structure. But as different brain areas are heterogeneous, both anatomically and functionally, such models cannot be overextended to other areas of process.
And in fact, when you see neuroscience, these mathematical models are in fact overextended often. And this is also one of the reasons that I always say that this overextending of models leads to aligning different levels of systems. Hence, philosophically, it leads you to claim to inflationary and deflationary claims very often.
Now, so despite the uniqueness of levels, there are nevertheless underlying general properties that all memory evolutive neural systems relies on them. And what are these? One, the graph of neurons at an instant t, its objects represent the states n t of neurons n existing at t, measured by their activity around time t.
Its link from NT to N' T represents the state of the synopsis from N to N', weighted by the propagation delay around T and by the strength to transmit an activation of N to N'. A synapse can be active or passive at T. The activity of N at T is the sum of the activities of the neurons connected to N by an active link, pondered by the strength of this link. The graph changes over time. Some neurons die, new neurons are formed, and the same for synopsis. The activity of a neuron varies, delays, and the strength of synopsis may also slowly change.
Two, the structural core. The graph of neurons has a central subgraph, subdiagram called its structural core, a complex of densely connected regions in basically posterior medial cortex, which is both spatially and topographically central within the brain. The core is thought to be an important structural basis for shaping larger scale brain dynamics and linked to basically thought to be linked to self-refersional processing and consciousness. Three, synchronous assemblies of neurons.
You know, already in Fortis, Donald Hepp, which I talked very briefly last session, had noted the formation, persistence, and intertwining of more or less complex and distributed asymmes of neurons whose synchronous activation is associated to specific mental processes. Hepp says any frequently repeated particular stimulation will lead to a slow development of a cell assembly as a closed system. And he gives the Hebb rule for synoptic plasticity. When an axon of cell A is near enough to excite B and repeatedly or persistently takes part in firing it, the axon of cell A's efficiency as one of the cells firing B is increased.
This is called basically a rule of synoptic plasticity or Hebb learning, Hebbian learning. And this rule has been experimentally verified. And that's basically the basic idea behind the idea of synoptic plasticity. Whereas we have the Hebbian learning, this idea of efficiency and persistence of one axon in firing another neuron creates this kind of learning of how basically things connected. There is also an anti-Hebbian basically learning. The idea that, for example, synoptic plasticity is able to prune itself by killing some of
its connections, disconnecting and basically form new connectivities. Four, degeneracy property of the neural code. Basically it says that more than one combination of neural groups can yield a particular output and a given single group can participate in more than one kind of signaling function. Therefore, the mental representation of a stimulus should be the common binding of the more or less different neural patterns which it can synchronously activate in different
contexts or at different times. Five, modular organization. The brain has a modular organization with a variety of modules or areas of the brain with a specific function from a small specialized part such as visual centers, processing colors, large areas such as visual or motor areas or nuclei of the emotive brain you know brainless and limbic system or the associative cortex these modules interact to direct the self-organized dynamic of the system now the neural Neural system can be represented by an evolution system denoted by newer.
It has for configuration at T the category neurons, newer T, which is the category of paths of the graph of neurons at T. Its objects model the states of the neurons n existing at time t. The links model the synoptic paths between them, labeled by the propagation delay and the strength, defined as the sum of those of their factors. The transition from t to later time t prime associates to nt at t of a neuron n, its new state being n t prime, a t prime provided n still exists at t prime.
And similarly for the links, the transition describes what has changed, but they do not indicate the kind of computation as processing of information which is internally responsible for the change. A component of Neur models a neuron through the sequence of its successive states. It is a cell called neuron or category neuron of level zero. Neur constitutes the level zero of memory evolutive neural system from which higher levels are constructed by iterated complexification processes, namely that we talk about in terms of core limits constructions. Category neurons and basically their links.
As mentioned, a mental object, for example, the mental image of a simple stimulus, synchronously activates an assembly of neurons P, and possibly several ones in different contexts. In simple cases, there is a neuron N binding the assembly, which becomes the core limit of P in NOR and will represent the mental object. For instance, there are neurons representing a segment of an angle or more complex, but very similar objects. I will talk about this later, how this has some, we can think of all this in the idea of Kantian imagination and Kantian intuition in modern neuroscientific category theoretical
sense. A metal object S which activates an assembly of neurons P having no cool limit in NUR will be represented by a conceptual object M called a category neuron, abbreviated basically as cat neuron of level 1, which will become the core limit of P in the larger system MENS and act as a memory of S. The construction of M by complexification process will determine what are the good links between M and other category neurons and will guarantee that M also becomes the co-limit of other assemblies of neurons which S can synchronously activate.
So basically throughout the levels is the idea of generative entrenchment, you know, construct structures of these neurons entrench the function of higher levels, basically. They support them. Having thus constructed category neurons of level one and their links, we can speak of assemblies of category neurons of level one and iterate the construction to obtain a hierarchy of category neurons of increasing levels, representing more and more complex mental objects binding together assemblies of simpler ones.
Formally, any assembly of category neurons is modeled by pattern P in memory evolutive neural system. For the assembly to synchronously activate the category neuron N, there must exist a collective link, SI, from P to N, allowing that all SI transmit an activation of P to N at the same time. In particular, this imposes that all the zigzags of links between P and P have the same propagation delay, basically have the same temporal window in terms of neural activation. A pattern with this property is said to be polychromous. If such a pattern
P is repeatedly activated, its distinguished links are strengthened via the Hebbian learning. And there is formation of mental objects. Basically, and this is something to do, we We can think of this in terms of encountering the same object and capable of recognizing it in an environment. And there is a good sense that's connected with Kentian idea of imagination and intuition and how basically concepts allow us to recognize objects in the environment regardless of their
specific particularities, basically their variations. For example, the concept of a house can basically act, you know, can be used to this link of intuition and imagination. This basically can be used to detect something that doesn't resemble, for example, a particular individual house that I have seen in the past, but nevertheless is in fact a house. Or for example, you can think of this also about shapes and colors of things.
If such a pattern P is repeatedly activated, its distinguished links are strengthened via HEP rule, and there is a formation of a mental object. This object will be represented by a higher-level category neuron M, which becomes the co-limit of P in memory evolutive neural system. It is important to note that the activation of P precedes that of its co-limit M. And this is kind of a co-limit construction of how all of this basically works and how you have the generative levels from level zero to level N and the transitions also between
the estates of structures distributed at different levels. This, sorry, this thing kind of hidden is, the arrow is T, you know, a state of the system at different levels, and the other one is T prime, later a state of the system at different levels. The degeneracy property asserts that the mental object can also activate simultaneously or at different times other patterns skewed, not necessarily connected to P by a cluster
of links. The representing category neuron M must also be co-limit of Q, so that M is a multiform category neuron which can be activated by any one of its different decompositions PQ, with possibility of switches between them. The existence of category neuron which are multi-form signifies the memory evolutive neural system satisfied the multiplicity principle. Once formed, the category neuron M preserves its identity up to its depth through its lower level decomposition. Though its lower level decomposition can vary more or less quickly over time.
The stability span of category neuron M at an instant T is the longest period during which M admits a decomposition P at T, whose successive states remain a decomposition of M. Now, memory evolutive neural system is an evolutive system. At an instant T of the life of the individual, the configuration category mends models the present state of the neural, mental, and cognitive system. Its objects are the category neurons of any level, from the level zero of the neurons up basically to level N, existing at T with their activity and their links with their
propagation delay and a strain. A link is active or not, basically at time t. The transition from t to a later time t prime points out the structural changes without accounting for the information processing at their origin. The changes are events of the following kinds, formation or preservation, if it exists, of a new category neuron binding some pattern p prime of already existing lower level category neuron, possibly loss or decomposition of some category neurons. In the categorical setting, the new configuration ments at time t prime is obtained as the complexification
of men's with respect to a procedure having objectives of the preceding kinds. Such a complexification is the solution of the universal problem of constructing a category in which the objectives of the procedure are satisfied in the best way. This is our universal property, the propagation delays and the strengths of synoptic paths level zero can be extended to the links of any level, as well as the Hebrew. Now I will just go on for 10 or 15 more minutes, and then I try to wrap it up. More explicitly, the construction distinguishes two kinds of links, simple links.
These are the links that bind clusters of lower level links in the following manner. Let M and M prime be two category neurons binding lower level patterns P and P prime respectively. If we have a cluster G of links from P to P prime, well correlated by the distinguished links of P and P prime, this cluster binds into a link from M to M prime called a P-P prime simple link. Such a link just translates at the level N plus 1 in our diagram.
The information that P can coherently activate components of p prime through the links of g. And this information is computable at lower levels. A composite of n simple links binding adjacent clusters is called n simple. In addition to the simple links, we have also complex links in basically in this schemata. Complex Complex links, they emerge at a higher level of basically this brain dynamic structure. They emerge at a higher level as composites of n-simple links binding non-adjustant clusters.
Their existence is possible because of the existence of category neurons m, which are multi-form. The following figure presents a complex link from N to M prime, composites of Q prime and Q simple link with P and P prime simple link, where P and Q are non-connected decompositions of the multi-form of category neuron M. Such a link represents information at the level, information emerging at the level n plus 1 by integration of the global structure of the lower level. So this is the local global structures more
formally put and not locally computable to lower level decompositions of n and m. end. So the complex, whereas the simple links at local level are computable to decomposition, complex links at higher levels cannot be computationally decomposed to lower level, basically, structures. The fact that the category neuron M is multi-form imposes global conditions calling out all
its lower-end decompositions and their collective links, could it be amenable to some kind of unconventional computation then? Now, here I've only talked about category neuron constructed by co-limits, but there are also category neurons obtained by projective limits, another concept of category theory, which arise, for instance, in the construction of a semantic memory. When the procedure asks also for the formations of such classifying cat neurons, it becomes a matter of a mixed complexification with a more complicated construction.
Here we have basically a modular construction based on co-limits. Whereas, for example, we have projective limits, we have basically mixed complexification processes. The construction of category neurons of higher levels allow making more precise the brain-mind correlation. A category neuron M of level 0 represents a mental object and not a physical neuron. However, the activation or the recall of M has neurophysiological consequences. It consists in the unfolding of one of the ramifications of M down to its neural level zero, first activation of one of its decompositions, P into synchronous assembly of category neuron
of lower levels, then decomposition of each component of P and so on down to the physical activation of synchronous assemblies of neurons. Because of the propagation delays of the links, the unfolding has a certain duration. At each step, there is a choice between various, possibly non-connected, decomposition so that the activation of M has several freedom degrees leading to multiple physical realizabilities into hyper-assemblies, assemblies of assemblies of assemblies of neurons. The ramifications of M have not all the same length. The complexity order of M is the smallest length of a ramification. It is less or equal to the level of M. The level indicates the number of steps in which
M has been constructed, while the complexity order measures the smallest number of steps sufficient for its later activation. Now, based on these observations, a conclusion can be drawn. Iturated complexification preserve the multiplicity principle and lead to the emergence in memory evolutive neural system of category neurons of increasing complexity order, representing more and more complex mental objects or cognitive processes. Finally, local and global dynamics of memory evolutive neural system, as any memory evolutive system, MENS has a multi-scale self-organization.
It is modulated by a network of core regulators which rely on modular organization of newer and help developing a central long-term memory. The memory itself is a hierarchical subsystem abbreviated as Mem, of Memory Evolutive Neural system which develops over time. It models the innate or acquired knowledge of any modality and the information of any kind which the individual can store and later recognize and or recall. A category neuron M in MEM, called record, represents the mental object associated
to an item S. And this S is external object, signal, past event, internal state, sensory, corresponding neural activity, etc. Sorry, sensory, motor, cognitive process. Now initially S activates a particular pattern P of category neurons if the corresponding neural activity persists. The links of P are strengthened by Hebbian learning, synoptic plasticity, and P binds into a category neuron M, which becomes its core limit in men's memory evolutive neural system. Over time, M takes its own identity as a multi-form category
neuron and can even dissociate from P at a later time to adapt the changing situation as long as the change is progressive enough. S can be recognized and M recalled through the activation of any of the ramifications of M down to the neural level with the possibility of switches between them. So M is a robust memory but not a rigid one, as in computers, since it remains flexible and can be constantly revised to account for changes. And this is basically one of the things that artificial intelligence is trying to model these days, basically how to model a constructive memory, a memory that evolves, a memory where
basically is not simply has this, is not defined as an existing storage, it doesn't have simply a recall function, but every time that it recalls something, the memory itself is being reconstructed. The impression of things are being reconstructed, the impression of past events or experiences. Memory contains a subsystem proc, the procedural memory, in which the records called procedures have links or commands toward the pattern of their effectors, for example, motor commands
of a specific movement. These category neurons are based on multiple brain regions. Memory also contains a subsystem, the semantic memory, in which records are classified into in various classes with respect to some attributes. The memory now plays an important role in the dynamics of memory evolutive neural system, which is modulated by the cooperative and competitive interaction between functional subsystems, core regulators, related to modular organizations of the brain. A core regulator based on a specific module of the brain, meaning that its category neurons have ramification down to its module so that they model hyperassemblies
of neurons of the module. It has its own differential access to memory, in particular to procedure, to recall its admissible procedure as specific of its functions. Now the dynamic of memory evolutive neural system must account for both the local information processing of each core regulator, which operates with its own rhythm and function, and the global dynamic, which results from an interplay among these core regulators. While the local dynamics are amenable to conventional computations, the emerging in the global structure raises computational problems. Now, the global dynamic must also take into account different local dynamics of the core
regulators. At a given time, the commands sent by the various core regulators should be realized by the effectors of the system. Since the core regulators have different functions and rhythms, these commands can be conflicting, There is a need of an equilibrium process to ensure the correlation of different commands, possibly neglecting some of them. For instance, to seize an object, the visual and motor commands should fit together. This process called the interplay among the coregulator leads to the operative procedure, which will be implemented on the system. The interplay searches for the best compromise between the more or less conflicting commands, keeping as much of them as possible.
In particular, it takes advantage of the degrees of freedom of multi-form command, which can be activated to any one of its lower level decomposition with possible switches between them. Decomposition allowing for a better coordination are selected through a kind of Darwinian selection process. For instance, depending on the context, we can seize an object in the right or the left hand. And if there is basically no stable equilibrium within these commands, we have sensory motor dysfunction. Now, to end up the discussion, how this is related, this category theoretical, hierarchical
description of brain information processing can be thought in terms of basically higher cognitive processes, creative cognitive processes. The core regulators jointly participate in the development over time of an important functional subsystems of the memory. The so-called archetypal core, AC, which will act as an internal model essential for the emergence of higher cognitive processes. Now, as it was mentioned, the brain has a structural core which plays a main role in the shaping
of large-scale brain dynamics. a category neuron in AC, the so-called archetypal core, is a higher order category neuron, often activated and with ramifications down to the structural core. Due to the rich organization of this core, the hyperassemblies of neurons which it binds are largely distributed in different brain areas. Therefore, an archetypal record integrates and interprimes recurring memories and experiences of different modalities, like sensory, motor, affective, etc., as well as notable events with their emotive undertones.
So archetypal records are connected by complex links which become stronger and faster because of the Hebbian rule. long time. These links form archetypal loops which propagate very quickly the activation of an archetypal record A back to itself and so doing maintaining it for a long time. The activation of A resonates to lower levels via the unfolding of ramifications of A and switches between different decompositions, between different states. It follows that an activation of part of the archetypal core extends to a larger domain, D, of memory evolutive neural system, both in depth, namely lower level decompositions that are activated, and
in duration. For example, if A is activated at T, it means that P has been activated earlier, and since Since the activation of A is self-maintained by the loops, the activation of P will be maintained in the near future. Now, AC represents an, or the archetypal core, represents an internal model of the self, self as an enabling structure, reflecting a personal vision of the world since each ramification of an archetypal record represent a specific association of mental objects dependent on the former experiences of the person. It plays a motor role in the development of higher cognitive processes through information
processing by higher-level core regulators based on associative brain areas and directly linked to archetypal core. These co-regulators are called intentional co-regulators, can be compared to the conscious units. For example, an arousing situation or an unexpected event S, such as, for example, a perturbation, a disturbance, leads to the activation of some archetypal category neurons. It triggers, through archetypal loops, an extension of the activated domain D. This activation is transmitted back to the intentional co-regulators, which can cooperate to construct
a global landscape, uniting their respective landscape and extending them in depth and duration. Now, global landscape assembles information related to the present state, reinforces evanescent traces recently accumulated in lower levels of the working memory and even anticipate some future trends. Successively, global spaces partially overlap each other. Now this global landscape can be compared to the concept of global workspace that I mentioned last session proposed by Jean-Pierre Changjou and Stanislas Duhain. I remember I talked about it when I was answering Adam. It gives a frame for the development of higher
cognitive processes, in particular conscious processes characterized by an integration of time dimension to possibly alternative and or intermingled processes which extend what Husserl calls retention and pretension. A retrospection process toward the past that proceeds by abduction in the sense of Peirce to recollect information back in time thanks to its reinforcement in global landscape. Processing this information allows for analyzing the event which has triggered the formation of global landscape and finding its possible causes, therefore sense-making of the present. And then the, you know, protention, a prospective
process toward the future is then developed in global landscape to try and select long-term strategies. You know, that this memory is not simply a recalling, you know, basically structure or does not have only a recalling function, but it also has a prospective function. It simulates basically future, possible future anticipations. And basically it creates models of expected actions according to, you know, that's the
role of anticipation according to what has experienced in the past. So this is called the prospective process. It is done through the formation inside global landscape of local virtual landscapes representing mental spaces where successive procedures can be tried by internally constructing the corresponding complexifications with evaluation of their benefits and of the risk of dysfunction. A sequence of alternating retrospection and prospection processes accordingly leads to various scenarios. Once a scenario is selected, the retrospection process allows backcasting or
or backchaining, to find sequences of procedures, basically implicating co-regulators of various levels, able to realize this long-term program. The formation of scenarios is at the root of anticipation and creativity. Scenarios directly inspired by the contextual environment and current trends are obtained by simple complexification of successive virtual landscapes that add, delete, and our combined components, for example, combinatorial and exploratory creativity, metaphors, and conceptual blending. More innovative scenarios make use of iterated complexifications, which are not reducible to a unique complexification,
and lead to the emergence of mental objects of increasing complexity, so that these scenarios transcend the current situation. Successive global spaces can consciously process information coming from higher order category neurons while they automatically keep traces of the operations of lower level co-regulators, which are recruited by retrospection processes. It explains how a creative process can go through an incubation period during which The person consciously performs unrelated operations, followed by an insight with emergence in the global space of new ideas for the creative scenario.
So this was kind of like category theoretical, and also an example of brain as a complex hierarchical system, and basically how it all come together with these different processes of complexification that account for basically different cognitive processes of lower and higher dimension. Now, we have gone way over our time. So any questions and stuff would be appreciated.
I know that this was a bit technical, but Tony triggered me to a little bit up the degree of complexity. Nevertheless, I just wanted to, you know, we talked about the intuitive stuff about complex in a hierarchical system. But I think it was important to talk about a little bit about formalism of how to model these things logically. Yeah, I guess I'll have to go through the paper to get sort of the singularity thing,
but the convergence, so the limit property that you're talking about and sort of the formulations of category theory. Like that alone was sort of unfamiliar. So then when you used it to sort of explain something as complex as a model of cognition, I kind of fell behind. So maybe I'll catch up. OK, I wanted to mention, yes, you reminded me of two things that might be helpful for you guys. There is this guy, I have forgotten his first name. His last name is Healy. He's the first person who actually attributed co-limit constructs to basically neural systems and neuroscience.
I will find that. But there is also a book by Ramirez called The Categorical Imperative, where he really details application of category theory to neuroscience. And it has some really kind of interesting, simple, very lucid arguments that are a stepwise develop to kind of reach these more sophisticated stuff like call limits and projective limits, et cetera. I mean, one of the things is that I'm very interested in, for example, when it comes to the Kantian intuition and threefold synthesis, synthesis of intuition, synthesis of imagination, synthesis of recognition
in concept, is that obviously concept can do something to the brain. They activate their higher cognitive processes. They activate lower cognitive processes in basically a constructive manner. Now the thing is that formulating this link between lower level and higher level structures and cognitive processes allows, by way of core limits, allow us to basically recapture the fundamental idea of Kant. That's what is exactly intuition. is basically an image of individual items situated in space and time.
For example, an organism sees, that does not have the concept, only sees the facing side of an object and basically the color facing side. So this object basically doesn't have any connotation for the organism. Basically, the color itself is the object for the organism. If you are going to properly make an analogy here, we can talk about that it sees an object. The stuff of the facing side and the front shape of it are the object-quas-stuff for the organisms. Now for us, it's a different thing. We have memory.
Basically we have a constructive memory. We see the object, not only the facing side and basically its color, stuff, in contrast to the organism that does not have concepts or imagination. We see basically the objects all around. We have the memory that has constructed, put together these various basically faces of the objects and color of it through and through. But not only that, we recognize and synthesize these basically memories of different basically
parts of an object, but also a stimuli corresponding to these basically memories, namely shapes and colors, so on and so forth, we synthesize them inside the concept. So the concept is that higher-level co-limit construct that allows us to basically synthesize not only differential responsiveness to stimuli, shapes and colors, but also different sensory impressions of the past. Basically different levels at the level of stimuli, at the level of memory, impressions are being combined. So this is kind of like a very in tandem with the Kantian program, this kind of modeling
brain dynamics. Actually, that's pretty helpful. Thank you. And there is a good, I mean, are you guys familiar with Dennis Hassabis? Demis Hassab, he was like this gamer geek from the 90s who created some fantastic video games like Evil Genius, Roller Coaster Tycoon, and Republic, which you guys should definitely check out, absolutely hilariously good.
And then he became, you know, and he was the youngest, you know, basically chess master. Then he left, you know, game design and he moved to AI and then he moved to neuroscience. And he wrote this revolutionary paper, you know, in like mid-2000s about the simulating role of memory in neuroscience, this idea of retention and pretension, basically that the memory, a dynamic model of memory, needs to take into account how it simulates not sensory impressions of the past, but also simulates them according to future trajectories,
It's basically the role of anticipation, the prospective role of memory as a simulation engine. And this is a really, I think, this has put AI in a good orientation. That one of the concerns of today's AGI and development of human-level AI is really working out and modeling an account of constructive memory, because memory plays a key role in basically sense impression, what Kant calls inner and outer sense, plays a key role in
imagination, combinatorial synthesis of basically sense impressions as distributed in space and time, and the recognition in a concept, recognition of discrete entities attributed to an object, and being able to use concepts to cohere them. So basically, particularities fall under the generality of the concept, particularity of different variations of houses, of concrete houses, become fall under the general abstract domain of the concept house. So this idea of constructive memory
and how you can model it via category theory with understanding memory is not just a recall. It doesn't have a recall function, but it's also a simulation of anticipation and higher cognitive processes are future oriented. I think it's really a very key topic in developing a human level AI, precisely because of its rule that it connects lower level cognitive processes with higher dimensional, higher level cognitive processes. Reza? Can you hear me? Yes.
Would I be correct in thinking that this is directly connected to combinatorial entrenchments, the kind of reusability of all these bundlings of low-level patterns, reusability, redundancy, as part of... and all these possible arrangements as part of this kind of synthetic, I don't know, reconstructability and memory? Yes. Yes, absolutely. And that's the whole point of memory.
You know, I mean, evolutionary speaking, from an evolutionary perspective, memory came to existence not to recall, but recalling the rudimentary sense of memory, what it comes they call memory as simply playing a recall function. Recall function serves another purpose, and that is basically reconstruction of past impressions so as to anticipate of how organisms should function in an environment, namely develop
an internal model of expected action in an environment in which it is either unfamiliar with or basically it is familiar with. Nevertheless, that's the whole point, that it synthesizes past impressions, it constructs them synthetically in order to construct a model of expected action, basically a simulated model of a future world in which the organism inhabits. Could I see that as a space of possible rearrangements of those...
Sense impressions? Yeah. You see, one of the key concepts in talking about constructive memory is the concept of situatedness. The idea is that an organism basically has a duration of interaction with the environment. So every time that, for example, you have experienced a sense impression A at time t, then currently you are experiencing another sense impression at time t prime. Now in order to construct the memory or recall the memory of your past impression, the past
impression that is being reconstructed is not simply being resurrected as if intact, as if you can simply recall it and it will be left intact. It will be rearranged, namely reconstructed according to your current experience, according to your current situatedness, to your interaction with the environment. So there is no such a thing, and that's I think one of the intuitive ideas, not intuitive ideas, kind of a technical, in fact, technical explanation behind the intuitive idea of Freud, account of trauma.
That traumas are like this, that you have a past experience. What a past experience is never resurrected intact. Every time that you re-experience a trauma, trauma is distorted more and more, namely is rearranged, reconstructed according to your current situatedness and your current, basically, registers of interaction with the environment, your current experience, your current sense impressions. And this leads to this idea that as you re-experience trauma, namely past experiences, trauma becomes this twisted journey of bad experiences in the past,
you can never, and the task of psychoanalysis in terms of Freud is to unravel what was exactly the past experience, because the past experience cannot be resurrected intact. That past experience has been reconstructed, or in Freudian sense, distorted beyond recognition. It's precisely because of this function of the memory. So there's a kind of dynamic decoupling of each reduction. Sorry, dynamic decoupling? Each kind of, each level of bundling has its own kind of autonomy.
Each level has autonomy, but nevertheless, as we talked about, there are, again, functional links that basically intervene with these autonomies at the specific levels. And these functional links are usually coming from higher dimensions, higher levels, at later times. Questions, comments?
Adam, Tony, Stefan, any of you guys? I have an admin question. When is the recording from the last session going to be uploaded? Tony. Oh, sorry? I missed the last one, so I've been waiting for the recording to come up. Oh, okay. Yes, actually, yeah, that would be helpful for me so I can get some of the stuff and transcribe them so I can upload the slides and my comments to the classroom page. Questions, comments, nothing?
If there isn't anything, we should go for lunch. We are already more than an hour over the limit. So Reza? Yes. You said that you were going to be able to upload the slides to the classroom page, yeah? Yes. OK. Yes, I will definitely do that. Sorry. It's just, I mean, as I said, I lost the two first slides. So I'm trying to kind of slowly transcribe some of the stuff. But I have the slides of the current session. I will upload them, which I think is even more appropriate because I realize that some of the stuff we talked today are a little bit out there considering
our previous sessions. JOHN MUELLER, I have a quick question, if we still have time. JOHN MUELLER, Sure. JOHN MUELLER, I guess a bit of an observation as well. but I was wondering if you were familiar with what are called Lancaster equations that were these models of like attrition for air combat produced during the First World War. And they kind of jumped to mind when you were talking about developing these clunky models for really complex systems or organizations. Yes, actually I'm not familiar with this equation, but yes, actually isn't he the guy
who was an engineer, or maybe I'm confusing with another person, who was an engineer and then moved into military science, and he started to develop a model of basically battlefields and combats. Or maybe I'm confusing. I need to look it up. But yes, yeah, basically, yes, there is, I would call some of these are quite clunky. But basically, and there is a reason for that when you look into, for example, extend these kinds of equations, for example, especially in military science, but also in economy,
when you have basically players, player opponent situation, you have a, when you extend these you basically come up with these kinds of very almost rudimentary accounts of equilibrium and non-equilibrium. And as you say, attrition, reinforcement, which are basically the fundamental concepts of game theory. And I think game theory is a very, very impoverished field for looking into interaction, basically
dynamics of complex systems. They are very useful. They are very useful for studying the local interactions. But when we talk about complex systems, there are not only interaction with the environment, these non-linear interaction between their own components, I think game theory massively fails. And any kind of, basically, theory that is based on explicit equilibrium, non-equilibrium formulas. That's great. That's really helpful. What's interesting about the equations was that they gave a mathematical basis for some of Germany's principles about concentrating effort
at a focal point, because they sort of showed like a, through quadratic equations that, like increasing the initial size of a sort of force concentrated in one area would lead to the kind of exponential improvements in things like firing rate. But yeah, I was kind of thinking about looking at something like that for the, also like I haven't, yeah I'm still behind on the first proper. Yes, well, I mean, the thing is that some of these kind of military things have been extensively revised by kind of a more network-centric, you know, schemas of warfare. And usually they are coming from Rand Corporation and by people from Lebeki and stuff.
It's definitely been updated. Yes, but even those, I think, even the network-centric ones are quite even rudimentary. What I think, in fact, when it comes to warfare, I think, in fact, the network-centric ones are even more impoverished than these classical ones, because the classical ones, in fact, give you a very local edge in warfare. For example, you say focusing and artillery, these kinds of stuff are extremely useful for tactical combat. Whereas network-centric, when it tries to kind of mix up these kinds of tactics and strategy, schemas of warfare, they become very sloppy.
And hence, you get the whole really kind of out there warfare at the age of chaos, schemas, which are very bad. A good critique of them has been provided by, I mentioned, Manaparata Guha in his book Network Centered Warfare. But yeah, I think... Yes, yes, it's a very good book. And Manaparata Guha teaches at Bath University. He has done some, he's doing some very cool research on warfare, especially this kind of the tactical side of it. Great, thanks so much Reza.
Welcome. I have just one more thing, it's not so much a question, more like a comment in that I'm so excited to kind of see the stuff that we covered today for instance, or being extended to our symbolic systems in terms of this kind of modular, reconstructive capacity in such combinatorial systems like neural networks? Yes. Are you familiar with Peter Wolfendale? No. But you know him, right?
Have you heard of him? No. Pete is an astonishing philosopher. He's superb. He's absolutely fantastic. One of Pete's current preoccupations is basically artificial neural networks and neural systems. And he basically is very interested in this idea of language and neural systems and memory and the Kantian program of AI construction. So you should look into his stuff and be in touch with him. He's very generous with his responses. But nevertheless, yes, I think neural systems, we'll talk about them, are very, very deeply
connected with basically syntactic combinatoriality. Familiar with, I mentioned Trent Stiakhan's book, Symbolic Species. I think first chapter, the second chapter of it is, if I remember correctly, he talks a lot about the connection between neural system, neural networks and basically syntactic combinatoriality and how you train a neural network to come up with a proper syntactic parsing and syntactic learning. And these are drawn on, you know, real empirical experiences
done with primers, especially bonobos. The most famous example of it is Kenzie the bonobo who basically has a very rudimentary, in a genuine sense, of language, concept learning. language, concept learning, or basically at least symbolic manipulation rather than concept learning, let's put it that way. But I think at the level of semantics, it becomes extremely difficult to move from the combinatoriality and statistical inference that involves neural networks to kind of a semantic complexity.
I think semantic complexity requires a different computational information processing than those available or prevalent in artificial neural networks and syntactic manipulation and syntactic training. This is one of the things that we talked about and I mentioned very briefly that different levels of constructs, different levels of this brain dynamics, brain-mind interface need to be understand as different computational strata, different, and they require, in fact, I would say, and this would be, you know, part of the end of our second module and our
entire module, in fact, we need to be able to develop different paradigm of computation these levels because they are extremely asymmetric in how they function and how they compute, they process information. OK, with that, should we call this session? Yes. Thank you so much, Dominique. And thank you everyone for bearing with me.
Very appreciated. Thank you, Reza. REZA, sorry. Thank you. Thank you, Chris Lee. Thanks, Chris. Yes. Do you have, it's a bit naughty, but do you have the book by Manav Guha on PDF or something? I just looked on book depository and it's ludicrously expensive. Yes, yes, I know that. I think I have it. I will look into my drive and see if I can find it. I think I have it, yes. I will send it to you, or just send me an email, and I will reply back. I will just . JOHN MUELLER 1- That'd be fantastic. I'll get my library to order it in the meantime.