Complexity & Computation (Session 4)

All right, hello everybody and welcome to the fourth segment of the first module of complexity and computation. And here we have our instructor Reza Negastani, who seems to be a little under the weather, I'll pass it on to him. So thank you Reza for being here and we'll start today. Thank you very much everyone. Sorry, I have kind of difficulty to raise my voice. Okay. So before starting, does anyone have any questions, any discussion? Would be great to kind of go over very briefly over the stuff that we talked about in the

previous session. Aaron, do you have a question? Sorry, I'm chewing. Yeah, I would like to go a little deeper into the logical depth thing or just kind of review it. Yeah, we are going to go into today and unfortunately it has some math this time. So I will give some formalism behind it, because I think it's necessary to not just with some of this stuff to go over the intuitive ideas, but also give a minimal formal definition of them. Yes, I will go over basically at least three things.

You know, the Kolmogorov complexity, Advanced logical depth and Crutchfield statistical complexity. I give formal definitions for each of them. And I guess if we wanted to go over something, maybe you could do like, because we talked a little bit about German idealism last week. Maybe you could do like a bit of like an overview of what you see, like the crossovers between sort of both bringing up the critique of pure reason, bringing up the phenomenology of spirit, what it has to do kind of with this material, AI as like AI or like spirit as self-construction, or I don't know, like the way your conceptual reasoning

is about this material and what that crossover is. Sure. I mean, well, you see, I mean, most revolutionary accomplishments of German idealism is basically Basically it brings three fundamental types of philosophy together, philosophy of knowledge, philosophy of mind, and philosophy of action. I mean, basically the entire, you know, I think it would be great to kind of like very briefly talk about what is exactly the moment of German idealism and exactly what is the accomplishment of Kant and respectively Hegel, and then back tied to basically contemporary

theoretical computer science and AGI construction. So basically, you see very briefly, I mean, most kind of striking feature of Cartesian rationalism is that Descartes embarks on this project of basically constructing an account of theoretical subject. But of course, he falls short. Then Hume tries to refine it a little bit. But also Hume fails, mostly, it's basically that's the reason why he fails.

There are many aspects to it. But generally, I mean, this is part of the entire Kantian philosophy. The way that he constructs the critique of pure reason is out of basically the critique of Descartes and Hume. Descartes understanding that mind does not have access to its internal states. for Kant that this is not, he shows why that's not true. And for Hume, it's basically what I would call Hume is like the archetype of a global reductionist skeptic. Basically, Humean project is a reductionist skepticism. And one of the main, basically, arguments

of Hume in the construction of subjectivity is that it simply, and this is what is basically the extension of what is usually called the principle of separability, Humean principle of separability. Roughly speaking, it's the idea that the principle of separability leads to another principle that's called the principle of basically integration or combination. The idea that the awareness or consciousness, subjective consciousness, can simply decompose to its discrete instances. So basically all you need to have a complex, basically in order to have awareness, basically

he tries to reduce the awareness of complex, and this complex can be so many things. whether, for example, temporal awarenesses, spatial awarenesses of items in the environment or objects in the environment. He tries to decompose it into basically a complex of awarenesses. So simply you have discrete awarenesses, and basically that is sufficient for you. For example, I think X at time T, I think Y at time 2, I think Z at time 3, so on and so forth. If you have these discrete instances of awareness, and this awareness can be awareness of your temporal, awareness of items in the environment as such, basically, like for example, any sentient entity

can have that kind of awareness. So if you have these discrete instances of awareness, that is sufficient for you to construct a subject, basically a self, a perceptive self. But Kant shows that a complex awareness of complex, basically, and this complex is basically the manifold of sensation or the manifold of intuition, so on and so forth, is very different, it's fundamentally different from a complex of awarenesses, of discrete awarenesses. And respectively, for example, a sequence

of temporal awarenesses, which is basically Hume's position, is very different. It's not sufficient to construct a temporal awareness of a sequence. So basically, this means that there is this gap between this parochial, discrete combinatorial schema of construction of subjectivity out of discrete instances of awarenesses, and awareness as an integrated generative framework of a perception, namely self. And that basically starts from this moment to build basically the idea of the synthetic

unity of apperception. And the synthetic unity of apperception or discursive apperception is basically a model of what we would call a sapient intelligence. and intelligence whose main criteria of distinction is basically its ability to engage in discursive practices, but also is endowed with an integrated and generative framework of subjectivity.

So Kant has started to build this. I mean, I think a very good, basically, it's really hard to track down, basically, the scope of this in Kant's own work. I think the best person who has done this, in order to show the stages of this construction, how basically Kant's try to rebuild theoretical subject, but on top of it basically constructs a practical subject. So basically Kant's idea of sapiens intelligence is not only to reclaim theoretical subjects

from the Cartesian and Humian, basically, philosophies, but on top of it interpose basically a practical subject. And this practical subject and theoretical subjects are basically unified, integrated within this perceptive self. So basically, first thing that Kant needs to do, as I said, needs to construct a stage by stage, you know, an aperceptive self and shows that how these theoretical and practical dimensions are unified within it. The person who is basically kind of clarifies these stages of construction, kind of rectifies some of the implicit Kantian errors is Szilard's, and the best point of entry to this kind of

Kantian reconstruction is lectures on pre-Kantian things, which basically starts from the Cartesian, human, then moves to Kant and tries to get everything into one picture. So basically the first thing that Kant does and also Solor makes explicit is that first First thing is that sapience intelligence is, before it becomes sapience as such, namely becomes a distemporally discursive, apperceptive intelligence, is that it needs to have basically

the minimum conditions of realization for, one, a unified field of subjectivity, basically non-conceptual awareness, and basically at the end, realization of conceptual activities, basically the power of judgment. So first things that Kant strives to do is to construct what he calls the structure of the outer sense. It's basically the form of awareness. This simply, and what is this? It's basically our spatial, perspectival awareness.

And this spatial perspectival awareness does not require basically conceptual, explicit conceptual tools to function. You can see this in basically all of sentient entities, and especially the ones with brain. That the first things that you need for spatial awareness are capable of basically a reliable differential responsiveness to the position of objects situated in a space. And by virtue of basically having a minimum capacity for differential responsiveness and

also an adaptive behavior, then you can have an egocentric frame of reference. Once you have this egocentric frame of reference, then you are capable of, as a sentient, then you are capable of basically arriving at this, what in neuroscience they call it, an allocentric frame of reference, namely telling, basically differentiating the relations between objects in the environments, and then becoming prospectively, spatially aware of these objects as situated in a space. So this is basically the reason that why Kant starts with, basically, a space and time in his scope of project,

intuitions of space and time. And the first criteria for this, basically for Kant, is to come up with the form of this awareness, of this rudimentary sentient awareness. One is the outer sense, a spatial perspective awareness. And the other one is basically temporal perspective awareness, namely what he calls inner sense. And what is really this inner sense? Inner sense is the most basically fundamental prerequisite for the construction of a discursive, a perceptive intelligence. Why is that? Because inner sense means that the mind,

the mind of sentience, does have access to its inner state, contra the Cartesian, you know, basically claim that mind does not have access to its inner state. So Kant wants to construct this, basically, a structure, this coherent structure, wherein a sequence of temporal awarenesses becoming awareness of a temporal sequence. Namely, for example, I think X at time T, I think Y at time 2, I think Z at time 3.

He wants to integrate this, so it becomes, I think, bracket, X, Y, Z, close bracket. So they become basically integrated within one unified field of awareness. This one unified field of awareness, which can only be constructed by a temporal awareness, namely the structure of inner sense, is the prerequisite for the construction of discursive aperceptive intelligence. And in fact, Kant in Critical Pure Reason, I think it's around page 200-something, he says something very important. He says something that basically this is the most fundamental

claim that any attempt to tackle with subjectivity needs to take this into account. So this is basically the inner sense, the form of this rudimentary awareness that is required and necessary for the construction of this curse of aperceptive intelligence is basically the idea of temporal awareness. So I don't get into details, but basically Kant, the way that Kant construct this temporal awareness, and that basically is assault on human reductionism of subjectivity, is that he introduces something, a functional item into the sequence of awarenesses called basically

a meta-awareness, that for a subject to be prospectively, temporally aware of worlds, of items situated in time, is not sufficient for it to be able to represent discrete instances, a sequence of instances. But it needs to be meta-aware of its own representations, of representations of items in the world. So that meta-awareness basically creates this link of nestedness between the sequences and then cohere and integrate them. So he does this.

Then the second stage, you know, basically is the understanding of intuition. in Kantian sense is the understanding that, you know, you—it's the relation of cognition with individual items in the world, provided by receptivity of sensibility in sentient intelligence. So once you have this sentient intelligence cohered by a perspective or a spatial awareness and the perspective of temporal awareness. And this is basically where the majority of our current AI

understanding at this stage is, really. Not even temporal awareness, but simply perspectival spatial awareness, what Kant calls it, outer sense. We haven't even gone into basically temporal perspectival awareness. I mean, in AI, for example, people work with idea of constructive memory, modeling constructive memory. But this is the fantastic revolutionary insight of Kant, the contra-Hume, which shows that memory by itself, even if it is constructive, namely, basically it creates impressions of sequences, memory by itself does not supply a temporal awareness.

You need something more than memory, and that would be basically Kantian's construction of meta-awareness as a fundamental awareness that coheres and integrates the sequence of basically discrete instances. So past, present, future, so on and so forth. So once basically he has this, he moves to intuition, construction of intuition, construction of imagination. And the ultimate aim is to provide this basically prospectively aware sentient intelligence

and turn it into sapient intelligence by creating this integrated framework through this, what is basically called Kantian threefold synthesis. The idea of apprehension, synthesis of apprehension, synthesis of imagination, and synthesis of reproduction in the concept. So once he has this threefold synthesis in place, which are basically the stage of construction of apperceptive intelligence, he arrives at what he calls a synthetic unity of apperception, basically a unified field, an integrated generative framework of intelligence called sapient,

what we call a human-level AI. Now this is basically very briefly the picture of Kantian, why is that it's important to engage with Kant. Because Kant has this extremely sophisticated, constructive method to show that every time that, for example, we think a rudimentary, sensory inductive form of intelligence is sufficient for the construction of sapient intelligence, that's not true. He shows that basically, and that's basically his rectification of his prior philosophy, namely Descartes and Hume especially.

So Kant makes this a schema of the construction of temporally discursive apperceptive intelligence. Now, Hegel's basically addition to the Kantian schema is that in Kant's philosophy, apperception, which is distinguished by the power of judgment, apperceptive intelligence, and that's basically concept using, simply. Concepts are still rudimentally and parochially defined. Concept basically just serves two primary functions. One, the intuition of an individual item in the world

falls under the domain of the concept. This is the first application of concept in Kanton system. Then the second application, which is more fundamental, and that's where Kant introduces the categories, basically, into the system, is that a concept is a rule for construction of a manifold of intuitions. And basically, for example, a concept of a line, how you can, through these dots or points, you can, by way of a rule, exercising a rule, you can construct a line.

For example, how you can construct an image of, for example, a car, a house. Obviously, a sentient intelligence sees, And that's basically the solarze and adendo. Sees only the facing side when it looks at the house. It only sees the facing side of the house, its color and shape, basically, that it has. It's in the field of vision. It's the seeing what it sees of the object, basically.

But discursive aperceptive intelligence is not only seeing of the object, but also seeing the object as a house, as a concept, basically. And that concept, the second application of the concept, the categoric or categorial function of the concept, namely its application as a rule to these discrete images of an object in the world and how it integrates and construct them is basically this idea of concept. You can see, for example, when we look at a house, regardless of what we see, we have

a concept. This concept is basically linked to these discrete images of different sides of a house with any particular instances of something like a house. It's connected by way of basically intuition and imagination. So basically, they are integrated, these different instances, these different images of particularities of an object called a house are basically linked together and constructed coherently, put together in the right way, according to a rule, which is in this case a concept of

a house. Now, this is basically Kant's idea of a concept. And it doesn't go too far to just elaborating what a concept is. Now, the Hegel's revolutionary thing is that for him, the concept, which is the fundamental, basically, feature of discursive, apperceptive intelligence, is that concepts are linguistic based or language based. And that's basically Hegel wants to argue, and that's basically his main thing in Phenomenology of the Spirit, that language is the design of sapient intelligence.

Language is the design of Geist. And if you do not have language, you absolutely do not have any of these kinds of extra cognitive abilities that you basically ascribe and attribute to discursive and perceptive intelligence. So Hegel's, and that's basically, so Hegel's tried to rescue or rectify or kind of develop the idea of the concept proposed in Kantian system by adding the component of language, by adding the component of sociality. And that's a very, I think, a very key aspect. And that's only has been very seriously

been discussed in theoretical computer science recently. That in order for you to construct a robust human level AI, you cannot make a machine. You need to have a system, a multi-agent system. And this multi-agent system needs to have different levels of interaction between its agents. And these different level of interaction are not only the kind of rudimentary machine learning of a statistical inference of how agents basically interact with one another at the level of statistical inference and learning, but also a different level of interaction, a linguistic interaction. And linguistic interaction has its own kind of logic of computational complexity.

And that's basically what we are going to talk about, we are Girard and Abramsky and Andreas Blass. It's the idea that linguistic interaction basically enables a semantic of information processing. Semantics, basically, this whole, and we will try to build basically a language simply from its basic core symbolic repertoire towards its syntactic structure and towards its basically semantic complexity. And that semantic complexity

of interaction, which is basically the base of Hegelian sociality, the idea of language Language being the design of Geist, Geist being a community of discursive, apperceptive intelligences. Language in this sense is basically this platform, this framework through which agents are endowed with a generative cognitive complexity unavailable to forms of intelligence or agencies that function individually or basically simply have a private, a spatiotemporal awareness

of items in the environment. So this idea of language, and language has this computational capacity and it plays roles, I think is a very good candidate for being cited as the implicit idea behind Hegel's emphasis on the sociality of basically rational self, that there is no such a thing ultimately as an individual self. Self in the Kantian sense, namely being a rational self, not a phenomenal self. Phenomenal self is simply temporal, a spatial awareness. A rational self is ultimately a social agent, and this social agent is only social by virtue

of language. What precisely language provides this individual agent that ultimately renders it social in the last instance is something that we will talk about. So basically this is the whole, you know, the schema of Kant and Hegel with regard to construction of a robust account of HGI, that you need to have these rudimentary conditions of realization, what Kant calls conditions for the possibility of cognition, namely the doctrine of transcendental psychology, simply arguing what are the minimum conditions necessary

for the realization of a perceptive intelligence. So this is Kant. Then Hegel addendum would be how to supplement this realization of discursive aperceptive intelligence with something that allows it to function as it does, basically. it engages in conceptual activities that enables it to cohere all of its spatiotemporal, perspectival awarenesses. And basically, ultimately gain an objective traction upon the world. Not only becoming a theoretically, basically,

robust subject, but also a practically enabled subjectivity. The idea that how philosophy of mind, philosophy of action, and philosophy of knowledge basically become the idea, are used and put in the service of construction of a theoretical subjectivity that is endowed its practical abilities. Great. You go ahead. Yeah, no, no, it's really interesting overview.

Actually, there's a few things to get a grip on. I was curious if you had a few we're talking about on the can't an idea that so sorry this conceptual or a apparatus it just arm reminded me of these modern all interpretations of human evolution all where you have the concert the ability to plan being very connected to the uh... mechanisms in the brain required to throw

so on so mechanism required to like throw something like a a rock or barren or whatever all seems to be very implicated in the same mechanisms archaeologically or in the brain as the mechanism for planning. And just the description that you were giving of Kant's sort of requiring certain mechanisms in a concept of space and time just seemed very related to that to me. Yes. Yeah, okay. Yes, absolutely. Yeah, absolutely. I mean, that's, for example, the reason that I suggested Stanislav Dohan's work is precisely

because of this. I mean, when you look at the stuff that is being done in neuroscience, especially the philosophy of neural materialism put forward by the likes of Shonjoo and Dohan and Allen So is that basically the idea of the global workspace in the brain, you have basically some modular information processing units. Then these modular information processing units are connected, but also they have their own groups. On top of them, they are connected, giving rise to a level of these local work spaces, neural work spaces, then you have on top of them a global neural work space.

So the idea is that once an organism, once an intelligence focuses on a particular aspect, basically engage what Kant calls engaging basically behaviorally with the environment, purposefully focusing, like a predator focusing on the prey. This act of focusing basically mobilizes all this information provided by these unconscious modular information processing units, mobilizes them and brings them to the level of the global workspace. So what Kant calls the sentient consciousness, or basically this outer sense and inner sense, spatially and temporally prospective awareness or a spatially and temporally consciousness

is simply what in neuroscience they call a global neural workspace. A global neural workspace is simply the idea of nominal consciousness, the idea that basically once you focus, once you engage in an embodied activity in the environment using, you know, putting your body in motion toward a prey using a tool, for example. These attentional activities basically activate this global workspace. But you never have access to those basically modular information processing units. And you don't need to have. And the thing is that Kant wants to basically construct,

and this is also CELAR's. to show that once you have a concept, concept in the Kantian inferentialist sense, in the linguistic sense, concept as a rule of inference, as a linguistic entity rather than as a natural object. Once you have that, then you are capable of bringing not only the items in the environment, your awareness of items or representations of items in the environment, but even you yourself under the power of judgment. Namely, you are capable of breaking into the realm of appearances and deepening into the intelligibility of the order of reality.

And that's the power of judgment. That's only basically is part of, is basically, is something that is exclusive to creatures who are capable of language use. And by language use, I do not mean it simply as a symbolic manipulation, but language in the sense of an integrated framework of basically symbolic combinatoriality, syntactic structuring,

and semantic complexity. Another great work on this front, I noticed that Stefan had put something, what was that, Symbolic Species Renewed, something like that, in the Google Drive. I noticed that basically, yes, it had like basically Trent Stiakhan's essay in it, and Basically, the title refers to a book written by Trent Stiakon called Symbolic Species. It's a very groundbreaking book when it came out. It actually has a really great understanding of some

of these kind of things that you just talked about in terms of this rudimentary, especially temporal awareness, and how basically they are connected to language, and how basically linguistic evolution basically uses all these different levels of awareness and basically ultimately links them to something called semantic complexity. Cool. OK. It's a superb book. Definitely everyone is interested in these things. I mean, I'm not really fond of these evolutionary books done, because with evolution you can basically see yourself entitled to so many metaphysical

claims, but these metaphysical claims have so many assumptions that everything can go wrong. But I think Diakhan's understanding of how he evolutionary lays out the structure of language and development of language is absolutely brilliant. One especially particularly striking feature of that book is that the Akhan explicitly refuses the idea that linguistic items are connected to natural items, basically objects in the world, that there is no relation between worlds and worlds. Hence, he implicitly affirms a coherentist, a word-word view of language.

I want to try to relate this back to logical depth, I guess, because there were some interesting themes in overlap. And I guess it relates to what you were just saying as well with sort of a coherent account, and that the thing being tracked here is sort of the inferential relations between terms rather than, or the inferential relations between moments in time are being tracked as well, rather than linking the moments themselves, and that's what makes the synthetic unity of that perception sapient rather than just sentient? Absolutely, yes, yes, yes, yes. Yes, ultimately what unifies basically the field of experience, the field of discrete

awarenesses is basically this inferential network that is implicit within concepts. And so I thought this was really interesting with, I guess I read the Bennett, Azura A. article on logical depth and computational depth. And sort of the way I guess we've been talking about complexity also has to do with history, right? So like a sapient entity, one that is a synthetic univial perception, has a history because it's able not just to link moments in time, but to understand sort of the inferential relations between them. Yes, absolutely, yes. Yes. Computationally deep object is one. I guess this is where the difference between like a physical depth or depth in a physical system and computational depth is a little different.

But understanding what logical depth is, I guess it's a way of measuring or tracking that something has like a complicated causal history as opposed to an inferential one, I guess, but it's the same. Basically, it has a degree of sophistication. has a degree of sophistication in the sense that something that takes a very long time to construct, computationally construct, basically requires an excessive run time, you can have this, compress it in the shortest description, basically. So for example, I will talk about this shortly.

For example, do you guys remember the movie Contact with Jodie Foster? Yeah, basically it has that they got this message from the alien civilization that they noticed that it was primes basically, prime numbers. They contacted them with prime numbers. With prime numbers, one of the things that's very interesting is that logically deep objects are artifacts of intelligence, of advanced intelligence. Basically it means that an intelligence has become capable of compacting something that takes massive scales of time to simulate, to construct, in the shortest possible description.

So prime numbers are logically deep objects. If you want, for example, relay a message to another civilization concerning that you are basically, you stand out as an intelligence. So basically other civilizations can detect you as an intelligence in time, you know, this basically register of basically development of intelligence in time, you for example communicate them with prime numbers because prime numbers are logically deep objects. It means that you basically, something that takes a long duration of time to construct, you have compacted in the shorter description a prime number.

And basically prime number is a logically deep object, a register of an evolved intelligence. Yeah, and it's sort of tracking this, I guess it's tracking the fact that sort of inferential awareness had to go into crafting this and sort of compressing the data itself into a form that tracks that sort of set of inferential steps. Yes, well it has so many basically aspects. It has the aspect that means that basically you have come up with an extremely sophisticated form of language. It means that you have come up with an extremely sophisticated form of compression, compression

of regularity. So simply you do not, no longer communicate these simple patterns, but a compressed pattern. a compressed pattern and a sophisticatedly compressed pattern, namely a logical depth, in this sense becomes a sign of an evolved intelligence, basically. An intelligence that has massive computational capacities, either in its head or on its own machines. Yeah, I guess the connection I'm trying to make here is we're just trying to link this idea of computational depth to the synthetic unity of perception and that both are trying to in some way develop a form of tracking, right, the inferential steps that go into

producing any kind of… Yes. …todestination. Yes. One is a computation, one and one is… Yes. In fact, a perceptive self is a massively… is probably the single most deepest object that we know, because not only it is a sentient organism, but it's also a language. And language is a massively deep object in a Benetian sense. So yes, I mean, basically self is itself an artifact. Rational self is an artifact. And that's really one of the genuine, I think,

basically abilities of a perceptive intelligence that is capable of treating itself, its own field of experience, as an artifact of its own ends, basically, its own reasons. Another thing that, you know, a kind of ramification of this idea of a perceptive intelligence and this unification of this discrete awareness via the inferential network of language, is that once it becomes unified, once it becomes unified, then you are capable of talking about individual subjects. So in this sense, individual subjects,

the unified field of the individual subject, in the domain of private thoughts, is in fact a consequence, not the premise, of sociolinguistic practices. So that's the kind of counterintuitive, that individuality in the robust sense is not the premise of sociality. It's the consequences of it. Because once you have this inferential network in the sociality of language, then you have the concept. once you have the concept you are capable of unifying your experiences for

one the second the most important thing you are capable of attributing this unified field of experiences as the experience of your own your own experience and that's the whole central core of what a perception means a A perception means that not only that you have a unified field of experience, but you are also capable of attributing that experience and understanding it as your own experience. Namely, you have a sense of individuality. Yeah, great, thanks.

I feel like there are a lot of places we could go from here, but I'm not sure what. Maybe we should go back to logical depth, or where were we planning on going for today? Yes, I'm going to start with the measures. I'm going to share the screen. Oh, I had no idea that Contact is based on novel by Carl Sagan. Oh, OK. Very interesting. Makes sense. OK. OK. Okay. Can you guys see the screen sharing?

Yes. Okay. Yeah. Good. Okay. Okay. Okay. So you see, we talked about complex systems and features in the last session. And I briefly kind of forayed into the idea of measures and where basically the basic idea comes from. And I talked about Boltzmann's. I talked about a little bit of Kolmogorov and Shannon informational content as an entropic structure of the system,

and talked about Benet's logical depth. Now the thing is that those, the ones that basically Boltzmann put forward, the idea of the informational content, how to quantify the informational content of the system, are basically measures of complexity. And in this sense, I think it's very important to understand and differentiate complexity in the sense of measures of complexity and complexity in the sense of complex systems. Complexity in the sense of complex systems

is basically can be both ontic and epistemic complexity, as we talked about. But the strong emphasis is put on what? On basically ontic complexity, the complexity of the physical system itself. Now, complexity in the second sense, complexity in the sense of complexity measures is mostly epistemic complexity, you know, complexity in the sense of modeling, specifically made to quantify, basically, what counts as

the complexity of a complex system. We can define measures of complexity that obey the arithmetic properties of information, information measures like Solomonov, Kolmogorov, and Shaitin. This is our one side, basically the primary measures of quantification that specifically deal with the informational content of a system. In addition to these, you know, basically arithmetic measures of complexity that specifically deal

with the informational content, and, you know, for that reason they are directly coming from the Bolsmanian heritage of looking into how things hang together in the system, in addition to these measures, there are also measures that are not just information measures or maybe are not information measures at all. And these are usually the kind of measures that are either using basically have something to do with temporal understanding of complexity and measurement of complexity, or measurement

of complexity based on computational resources. So to summarize, there are also other measures which are not information measures, but very important and very they typically have something to do with time you know an example of this would be best logical depth steps of computation take to construct or generate an object basically here you know the main concept is the notion of runtime or resources basically the resources an observer needs to spend and needs to possess in order to be capable of inferring the state of the

system or basically have a model of a complex structure or any kind of, you know, be capable of basically detecting complex regularities. And these are usually the main person behind this, James Crutchfield, and his idea of a statistical complexity. I'm going to talk about all of these three today and kind of talk about their weaknesses and points of strength. This distinction between information measures and temporal measures, the arithmetic measures,

and basically temporal measures in the sense of computational steps, for example, indexes of time, so on and so forth, can be boiled down to the distinction between what we can call length of computation, basically a space criterion for complexity and depth of computation, time basically criterion for computational complexity. We will look into this whole idea of computational complexity in terms of space and time and resources in much more detail in our second module. But for now, it's important to just understand this, that the information, that the distinction,

main distinction between information measures and temporal measures of complexity can be seen as the difference between length and depth, basically. Complexity as a matter of descriptive length and complexity as a matter of generative depth. And so it's really important to note when we are using the notion of land, we usually use the notion of description. And when we use the notion of depth, we usually use the notion of generation or construction. And this is another distinction between, for example, Kolomogorov, you know, measure of

complexity and mass measure of complexity. Kolmogorov, length is descriptive, whereas Bennett's logical depth is generative, generative measure of complexity. So first, length. The intuitive, I mean, I'm going to talk a little bit about the intuitive idea of length and then go into a formal definition of length later. The most basic definition of description length is simply the number of words required to describe some object, event or sequence. The more words needed to describe an event, the more complex that event probably is.

At the first level of approximation, description length seems to work pretty well. A dead branch is much less complex than a living tree. The branch can be characterized in far fewer words. Similarly, a game of marbles is less complex than a game of soccer. This is true whether we describe the rules or the play of the given game. But there is a potential problem with description length. The length depends on the language used. And by language, I mostly mean a formal language. Fortunately, the problem of multiple languages

has been overcome. To show how, we first have to formalize notion of minimal description length. Now, it's worth working through this level of detail for two reasons. First, we can look at the subtleties involved in constructing a useful definition of complexity. Second, we can see the extent to which the complexity depends on encoding. First, let's see why choice of language, in fact, matters. Suppose we want to communicate that a period of human existence had a particular spirit. In the English language, we would have to say something awkward like the spirit of the age, whereas in German we can just say zeitgeist.

Now, we can take this same idea and make it more formal. Suppose that we want to describe numbers, but the only number in our language is 3. To express a number 5, we would have to write this formula. To write 6, we would only need to write 3 plus 3. Therefore, 6 would be less complex than five in this language based on minimum descriptive length. One has more words and the other has less, simply. To show that the language only matters up to the point, we first need to assume that the phenomenon of interest

can be written in a description language. We can then define Kolmogorov complexity as follows. Kolmogorov complexity is the minimum length of a program written in description language that produces the desired sequence of symbols. If we have two languages, A and B, then we can write the Kolmogorov complexity, K, of a sequence S relative to those languages as K, S, and K, B, S. The invariance result states that given any two languages, A and B, there exists some constant C that for any sequence S,

the absolute value KAS and KBS is less than the constant. This result implies that for sequence with high columnar of complexity, the choice of language does not matter, or does not ultimately matter. So at this point, we can say that if the thing we are describing is reasonably complex, if it requires a lot of words, then the choice of language does not matter much. This does not mean that minimal description length is a perfect measure, though. It has two shortcomings. The first is technical. Given a sequence and an alphabet, there does not necessarily exist an algorithm that

will generate the Kolomogorov complexity. And I talked about this very briefly, that one of the shortcomings of Kolomogorov complexity and logical depth as a measure of complexity that's directly drawn from Kolomogorov complexity is the uncomputability issue. The second problem is that the description length assigns high values to random sequences. For example, consider the following three sequence of zeros and ones. To intuitively grasp this, think of ones as days the stock market goes up and zeros as days that the stock market goes down. So we have these two patterns.

Or sorry, sequences, strings. The first sequence can be described as 10 zeros followed by 10 ones. That's a total of six words. The second sequence can be described as k0s, then k1s, k1 to 4. That's nine words. The third sequence can be described as 0, 2, 1, 0, 4, 1, 0, 1, twice, 1, 0, 0, 1, twice, 0, 0. That's a lot of words. I think it's 16 words. Yet, if you look at the second and third sequences, we see that the second string has patterns,

and the third is random. Therefore, rather than giving a long description of the third, we'd like to just say it's random. And basically, that's only three words. As the sequence above made clear, description length conflates randomness and complexity. But as we have discussed, complexity is not the same thing, and it's a randomness. Sorry. OK, yes.

OK. Instead, complexity lies, as we talked about, lies in between order and disorder, between basically a structure and randomness between these two extremes. One approach to making randomness less complex is the measure known as effective complexity. You know, I will talk about this later. This concept amends the idea of minimal description length by considering only the length of a highly compressed description of its regularities. In other words, it strips away randomness and only what is left counts.

Alternatively, complexity can be measured as the difficulty of generating the sequence. This can be calculated either as the minimal number of steps, logical depth, which Charles Burnett, which I will talk about in length later, or as a number of steps in the most plausible sequence of events, the so-called thermodynamic depth. The logical depth of a human being would be huge, but it's relatively small compared to the thermodynamic depth, which also takes into account the most likely evolutionary history that resulted in humans. Measures that capture the difficulty of generating a phenomenon are much theoretically robust,

but they too have shortcomings. First, they have computability issues, just like description length. So the measures were great in theory, but they are very difficult to apply in practice. A measure that we cannot calculate quickly has limited practical value even if it's theoretically sound. A second problem with this whole class of measures is that they apply to a fixed sequence of events or outcomes. The system being studied must have a beginning and an end. You might instead consider a sequence that continues to grow over time and ask how difficult it is to predict that sequence as accurately as possible.

This idea underpins Crouchfield and Young's concept of a statistical complexity. With a statistical complexity, a random sequence would have low complexity because a machine, the so-called epsilon machine, which I will talk about, that generates that sequence would be relatively simple. To calculate a statistical complexity, an algorithm classifies past data into categories or groups called causal states so as to produce patterns that are statistically indistinguishable from the real data. Though not easily calculated, basically

Crouchfield and Young measure has been derived in some cases. they have actually applied in some physical complex systems, like crystals and neural firing patterns. Now, let's get into the formalism behind the concept of length. One of the proposed measures of complexity has been algorithmic complexity, which we briefly talked about in the last session. This was independently defined by Solomonov, Kolmogorov, and Chaitin, but it is almost universally named after Kolmogorov. Kolmogorov complexity, K, makes use, sorry, some of these, I noticed that some of these

math formulas are kind of like out of whack. It's just hard to use math type in word. Homogorof complexity k makes use of a universal or general purpose computer, basically a universal Turing machine, u, which has a language l in which program p are written. Programs output sequences of symbols in the vocabulary, usually binary of l, and l needs to be again in frank terms. Let x power n be one of such sequences. The cosmogorov complexity of xn is defined as follows.

Now the thing is that this equation states that the complexity of xn is the length absolute value p of p, basically the program, in bits of the shortest program that when run on general purpose computer outputs x to the power n, and then it halts. Generalizing this, we can consider a generic object x described by a string generated by program p on u, again, universal Turing machine.

There are small variants of formulation, but in all of them, the complexity captured by this definition is a descriptive complexity quantifying the effort to be done for identifying object X from its description. If an object is highly regular, it is easy to describe it precisely, whereas a long description is required when the object is random. In its essence, Kolmogorov complexity KX captures the randomness in X. As all programs on a universal computer V can be translated into programs on the computer

U by Q, one of U's programs, the complexity of KVX, will not exceed the complexity. Actually the second one, sorry, the second one needs to be KUX. I'm mistaken typing the notation. Will not exceed K u x plus the length of Q. This is basically the equation and formula for what I just said. Even though the absolute value Q may be large, it does not depend on x.

Then we can say that the Kolb-Mogorov complexity is almost machine independent. Then we may omit the subscript indicating the universal machine and write simply kx, whatever the machine kx captures, all the regularities in x's description. Now, a good example of the difference between what is exactly a commover of complexity is that to the left, we have a very regular pattern consisting of 48 tiles. To the right, we have irregular pattern, which of course is not random, consisting of 34 tiles. Even though the left pattern has more tiles, it's called more of complexity, is much lower

than that of the right pattern. Now, the main criticism to our Kolomogorov complexity is that it is not computable. Nevertheless, computable approximation of kx, even if basically the Kolomogorov complexity of object X, if not uniformly convergent, can be used, which are sufficient for practical purposes. Another criticism is basically its monotonic increase with randomness. Hence it is unable to capture the internal structure of objects.

That's important. And this is basically one of the things that Crouchfield's main motivation behind Crouchfield's development of a statistical complexity measure was to overcome this, basically, lack in Kolmogorov's measure of complexity. The idea of coming up with a measure of complexity that is, in fact, capable of telling us something about the internal structure of an object. So Kolmogorov complexity is, in some widely believed, is a suitable measure of complexity. Sorry. It looks like my screen sharing and my notes that I have here

are not the same. So OK. Another criticism regards its monotonic increase with randomness. Hence, it is unable to capture the internal and structural objects. It is instead widely believed that a suitable measure of complexity must have low values for both totally regular and totally random patterns. When we want to model a set of data, there is usually some noise affecting them. Kolmogorov complexity captures the regular part of a data, i.e. the model. We have then a two-part encoding, one describing the regularities, namely the model in the data, and the other describing

the random part, namely the noise, as schematically represented in the following figure. The regular part describes the objects that are not represented by the regularities, simply by enumeration inside the set fracture x. The Kolmogorov complexity captures the regular part of data, whereas the irregular part, the noise might be described separately. The regular part inside the box requires Kolmogorov complexity of object X, bits for its description, whereas the sparse objects require logarithm 2 of the object X bits, where X basically

a member of the sets of fracture X. Now, this idea of basically a two-part encoding, the ordered part and the disordered, the unordered part, the model and noise. How is basically a main idea on which Solomanov came up with his own measure of complexity. And ultimately, this is something that we will talk about in the second module.

And it's basically the main idea behind the idea of compression computation. Specifically, what is usually called the duality of compression and regularity, proposed by Solomanov. The idea is that the duality of compression and regularity means that anything that can compress data is a type of regularity. And any regularity can compress the data. So this is something that we will talk about. This is basically called a two-way compression method

proposed by Solomanov. It's a very significant idea in computation. But it has its own shortcoming. Plus, it's something we will talk about. I don't want to get into more details about this right now. But this is something that we will talk about, especially with regard to, for example, inductive inference as a Solomanov two-way compression. And for example, a conceptual material inference as a different kind of compression, something that a Solomanov compression cannot really capture. And we will talk about, for example,

conceptual material inference as a different kind of compression. Usually it's called basically an entry-exit form of compression, two-way compression in the sense that two agents are capable of accessing basically regularities of a given set of data. So I don't want to get into details right now about this, But it's kind of useful to know that Kolmogorov complexity and the idea of this regular part of data and disordered part of data is also basically

kind of a main intuition behind another measure of complexity. And that's Solomanov's measure of complexity. And Solomanov also used this idea in order to propose a principle of computational compression. understood under the principle of duality of compression and regularity. As I said, anything can compress data as a type of regularity, and any regularity can compress the data. Now the two-part encoding of data is a starting point of another measure of complexity called meaningful information.

This is proposed by Paul Vittany, which is the only part encoding the regularities. Vittany claims that this is the only useful part, separated from accidental information. He has introduced a relative notion of complexity, the normalized compression distance, NCD, which evaluates the complexity distance NCD of X and Y, X and Y being two objects. The measured NCD is again derived from Kolmogorov complexity of objects and can be approximated by the following formula.

Now, in this formula, kx, Kolmogorov complexity of object x and respectively the Kolmogorov complexity of object y, is the compression length of a string x and respectively y, while kx and y is the compression length of the concatenation of the strings x and y. Now, similar to Whitney's idea, another measure, the effective complexity, EC, proposes

that complexity should only be related to the part of the description that encodes the regularities of an object. For this reason, effective complexity of an entity at the length of a highly compressed description of its regularities. Now, the notion of complexity is basically in the sense of effective complexity is presented as context dependent and subjective, and that it depends on the description, granularity, and the language, as well as from a clear distinction between regularity and noise, and between important and irrelevant

aspects of data. In this sense, EC is the definition that most closely corresponds to what we mean by complexity in ordinary conversation and in an intuitive sense. Now, so this was Kolmogorov complexity and with the kind of idea of a formal description of a descriptive length. Now we can go into logical depth. So logical depth is a measure of how difficult an object

difficult an object. Actually, the best way to talk about this difficulty is not difficulty in terms of basically difficulty of resources. So I think this sentence that I have written needs to be clarified. I don't mean it in difficulty in the sense of resources. Because in fact, one of the reasons that, one of the shortcomings behind logical depth is that it's a resource blind. I mean, for it's the main resource is not really computational resources, but it's time resources. It's in a kind of a strict sense, technical sense, it's resource blind.

So the difficulty of constructing an object in the benetzological depth is really the difficulty of basically temporal construction, as steps of construction, namely the runtime. In Benet's words, logically deep objects contain internal evidence of having been the result of a long computation or a slew to simulate dynamical processes and could not plausibly have originated otherwise. Now, we talked about the intuitive notion of length before we gave the formal definition of length.

Now we can do the same thing about depth. The intuitive notion of depth is basically has proposed as a technical refinement of the concept of information, particularly the informational content of a system or an object. Information can be greater or lesser depth. Depth can loosely be defined as the number of steps or runtime in an inferential process or chain of cause and effect, linking an object to its probable origin. The logical depth of a statement is therefore the expression of its content. The more difficult it is for the sender to arrive at a statement, the greater its logical depth.

The more working out time, or the slower to simulate, The sender has used more time it has to simulate an object. The sender has used or has spent to simulate this object in its head or on the computer. The greater its value becomes because it has saved the recipient from having to do this work. For example, allowing that a random string S can be replicated only by a program of roughly the same length as S. Instructions, copy S, can nevertheless be carried out efficiently in a few steps.

Similarly, a totally ordered string, such as this one, requires this. requires only few computational steps to execute. Copy 01, n times. Since a simple copy instruction is repeated many times over and again, the strings like these that entail very large or a small algorithmic complexity are shallow in the sense of having meager logical depth, i.e. little ingenuity is needed to execute the steps of the program.

Therefore, strings that possess logical depth must reside somewhere between these extremes, order and disorder. The value of message, as Bennett suggests, appears to reside not in its information, its absolutely unpredictable parts, not in its obvious redundancy, verbatim repetitions, unequal digit frequencies, but rather in what can be called its buried redundancy, that is, parts predictable only with difficulty. Now, the advantage of Benet-Overkul-Mogorov complexity is that it's overcome that kind of polarized

concepts of measure of complexity, polarized identification of complexity that was basically the characteristics of Kolmogorov complexity, the idea of randomness. So Bennett situates complexity between order and disorder, basically between a structure and randomness. Now, he He also, basically his measure of logical depth can also gives us insight to the internal structure of the object in terms of its, not in terms of descriptive length, but in terms

of generative, basically, structure. How much time it has taken for this object to be simulated or generated by this runtime, by this amount of time being spent to compute various instructions, various steps of the program. In developing the concept of logical depth, Charles Bennett starts by distinguishing the the concept of a message from its value and identifies this last with the amount of mathematical

or other work plausibly done by the message originator, which its receiver is saved from having to repeat. In other words, something complex contains internal evidence that a lengthy computation has already been done. Such an object is said to be deep. And that's what we talk about in terms of the movie Contact. Basically, a logically deep object as a message is basically can be said to be an artifact of an evolved species, because it shows that, you know,

it basically is itself an internal evidence of a very long computation, a very sophisticated form of computation. The logical depth is defined as the number of steps that it would take for a properly programmed Turing machine, starting from a blank tape, to construct the desired sequence as its output. Since in general there are different properly programmed Turing machines that could all produce the desired sequence in different amounts of time, Bennett had to specify which Turing machine should be used. We propose that the shortest of these, i.e., the one with the least number of states and rules, should be chosen in accordance with principle of Oukam's razor.

One may consider a book on number theory difficult or deep. The book will list a number of difficult theorems of number theory. However, it has a very low Kolmogorov complexity, since all theorems in that book are derivable from the initial few definitions. Our estimate of the difficulty or depth of the book is based on the fact that it takes a long time to reproduce the book from part of the information in it. The existence of a deep book is itself evidence of some long evolution preceding it. So in Bukhin-Namur theory Kolmogorov measure

treated as not complex, whereas the logical depth treated as a very complex object. An interesting aside, Sure. Bennett, in the piece, Bennett gives some very interesting examples. The first two are, in terms of non-trivial causal history, the human body, the digits of pi. and then he goes on when talking about books and he compares this to Borges story

about Pierre Menard the author of the Quixote where someone sort of tries simply from memory to rewrite Don Quixote I don't know if people are familiar with this but it's a helpful example yes, yes, okay, excellent No, where was this? Was it in the piece that I shared? I don't remember if you, I'll share this if not. It's a logical depth and physical complexity. It's a journal article that Bennett wrote in 1988. Oh, okay, okay. I don't know if it was the first mention of this or an earlier one. Interesting. Bennett is a really interesting person.

I mean, like, you know, it has coming from a very kind of an intriguing background. He was a chemist, basically, and then he was recruited by IBM, and he became a mathematician working for IBM using, you know, statistics and stuff, and then he wrote these papers for IBM, actually. Yeah, that is what this one is. And I'll share it to the draft that Stephen set up. Sure, superb. Excellent. It's also interesting that you work at IBM. IBM has all these AI computers, Deep Blue. It's deep, I guess. I don't know. Yes. It's depth. No, I know. IBM is, I mean, first of all, if you are familiar with IBM,

the company is the idea that they have, They kind of put the whole idea of computer production on the side, and they have been working on these massive projects on AI, but also for military. It's in fact a company that works massively for the military sector. And one of the reasons that, for example, advanced military studies was saturated with complexity sciences was through IBM actually. This whole idea of network-centric warfare, you know, warfare at the age of chaos and these kinds of stuff are coming from this

direction actually. Okay, so that was the intuitive definition of depth. Well, it turns out that it's quite subtle to give a formal definition of depth that satisfies our intuitive notion of it. In order to arrive at a formal definition of depth, some notions are to be introduced. Given a string of n bits, the string is said to be compressible by k bits.

If its minimal program in Kolmogorov's sense is shorter than itself of at least k bits, a simple counting argument shows that at most a fraction to negative k of strings of length No greater than k bits can have this property. This fact justifies calling the strings that are incompressible or nearly so algorithmically random. Because of the ability of universal Turing machines

to simulate one another, the property of algorithmic randomness is approximately machine-independent. Logical depth is the necessary number of steps in the deductive or causal path connecting an object with its plausible origin. Formally, the notion of logical depth of an object is the amount of time required for an algorithm to derive the object from a shorter description. In fact, with some probability, we can derive the object by simply flipping a fair coin. But for long objects, this probability is small.

If the object has a short description, then we can flip that with a higher probability. Ben's proposal tries to express the trade-off between the probability of flipping a short program and the shortest computation time from program to object, to constructing an object. In order to solve some stability problems, Ben's definition considered not only the object's shortest description, but every description of the object and its computation time. It turns out that it's quite subtle to give a formal definition of depth that satisfies our intuitive notion of it. So even acknowledging, now getting back again to this formal definition and this relation

with Kolmogorov complexity, even acknowledging the merits of Kolmogorov's definition of complexity, Bennett invokes as a better notion of an object's complexity the difficulty or the length required for the Turing machine's program to actually generate the object for its encoding. So when I'm saying length in this sense, it shouldn't be confused with the length in the first sense that was basically part of the Kolmogorov complex, the descriptive length. This length is basically the steps. The notion of complexity defined by Bennett is actually at odds with Kolmogorov's ones.

In fact, where the one shall increase, the other may decrease. A relevant question is what program one should consider for generating the object. At the beginning, Bennett considered the shortest program in Kolmogorov's sense. However, he realized that the shortest program by no means is bound to provide the shortest computation time, i.e., the minimum work. After trying more than one definition, Bennett settled for the one described in the following. Let X and W be strings and S a significance parameter.

So X and W be strings or objects and S a significance parameter. A string's depth at significance level S, denoted DSX, is defined as the least time required to compute it by an S incompressible program. At any given significance level, a string will be called S deep if its depth exceeds S, the parameter of significance, and S shallow otherwise. In short, Bennett starts from Kolmogorov complexity, but instead of considering complexity as the length of programs on a Turing machine,

It considers the time these programs take to reproduce the object in question. Then, its definition of complexity, which is the logical depth, is as follows. Let... Sorry for sniffing. Let x be a string, u a universal Turing machine, and s a significance parameter. A string's depth at significance level s, denoted d as x, is defined by the following formula where pi star is the shortest program generating x on a universal Turing machine.

And t pi is the time taken by program pi. As it is possible that the machine u needs some string of data in order to compute the object x, the above definition can be generalized to the following one. Now let x and w be any of two strings, u a universal machine and s significance parameter. A string's depth relative to W at significance level S, denoted dS for X and W, is defined

by following the equation. Basically, all it says is that X depth relative to W is the minimum time required to compute x from w by an s-incompressible program relative to w. Now whereas algorithmic information, called Magorov complexity, is essentially a measure of randomness of dynamics, Bennett has made the interesting suggestion that a better measure of organization should involve the value of a message rather than its information content.

He noted, a typical sequence of coin tosses has higher informational content, but little message value. An ephemeris giving the positions of the moon and planets every day for 100 years has no more information than the equations of motion and initial conditions from which it was calculated. But saves its owner the efforts of recalculating these positions. The value of a message thus appears to reside not in its information, its absolutely unpredictable parts, nor in its obvious redundancy, verbatim repetitions on equal digital frequencies,

but rather in what might be called its buried redundancy, parts predictable only with difficulty, things the receiver could in principle have figured out without being told, but only at considerable cost in money, time, or computation. In other words, the value of a message is the amount of mathematical or other work plausibly done by its originator, which its receiver is saved from having to repeat. So Bennett suggests using the term logical depth to measure this value of a message. He again says, a message most plausible cause is identified with its minimal algorithmic

description, and its logical depth, or namely, plausible content of mathematical work, is roughly speaking, identified with time required to compute the message from this minimum description. Now, there are at least two important aspects of the concept of logical depth. The first is that it involves the importance or value of a message to something. If this message is to have value, the something cannot be part of the system's environment, as defined but rather must make use of this value content, which stochastic and predetermined environments

cannot do. So Bennett's logical depth might be important in hierarchical systems, where dynamic output of one level can be used as an input of another level, or in compound systems formed from common zeroed parts. In this sense, the concept of logical depth is an important attempt to address organizational concerns of complexity, in addition to order measures. A second point to note, however, is that to obtain a measure of value, the logical depth is based on the idea that a message most plausible cause

is identified with minimal algorithmic description. Here we see an essentially non-empirical and non-physical element enters the picture. The minimal algorithmic description has no apparent connection with possible physical dynamics. And that brings the whole idea of complexity in the sense of complexity, measures and complexity in the sense of complex systems. Indeed, a minimal description looks most completely random. For if X star had a significant regularity, it could be used to encode it more concisely. O'Connor's razor was never intended to cut this deep.

Thus, Wild Bennett approves of the fact that the use of universal computer frees the notion of depth from excessive dependence on particular physical processes you can think of, for example, prebiotic chemistry, and allows an object to be called deep only if there is no shortcut path, physical or non-physical, to reconstruct it from a concise description, there would appear to be good reasons to attempt to find another value which is not as excessively independent of physical processes. And that's why we move to basically the statistical complexity measure proposed by Crutchfield and Young. Neither of these algorithmic concepts,

Kolmogorov complexity and vanace logical depth, As it might appear, generally evaluate the complexity of a system directly from its output x. As Bennett has pointed out, neither the algorithmic information nor the logical depth are effectively computable properties. The difficulty is associated with the unsolvability of the halting problem, which makes it impossible to prove that an output is random, although it may be proved to be non-random. In other words, it's generally possible to obtain either the algorithmic complexity of information or logical depth only if we have a model and the initial conditions which yield

the output to be analyzed. Even in this case, the logical depth is referred to as a minimal algorithmic description by making use of additional assumptions rather than the given known model. Part of the difficulty with the computational insights into complexity arises from their dependence on limiting concepts, outputs, for example, outputs of infinite length. Nothing in the empirical sciences depends on this concept. For algorithmic complexity, logical depth and such concepts as computational universality involve aspects which are too refined, basically too kind of too non-messy, too refined for

an understanding of complex physical systems. While there exist suggestions on how to define a complexity from finite sequences, there appear to be no insights yet developed into the value of such finite sequences. Now, it needs to be emphasized that the value or other organizational qualities of a system need not to be intrinsic to that system in its environment, but may well depend on the duration that the system is required to exist before it dies, thrown away, etc. But I think the most important drawback of logical depth is really in practice other than the issue

of in-computability, which also James Ladyman talks about. So, you know, the question of Yeah, the most important drawback of logical depth is its application to what we have so far examined as complex systems. One might try to substitute the computational effort required to simulate the evolution of the physical system or execution time to generate a given number. But doing so brings into play another variable. the efficiency and realism with which a mathematical model

simulates the actual evolution of the physical system. There are many examples of relatively simple physical processes that computers nonetheless have a very hard time simulating, for example, a low dimensional chaotic system. In such cases, question then would be are we measuring the complexity of the physical system, or are we measuring the mathematical use the mathematical model used to describe that system. So a statistical complexity, which was proposed by the

by physicists Crutchfield and John Young, measures the minimum amount of information on the past behavior of a system needed to make optimal predictions of the statistical behavior of the system in the future. Measurements have been taken of the statistical complexity of real-world phenomena, such as the atomic structure of complicated crystals and models of neural activation. In order to examine a statistical complexity and its relation to the complexity of structures and regularities, i.e., structural complexity, and that's something that's basically a very distinct feature of this measure of complexity,

that a statistical complexity implies a structural complexity, the complexity of regularities. In order to do so, to examine this, what is basically a statistical complexity and how it's related to a structural complexity, it would be helpful to look at some of the ideas behind it in computational mechanics. So what is exactly computational mechanics? Computational mechanics provides another way of measuring an object's complexity or regularities. It makes use of the models of formal computation to provide a direct, a structural accounting

of a system's intrinsic information processing capacities. In other words, it lets us how a system stores, transmits, and manipulates information. So, an example. Suppose we have a long sequence of symbols S1, S2, S3 from binary alphabet. Additionally, we assume a stationary probability distribution over the sequence. Now, the thing is that the reason that the binary alphabet and this discretized account of computation

is used in this schema, and basically you can justify it, is that Crutchfield says something in the calculus of emergence, that basically our measurements are essentially discrete. And that's, I think, a good point. When we measure a complex system, measure a behavior of a system, measurement always has a discretized framework. We basically measure at discrete time instances. All our basically measurements have discrete timestamps. And because of this, we are justified actually to use a discretized framework, a binary framework,

in this case. Now, our task is to observe a sequence and then come up with a way of predicting, as best as we can, subsequent values of the sequence. So we want to see this string and then design a machine or an intrinsic model, and basically an automaton, that is capable of doing pattern discovery simply, arrive at subsequent values of the sequence, patterns that can emerge out of this string, out of this regularity.

The sequence might have non-zero entropy rate, so perfect prediction might be impossible. We will begin by focusing at some length on the following string. Now, let's look at this string. you look at it very carefully, you notice that every other symbol is 1. The other symbols are 0 or 1 with equal probability. We cannot notice that every other symbol is one, the other symbols are zero or one with equal probability.

Now we have found a regularity, a pattern which is stochastic. Now the question would be how can we represent this regularity mathematically, and can we have a program, a computer, an automaton, a machine that does the pattern discovery business for us? The machine that can reproduce this sequence is basically this two-state machine. From a state A, one sees a 1 with probability 1. And from a state B, one sees a 1 with probability 1.

and a zero with probability again of the same value. This is basically a stochastic generalization of a finite state machine. Note that it is still deterministic in the sense that the output symbol, 0 or 1, determines the next state, A or B. Now the question is that why are only two estates necessary, and what exactly do we mean by state here? This is basically kind of a Markov model of the states

of that machine. First of all, there are many particular observed sequences which give one equivalent information about the future sequences. For example, if you see 1, 0, 1, 0, or 1, 1, 0, or simply 0, in all cases, you know with certainty that 1 is next. The idea is that it only makes sense to distinguish between historical sequences that give rise to different predictive information. But there will usually be many sequences that give the same predictive information. Now, this is very important.

We can group these sequences together into a state. These states are known as causal states. Now, I mean, initially, basically, our machine pattern discovery machine looks like this. I mean, the problem is like this. You know, we have many sequences that can provide the same predictive information. So what we need to do is that we need to group these sequences together into states, groups that we call causal states. Now, this is, as you see in the diagram, this is the space of all possible paths in our

string. What we need to remember in order to predict basically all these numbers. Now, because of this, you know, kind of that we have too many basically past the states and they're completely unorganized, you know, you do not have any kind of a grouping here And any of them can actually provide us with predictive information. This machine, how it models the behavior of the system, how to predict the future states, is extremely costly in terms of its memory, in terms of operations, so on and so forth.

When we actually group this into this, on one side A and on the other side B, as we saw in our basically A-B state machine diagram, when we group it like that, these past states into group A and B, here causal states partition the space of all past sequences. Now, all we need to remember are causal states A and B. So we have basically fulfilled the criteria of difficulty of operations and too much to remember, the cost of memory.

So this grouping is cost-reductive, but also, as we will talk about, is actually, because of this resource sensitivity, because of this cost reduction, it allows us to effectively predict future states. And this also ties back to the idea that I said that the aesthetic measures of complexity usually fall under either the arithmetic information content or temporal measures or resource measures. And that's the whole idea of resource or cost measures. But back to our string. How much of the left half of that string,

this is, let's look at it again. How much of the left half of a string is needed to predict the right half? And these are denoted by arrow to the left, arrow to the right. We only need to distinguish between the left half that gives rise to different states of knowledge about the right half.

Two left halves that give rise to the same state of knowledge are equivalent under approximation. So equivalent classes by approximation are causal states. Now equivalent classes of past states that can group together are called causal states. There are minimal sets of aggregate variables necessary for optimal prediction of the right half of the string.

So again, what are causal states? Causal states are equivalent classes of past states that can be grouped together. sets of agrival variables necessary for optimal prediction of future states of the system. Here the right half of the string. For example, when probability of the right half, starting with this state, with the state just zero.

Sorry. Something is not working here. Well, I just want to, if you can see my pointer, I want to just do this. This. This zero. You see it's in the left half. For example, probability on the right half. probability of right half and zero equals to probability of right half and 1011 state.

So these are, we can say that zero and 1011 are equivalent according to this causal grouping, this grouping of schema, to these equivalence classes. This means that the probability of over the future, basically the right half of the string is the same if you have seen zero or one, zero, one, one. The causal states, together with the probability of transitions, between causal states are an E-machine.

The causal states, together with the probability of transition between causal states, are an epsilon machine. A minimal model capable of statistically reproducing the original configuration. So basically, Epsilon machine is another version of Ocon's razor. It's basically this machine, this minimal model that is capable of inferring the future A state of the system, a future, basically, the right half of our regularity from the

left half of the regularity, future state of the system, by inferring from the causal state, namely the equivalent classes of past states of the system. Epsilon machine tells us how the system computes. Basically the symbol epsilon reminds us that the measurement symbols upon which the machine is formed may be distorted via noise or discretization processes. Now, a statistical complexity can be roughly characterized in the following way. First of

all, the statistical complexity is defined as the Shannon entropy of the asymptotic distribution of the causal state. It's basically, you can see its similarity with the Chalm-Gorch-Channon formula for entropy, the entropy-quo-informational content of the system that we talked about in the last session. The second characteristic of the statistical complexity is that to perform optimal prediction of the system, one needs only to remember the causal states. Rather than all of these past states, all we needed was equivalent classes. Basically, how we partition these

states into causal states, equivalent classes. These equivalent classes and these equivalent The third characteristic of a statistical complexity, insofar as to perform optimal prediction of the system, one can only, you know, one only needs to remember the causal estates. The third characteristic then would be the statistical complexity measures the minimal amount of memory needed to perform optimal prediction.

The fourth characteristic is that the statistical complexity is a measure of the pattern or structure or regularity present in the system. And basically, our regularities, our structures, and our structure-inferring, basically, mechanisms could be inferred by this simple machine, by a two-state, finite-state machine. They're not just equivalent classes of A and equivalent as classes of B, causal state A and B. Now, very briefly, when I talk, I mentioned

about these epsilon machines. Epsilon machines, which basically are these automatons, structure-inferring automatons. They are minimum models, minimum models in the sense that their size is minimum. But also, they start as a very minimum, basically, computational machine. If you remember, Logical Depth, basically Bennett's U machine, Universal Turing machine, is a general purpose computer. It's a very powerful computer. In terms of its computability, its computational capacities,

it lies on top of basically Chomsky hierarchy. You have at the bottom, you have regular languages. Then you have context-free. Then you have context-sensitive. then you have, you know, basically recursively innumerable languages which are computable by universal theory machines. So universal theory machines are powerful. They are not minimum. And basically because of that, as we talked about in the first session, if you remember, We said that if we go up this hierarchy, the cost of computation, the cost of modeling,

the cost of basically constructing an internal model for the observer in order to infer about complexity goes up as you move up this hierarchical ladder from regular languages to recursively innumerable languages computable by a universal Turing machine. So logical depth and Bennett's idea that in definition of logical depth, we have a universal Turing machine. He starts his modeling with a very powerful, very costly, basically computational framework. Whereas Crunchfield starts from a minimum machine.

Hence, his emphasis is on the cost. Epsilon machines are these minimum models, rather than these powerful models, universal theory machines. Now, so, epsilon machines can also be characterized by a number of, basically, features. The causal states are sufficient statistic. I don't get too much detail into what sufficient statistic means.

But sufficient statistic is actually basically explained in terms of local and global length scale. I talked about it very briefly in terms of the concept of length scales as the order of magnitude. And basically, sufficient statistic is something that is usually a term that is brought in when people talk about local and global order of magnitudes, moving from one state, from one equivalence class, one causal state to another, one basically level of a structure

to another level of structure. That is to say, so the causal states are sufficient as statistics. That is to say, all the information about the future is contained in the causal states. The causal states are minimal. Sorry? Sorry? Oops. Can you guys hear me? Sure. Yeah. I kind of like I could hear my own voice. Sorry, I think my mic was on. Oh, don't worry, don't worry. Okay. Okay.

Okay, I messed up here. Okay. So the causal estates are minimal. The causal estates are unique up to trivial relabeling. And the causal estates form a Markov process, as we saw. Basically, that was it. Statistical complexity basically measures the minimum amount of information about the

past behavior of a system that is needed to optimally predict the statistical behavior of the system in the future. This past behavior, as we saw, is basically the way it tries to do it. It tries to partition them, group them into equivalent classes called causal states. Then using the old comes principle of parsimony, comes razor, inferring the future state of the system, the kind of induction necessary to arrive at the right half of the string

or regularity or structure. A statistical complexity is also related to Shannon's entropy, as we just saw, in that A system is thought of a message source, and its behavior is somehow quantified as discrete messages. Predicting the statistical behavior then consists of constructing a model of the system based on observation of the messages the system produces, such that the model's behavior is is statistically indistinguishable from the behavior

of the system itself. That's why it's called intrinsic modeling. For example, a model of the message source of the string ACACACACACAC, think of it as a very short, artificial string of DNA could be very simple. Basically, it can be written with just one instruction, repeat AC. Therefore, it can be said that its statistical complexity is low. However, in contrast to what could be done with entropy or algorithmic informational content, a simple model could also be built of the message source that generates the string

ATC, TG, TCA, as you see, choose as random from ACG or T. The latter is possible because models of a statistical complexity are permitted to include random choices. The quantity value of a statistical complexity is the information content of the simplest such model, an epsilon machine that predict the system's behavior. Thus, like effective complexity, a statistical complexity is low for both highly ordered and random systems. So when it comes to extremes, randomness and order,

a statistical complexity is low. Instead, it is high for the systems that align between, basically, order and disorder, structure and randomness. Now the thing about, you know, Crutchfield's statistical complexity is that you have a a lot of great additions and benefits, even compared to vast logical depth. Now, for example, you get this complete cost-friendly,

resource-sensitive measure of computation, a measure of complexity. you get basically not only a predictive measure of complexity, but a generative, again, like Bennett's logic called it, a generative measure of complexity. Basically, an e-automaton, epsilon automaton, an epsilon machine, the prediction of the right half of the regularity from the left half of the regularity is not only a process of induction from the past causal estates to the future estates, but it's also a model of construction of the future

estates from past estates. So you have both predictive measurement and constructive measurement. Then you also have, like what you call, depth, you know, this oscillation, fluctuation between order and disorder. You have basically, with this measure of complexity, you can talk about the internal structure of an object in terms of basically the inferential link between its causal states, past states and future states. But one of the, you know, main issues and still, you know, this basically plagues all of the measures of complexity, is that

it's hard to apply to physical systems. Although, basically, Crutchfield and Young have applied this, as I said, to complex crystals and neural activation patterns and actually have managed to calculate the degree of their statistical complexity. So like the other measures, logical depth, Kolmogorov complexity, effective complexity, term of dynamic depth, so on and so forth, it is typically not easy to measure a statistical complexity if the system in question does not have a ready interpretation

as a message source. However, Crutchfield, Young, and their colleagues have actually, as I just said, measured the statistical complexity of a number of real world phenomena, such as the atomic structure of complicated crystals and the firing patterns of neurons. Nevertheless, despite this idea that all measures of complexity are plagued with application, and in some cases this complication between basically complication and complexity, as

we saw with Kolmogorov idea of complexity. Or for example, they have the resource unfriendliness aspect, as in the case of logical depth or in computability, which plagued both Kolmogorov and logical depth. In comparison to these, I think the statistical complexity can be said to satisfy the criteria required to characterize a good measure of complexity. Basically, and what is this basic, this hydratum, is that a good measure of complexity should not be maximal for either random or for highly ordered

systems. Namely, it needs to remain. It should reside between order and randomness. Also, the issue, as I said, also the issue of using a minimal machine, an epsilon machine, rather than starting from a very powerful class of machine, universal Turing machines, is basically satisfy another aspect of basically a good measure of complexity. And that's resource sensitivity, the cost of computation, the cost of measuring, the cost of modeling, the cost of observation.

So now, before just finishing the session, there is this, you know, one paragraph in James Ladyman was a complex system that I would like to read. And he says, the hierarchical architecture

of complexity linked to the statistical complexity as a measure of complexity. A data set is complex if it is the composite of many symmetries. A hierarchical structure possesses exactly the architecture which can generate many symmetries. Such symmetries are, for example, the low and high frequencies, which are associated with the upper and lower levels of the hierarchy. Thus, the definition of a complex system as one which has highest statistical complexity does overlap, if not coincides, with the definition of a complex hierarchical system. It is an open question whether any non-hierarchical,

non-engineered structure generates data with highest statistical complexity. So again, two of the great powers of Bennett's logical depth, even though Bennett's logical depth is incomputable and not resource friendly, is that logical depth and statistical complexity can be thought very coherently within our basically the picture of the complex system as hierarchical systems that we talked about. Whether by hierarchy we mean modular hierarchies

or we mean nested hierarchies. Because we see that in both of them, we have a different sense of depth, really. One is a temporal depth, logical depth, vanicological depth. And the other one is basically a predictive depth, an inductive depth, depth of inferential links, predictive inferential links between causal states, how regularities generate one another, how basically something new emerges from a

set of equivalent classes or causal states, past causal states. So this was just kind of a very brief introduction to three measures of complexity, standing for three main features of how measures of complexity are usually defined. Space, time, and resource. We get back to this whole idea of space, time, and resource in terms of computational complexity in the second module. For now, we talked about them in terms of length, temporal steps, run time, and basically

the cost of transition, the cost of observation, the cost of modeling. And we talk about these in terms of, you know, they are examples related to these measures and their defining characteristics was Kolmogorov complexity, logical depth, and Crutchfield statistical complexity. Now that we have this kind of very basic introduction in terms of measures of complexity, next session we can talk about a little bit about some examples that kind of put minimally some of the stuff that we have been talking in the first modules into perspective, both in terms of measurement of complexity and also in terms of the stuff we talk about, the epistemological

ontic problems of defining complexity, dealing with complex systems, modeling complex systems, on and so forth. So the next session, the first half, we talked about some examples. And then we moved to basically the topic of our second module, which is computational complexity, which we then tried to build from the stuff that today we talked about, computational models, computational measures of complexity toward kind of a fine graining of theories of computability and paradigms of computation,

emphasizing on Church-Turing thesis on one side, and basically Abramsky-Wegner idea of interactive computation on the other. So questions, talks, discussions, and it's still something that I haven't done, and I promise, is talking about time and how it's fundamentally related to the question of complexity. But that's something we can talk about another time. Questions, talks, comments? It seems like a connection to me between,

So you have Bennett's definition of complexity, logical depth here. But it seems in terms of computational work to me. But it's kind of weirdly insensitive to the idea of state, which was sort of why you were bringing in Crutchfield as giving a much stronger sense of state and history. Yes. and and this to me is very related to you yet your paralyze the ball calculations in on in computing basically because it's arm you're able to separate out the program into lots of little parallel programs the more you're more able to do that

on as if you can sort of feed each of those programs a minimal amount of data, right? Whereas you're doing... Yes, that's the scheduling problem. Right, so you don't have a big scheduling problem if it's massively paralyzable, right? So like if you're sort of exploring a large search space in a simple way, like a ray tracing problem, or data problems are of this nature. But if you have state, the more stateful it is, the state is an input to each new calculation of state. Yes, absolutely. And so the Bennett measure seems quite insensitive to this, but the Crutchfield measure will show it up, basically.

And it's very connected to paralyzable, which I think can be captured for one. I'd have to dig up some references. Yes. I mean, the thing is that this is something that we talk about in the second module in terms of resource sensitivity, both in terms of computation, but also resource sensitivity becomes kind of a key topic for us in developing a robust computational account of language, namely semantic complexity. Basically every phenomenon that is complex is resource sensitive. Resource sensitivity has actually a logical formalism captured by Girard for the first

time in linear logic. One of the things is that once you have a resource sensitivity, then you need to establish rules between your states, between how things interact with one another. And these rules are resource sensitive. You can't smuggle basically premises or propositions or formulas, you know, in your, for example, logical structure, which is, you know, the kind of characteristic of classical logic as being resource unfriendly. Or in terms of computational problems that you talk about, this is usually accounted not

as strictly in terms of parallel systems, but concurrent processes, theories of concurrency. Because concurrency is all about basically avoiding resource starvation. We can think of this about, for example, how ATMs work. to have your account, your ATM's programmed according to this concurrent logic in the sense that if you draw money from an ATM and bring it basically to zero, your account to zero amount, another person who has access to your account goes to that ATM and at the

same time basically draws that amount of money, it goes to negative. So ATMs want to have a kind of a concurrent scheduling that does not allow for this kind of resources salvation. But also there are other issues like deadlocking, so on and so forth. Yes, definitely I think concurrency, which is you have this account of estates and groupings is basically extremely resource sensitive. And we will talk about it. In fact, I would say that parallel systems, technically understood in terms of curing, basically parallel systems, are not also completely resource sensitive.

Only concurrent processing, concurrent computation, different from parallel computation, is concurrent resource sensitive in a very genuine sense. And we will talk about this in terms of the logical, exactly the fine grained details of how concurrent systems are different from parallel computation. And that's why basically I brought about it and you talked about it. The concurrent processing is something that actually computer scientists noticed. First of all, concurrent processing was proposed first by in chemistry in terms of how different you know, chemical processes uses, for example, resources of molecules and atoms and, you

know, chemical resources. The thing is that theoretical computer science, for the most part, until like 70s, was, in fact, did not have a formalism for resource sensitivity. It wasn't even, in fact, resource sensitive. Resource sensitivity arose in computer science because of the implementation of actual physical computer. The physical computer, like the CPU, RAM, all of these things, the scheduling of how these things are interacting is itself needs a formalism, a robust formalism for resource sensitivity. A good example of this is the allegory or the thought

experiment of dining philosophy problems, which is basically based on Adam Petri's concurrent processing. And we will talk about Petri Nets a little bit. Cool. And just so you can get this intuitive idea of why I'm saying that language is really source sensitive at the level of semantics, think of this as this brandomian game of giving and asking for reason, namely game of assertions. Assertions, propositions, concepts have propositional

content. Propositions are resource sensitive. In a dialogue, we are bound to resource sensitivity of our assertions. We cannot basically extend a proposition and copy them at whim. It's basically this whole idea of in sequence calculus, you have formulas in a sequence as premises, as set of propositions. And on the other side of the sequence, you have hypotheses. Now, in classical logic, which is a resource unfriendly, basically you can simply copy all of your formulas of your premises, and the hypotheses

won't be changed in classical logic. You can have different replicas of A as premise, and your hypothesis still will be the same. Or your hypothesis can be replicated, and it does not change the structure of your, basically, chain of deduction. Whereas in linear logic, anything that requires interaction, whether this is a computational interaction between systems, or whether it is interaction between, in terms of logical connectives, or whether it is interaction between linguistic agents, is a resource-sensitive, basically, interaction. It requires resource sensitivity.

And in fact, Girard's linear logic was devised precisely to address and reinvent constructive a schema of intuitionism and rectify the resource unfriendliness of classical logic by way of basically turning the monoidal structure of premises and hypotheses in basically classical logic into dualities. This is the whole idea behind this interaction and resource sensitivity is a concept of duality, duality as in mathematics and computation. You have basically, and the basic intuition behind duality is simply interchange of roles

between interacting states, interacting systems, or interacting basically linguistic agents. Me and you going back and forth when we're having a dialogue, taking turns in terms of asserting and questioning and counter asserting. Because there's a shared state there, right? So I'm dependent on the input of whatever your response is. JOHN MUELLER- Yes, absolutely. It's like a game of chess. My response. JOHN MUELLER- Yes, and that has- You can't just simulate both sides of the conversation in parallel. It's specific to the interaction between the agents.

JOHN MUELLER- Yes, and this has massive ramification. Because you see, it's like in the game of chess. Once this idea of pairing, two agents being constrained to this interactive framework, to this idea of dualities, once they become constrained as such, then moving inside the game of chess, every piece, every move takes a functional role within our game, within our interactive game. A pawn becomes simply constrained to the rule of our interaction in the game, namely, or basically, you use, for example, you, for example, I move a pawn, you counterpose it with, for example, a castle or another pawn.

So this counterposing, this idea that you interact with me and I'm constrained with your move, and you are constrained with my move, basically manifests or generates a function, a symbolic function, an abstract function to basically our pieces within the chess game. This idea that concepts have functional role within a system of dialogical inference basically has something to do with this computational resource sensitivity. That resource sensitivity is something that is at the deepest level computationally responsible

for making a symbolic or abstract functional rule within a dialogical interaction, a system of linguistic interaction possible in the first place. We will talk about this in terms of the notion of computational games and logical games, that how basically resource sensitive account of logical connectives allows us to develop basically a functional role semantics, an inferential role semantics. It's basically a good example of this. As you say, I move this.

I say x, y, y, y inside a syntactic formal language, which does not need interaction. Then if you can recognize my syntactic language, you counterpose it with, for example, xy, xy, another syntactic string. Now, you have taken my syntactic vocabulary as the input for your syntactic vocabulary. And I take your syntactic vocabulary as the output of my syntactic vocabulary. Namely, these two syntactic utterances stand with a functional relation to one another. This is the germ of a concept formation, namely functional rule of syntactic utterances that

are dead, that are meaningless, absolutely, stand in functional relations to one another. And this is basically the core of semantics, really. is standing in rule-based relations to one another. Syntactic, meaning sign designs, becomes semantically complex because they stand in increasingly complex relations to one another. Cool. Questions, answers, comments, critique?

I'm still wondering about the axiomatic system question in relation to this. Yes, I know this. I know it's a good answer. I'm still giving you time for one more week. I can do one thing. think I can chart out the basically different meanings of axioms and axiomatic systems. Then you can pick one of these axiomatic systems and then think about whether it is a logically deep object or not, and whether it is complex in terms of, you know, basically length or not, whether it is complex in terms of depth or not.

So we haven't really argued and talked about it. I'm still waiting for your input and discussions. But I will also put that thing on the classroom page so it kind of makes it a little bit easier to talk about. I mean, kind of talking about different axiomatic systems and what we mean by axioms. Because there are at least, I would say, six forms of how we use the word axiom, and respectively six major axiomatic systems. Very briefly, the Euclidean, which is the intuitive idea of axiomatics.

axioms are understood intuitively with regard to intuition, and exactly like principle of Euclidean geometry lines are intuitive objects. Then we have non-informal Hilbertian axiomatics. Then we have informal Hilbertian axiomatics. We have informal set-theoretical axiomatics. We have formal set-theoretic axiomatics. We have Carnapian predicate axiomatics. And then we have what is the most sophisticated one is basically due to Girard, Abramski, and Blas. The axiomatics as interact as basically acts.

Axioms are not objects, but are acts. And they interact. And they have no resource sensitivity. They are not bound by some pre-established rules. but they generate rules as they interact. And that's basically the programmatic understanding of axiomatics. What is that? Does that have a name, the Gerard and Bramsky? And who else? Who's the other person? Andreas Blass. Andreas Blass. They usually call it the geometry of interaction or interactive axiomatics.

. Questions? No one? Everyone silent? At some point, I should ask you questions, if you don't ask me questions. Yeah, unfortunately, I need to be going.

But yeah, I think laying out sort of basic what could be meant by axioms would be helpful. I kind of found sort of going through and just sort of looking for some kind of philosophical system or mathematics of propositions to be sort of complicated. Yes. Yes. You see, I think, and this is something that Lawrence Pantel talks about this, which is not really his, but he builds on this. The first time was proposed by Wolfgang Segmuller. He is an absolutely phenomenal—he is actually one of the best philosophers of science ever

existed in 20th century. His books haven't been translated, although there is a Wolfgang Stegmuller. Here. Do you have any friends? Wolfgang Stegmuller. I think his book called History of Epistemology, and Sciences, you know, his papers on epistemology, the history of science and axiomatics is published by Springer in two volumes. It's a really great book, but some of his main

books haven't been translated. Nevertheless, the thing is that Stegmuller and also Lawrence Ponto, I definitely suggest those of you who are into philosophy read his Structure and being. It's a monumental work. They talk about that really once you get into the nitty-gritty of stuff, there is not much distinction between philosophical language and mathematical language. The reason is that they are extremely theoretical. They are extremely, are basically removed are basically removed from ordinary natural language. They are theoretical languages.

And theoretical languages are usually both axiomatic and coherentist. Basically, they are networked. Yes, but nevertheless, there are fine-graining of the distinction between mathematical axiomatics and, for example, philosophical axiomatics, for example, especially in the case of, for example, formal Hilbertian and formal satirical. Could I ask, what do you mean, what's the difference between formal and informal? Yes, when we say formal, meaning that basically,

Okay, let's start with the informal. For example, informal Hilbertian axiomatics is different from Euclidean axiomatics in the sense that axiomatics of a Hilbertian system are abstract, are abstractions. They are not intuitions, whereas Euclidean axioms, axiom of a line, for example, and the relations between them, point, plane, there are intuitions and the axiomatic relations in the Euclidean systems are intuitive relations between intuition of a line, intuition of a point, and intuition of a plane. Whereas in the informal Hilbertian axiomatics, we deal with abstractions, no longer with

intuitions, abstractions. Now, abstraction is basically an informal Hilbertian or when we are talking about informality. There are abstractions but are not intuitions. So now the difference between informal and formal is that it is not only an abstraction but it is the form of abstraction. Namely, it can be, you can simply basically mechanize it, mechanize this form, mechanize this formalism. And this is basically the whole point of these formal languages don't require the constant

intervention of basically semantics. They are extremely, basically, dependent on the syntax of the formal language, of how you can automate and how you can mechanize the formalism behind this. So the thing is that this is something that Katerina Dolit-Novais brings out, that the The difference between informal, basically, axiomatics and formal axiomatics is this idea that in the formal axiomatics, namely formalism, they are de-semanified.

Abstraction is de-semanified. Abstraction is purely formal. And what is the form of it? is the form of its structure, the form of its syntax. Where you do not need to put on the rules. Sorry, Reza? Yes. What's the benefit of that, of de-semanifying something? Well, de-semanifying something means that it becomes, the language becomes extremely effective but also becomes extremely manipulable. Because of this manipulability, then you can re-semanify, then you can apply it in different contexts. So when you usually have this semantification, you have also a re-semantification, re-application. Basically you can manipulate and reapply, for example, the constructions that you have

made in formal language to a different field. Whereas when you have semantics, or basically this kind of intervention of semantic complexity, then your formalisms are very context dependent. And because of this context dependency, you can't really apply them. And that's basically one of the main reasons why the formal language of modern sciences can be applied to different fields. Basically, their theoretical core, their formal core, precisely because of this, you know, weak semantics. Because of the pure formalism.

Risa, could you say that also that's why scientific, like these de-semancified systems can't be applied to, because they haven't been fully de-semancified? So that's sometimes where the hang-up occurs, that something still retains a semantic context. So when you try to apply, for example, like I'm thinking in terms of mathematics and music, when you try to apply mathematics to actual sound, then there's one of the reasons why it doesn't transfer directly is because of it still retaining the mathematical concepts or axioms retaining some semantic information.

It doesn't. Yes. Well, it's, you know, when I say, you know, the semantified, I do not mean that they do not have basically semantics at all. I mean it in the sense that they do not have the semantics of the natural language. What is the semantic of natural language? Semantics of the natural language is characterized by one really main characteristics, And that's the feasibility. The dialogues responsible for the generation of semantic complexity in natural language are the feasible. Whereas this feasibility is being reduced, this feasibility of natural language and semantic

of the natural language being reduced in mathematical system. But nevertheless, yes, it has semantics. And it has basically every language needs to have the symbolic structure, the syntactic structure, and the semantic structure, and the pragmatic structure. And so yes, mathematics have semantic basically a structure of their own, and the semantic content of their own. But this semantic is, as you say, is strictly speaking the semantics of the purely formal language. Basically, it's a semantics that is tethered to the syntactic manipulation

and the syntactic capacities of that formal language. So when you talk about the benefits of abstraction, not kind of sort of unlinking or redefining a term as you move along, I'm kind of reminded of what, also what I kind of wanted, or what I was looking for sort of an example of axiomatic systems, and I'm going off, I previously, just before this, took Peter Wolfendale and Ben Woodard's class, so all my examples are coming from German idealism here. I was looking at Fichte's, there are various versions of

his Wissensches there that proceed sort of proposition by proposition or Hegel's logic also where he kind of starts with vaguely undefined terms and proceeds to try to define them and sort of dialectically build them up as he goes. Is that sort of what you're talking about, sort of the benefits of abstraction in that kind of sense? Or are those things kind of at all applicable here? Well, I mean, there are so many relations from abstraction to axiomatics, but I can't really single out one. I mean, these are all very kind of different from one another. For example, Hegel's dialectics, axiomatics is really something that I think that can

be talked about in terms of interactive axiomatics. as precisely premises as dialectics does not have pre-established rules. The only rule is in the interaction. And through the interaction, the rules are being built for axioms. And new axioms are being constructed. New premises are being constructed. And this is very different from, for example, the classic Tarskian axiom, the Urbian axiom, Carnapian axiom. I would I would say that Hegel is extremely sophisticated in this sense. And this is something that Jouard talks about when he talks about linear logic. He compares very briefly linear logic and the geometry of interaction and interactive axiomaticism.

Hegelian principle of dialectics. Yeah, I guess I want to share this book review because maybe you'll look at it and I'll link to it all. but sort of in looking around this, I found some interesting articles in this book that I actually had time to get into, but comparing sort of Fichte's, Isnchatz, Lera, and geometry, and sort of his mathematical theories to, like, basic geometric constructions, and how he compares sort of elementary transcendental philosophy to mathematics. Uh-huh. And a lot of overlap. I think generally... It seems like an interesting book.

GERHARD KALVANIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERIENIERI They are different, and there's this massive asymmetry in what is exactly mathematical axiomatic is, modern mathematical axiomatics, from the kind of, you know, Euclidean geometrical axiomatics

that is basically favored by German ideals, and because of that, you know, this whole idea of lived experience, the intuition of point, line, so on and so forth. Whereas the kind of the brilliance and kind of the brilliant formalism of modern mathematics is, I don't think so many people trying to bring this back to intuitions and stuff, but I think it's not possible anymore. We can talk about these in terms of, for example, category theory and, you know, Longo's critique of, you know, Hylbertian mathematics, so on and so forth. But at this point, mathematics

It's very different from that kind of picture of axiomatics that you see in the Euclidean system. Nevertheless, it has constructivity. I mean, all axiomatics are constructive objects, or axioms, constructive objects. The thing is that now how construction can be done, that's a different issue. Construction, I think, and one of the great things about modern mathematics is that it moves more and more toward this interactive idea of, basically, axiomatics. And axioms don't have pre-established rules, but the rules emerge as the axioms interact with one another. Things are constructed to interaction

of the axioms with one another, rather than just a set of axioms and then deductively in the classical axiomatic system trying to basically construct the stuff from them. Another thing with axiomatics is that so many people when talk about axiomatics think of that axiomatics essentially means a completeness. Well, some axiomatic universes mean completeness, but some don't. Interactive, for example, idea of axiomatics doesn't imply any completeness, so it's not basically susceptible to the charge of, for example, that kind of formal Gilbertian axiomatics where basically the universe that you have constructed, the objects, the universe of

objects that you have constructed from these axioms is a complete universe, and hence is susceptible to the charge of a Godelian or a Cantorian incompleteness argument. All right, great. Yeah, then I think laying that out and allowing us to kind of like read that over and then sort of try to come up with some writing or some thoughts on it will be really helpful. I will definitely write it on the classroom page in more detail what these things are. Another book that is very, I mean if you want this for your own research, which is great

in terms of, you know, development of axioms in philosophical languages or development of axiomatic systems is Jules Bouillemann's book. I think there is only one translation of it in English, and that's exactly the philosophical systems, which he talks about axiomatic systems. Oh, yeah. I do, but thank you, everyone. OK. Bye bye. I wanted to thank the class for the Hegel and Haiti recommendation. I really enjoyed that. That was helpful. Yes, I plan to read it. I definitely want to read it.

I just skimmed over some parts of it. It looks good. Nice. There is another Hegel book is coming out, or if it hasn't come out, and the person who has written it, I think she has a very good grasp of Hegel, Rocio Zambrana. It's about Hegel and the idea of intelligibility. It just, I think, was supposed to come like last month. I'm not sure if it's out yet or not. It's very good. I have to say part of it. What is the classical idea of the intelligibles? I'm trying to figure that out this week too.

Intelligibles are objects of understanding, very briefly. Hi, Jessica. You're echoing just a little bit. You're echoing just a little bit. So if you mute it after you ask your question, it will be easier. Yes, so intelligibilities are objects of understanding. And the intelligibilities can be, roughly speaking, can be compartmentalized to two categories, intelligibility in the order of appearances and intelligibility in order of reality, in order of being, basically. And this is basically one of the main themes behind Solarzian

idea of the manifest image and the scientific image, intelligibilities as belonging to the manifest order, the order of appearances, and intelligibilities as belonging, and then objects of understanding as belonging to the order of reality, which are mind independent. Another thing is that how you can commensurate these two order of intelligibilities. And that's basically the project of the synopsis. But intelligibilities, I mean, I will try to talk about them, okay, if this is something that... And yes, there's also a good connection between this idea of intelligence as being a theoretical,

practical subject that deals primarily with the order of intelligibility. Not only the intelligibility's core objects of understanding and the order of appearance and the order of reality, but also practical intelligibility's. So, and this is what is ultimately, as I said, The aim of the Hegelian and Kantian project of construction of a rational, a perceptive intelligence is the idea of a theoretically enabled and a practically enabled intelligence,

namely an intelligence that is capable of gaining traction upon theoretical intelligibilities, of understanding, in the order of appearances and the order of reality, but also capable of embarking upon practical intelligibilities. And what are practical intelligibilities? Practical intelligibilities are intelligibilities concerning what to do rather than what to think or what to understand. And what to do is basically the whole pre-neal questions of philosophy is what to think and what to do. So what to do is the order of practical reasoning, whereas the theoretical reasoning, which deals

with theoretical intelligibility, is basically the question of what to think, the order of practical reasoning. Go on, Jessica. It also sounds kind of like Parmenides, like to divide it into order of being and order of appearance. It's order of appearance and order of reality, really. It's simply being understood as the manifest, you know, simply the idea that basically we work through the categorical structure of the mind.

But we should not suppose that if we are working within this framework that the reality in fact has a structure. In fact, this structure has a categorical structure as such. This lack of categorical structure in the order of being, as in contrast to the order of thought, which has in fact a categorical structure, can be said to be the premise for the distinction between appearances and reality. it being simply something whose categorical structure, if it has any, has not been given to the mind as like the imprint of nature upon a block of wax, this block of wax being

our mind. And that's basically what Sellars calls the myth of the given. That's the whole point that reality is mind independent in the sense that it, in the first instance, it lacks and we should not be supposed as if it's imprinting a categorical structure upon discursive and perceptive intelligence, upon our intelligence, upon our cognition. It's the cognition that has categorical structure. And the task of the cognition is really going deep and to determine the structure of reality, with the idea that the structure of reality is mind-independent. The structure of being is basically mind-independent.

It's free of thought. It's the thought that gives categorical structure, not the other way around. OK, should we finish today's class? Tony? Tony? Yes, I'm here. Yes. OK, should we call today and... Tony? Sure. Okay, excellent. Okay, great.