Complexity & Computation (Session 3)

All right, welcome back everybody. This is the third session of the first module. So, yeah, so Reza is back and connected. So, we'll pass it to Reza and we'll begin. Excellent. So, before starting today's session, is there any question, discussion, talk about, about the stuff that we were talking about in the previous two sessions. Anything? I've been reading quite a bit of WIMSAT. Which ones? I went down the path of—well, I read re-engineering philosophy for limited beings, I think that's

this, and then went down the kind of like generative entrenchment path, so it took me out of the book into a couple of his papers where he talks about scaffolds. Yes. And scaffolding culture, and then, you know, kind of evolutionary development takes Yes, I have read that, okay. And I was just curious to know whether this notion of scaffolds might be linked to what you were saying about platforms. Yes, scaffolding his platform, yes. Yeah. Yes. Yes. Yeah, I mean, I think a really great philosopher of science. And it has actually contributed to this kind of revolution in modeling, that piecewise

model. The thing about him, if you have, I think the one that you have read is the one on cultural scaffolding and he talks, he kind of like, basically that paper is kind of like a critique of the linguistic term in cultural development. I mean, when people talk about these kinds of things from kind of a Darwinian position about scaffolding and, you know, entrenchments, it's very easy for them to dismiss basically the centrality of language in culture formation, in fact, as being the de facto platform for

cultural formation. precisely because they see language simply as some sort of ordinary language, or they kind of see it through the lens of traditional philosophy of language, which is more or less as credible or as sophisticated as classical logic. It's quite like very, very, has classical rules, classical sense of what language is, so on and so forth. But the thing is that, and this is something that we will talk about in the third module, that once you see language as this manifold of hierarchies, syntax, semantics, pragmatics,

And whether it is natural language or formal language, I mean, a theoretical language, artificial language, then you see precisely that it's impossible to have actually any kind of social or cultural formation without the cognitive tools provided by these syntactic semantic, pragmatic levels of language. And deep down, language is nothing but really this super vast computational system, which is absolutely essential.

I mean, it's completely unimaginable to think of any complex cognitive capacity without language capacity. Precisely because language capacity is, you know, has, not only it has different levels of, you know, complexity, but also has basically integrates different models of computation. And not all of these models of computation, the paradigms of computation, you know, doing computing are available to the kind of structural, functional picture that, for example, Winsat talks about. Usually all of these people who talk about kind of a Darwinian term to complexity sciences

or kind of a computational equivalent of it, by computation all they mean is simply effective the Turing Church thesis. But the whole point is that language is not just that. Language actually has different models of computation. And only at the level of the syntax, it's basically you can model it via Turing Church thesis. And this is something that, yeah, we will get to this at the end of the second module. And the third module will be this, which brings us back to this that I talked about, that ultimately we want to investigate is basically the deep picture of cognition. We want to prove a computational hermeneutic of sapience cognition as a deep

object, basically, as a logically computationally deep object. So yes, I will talk about this DeepObject a little bit today. Yes. So yeah, so just wanted to make this comment about WIMSAT. And kind of talked about it in the last session, when people talk about computation and stuff, about hierarchies. I mean, there are extremely strong thesis and very, very powerful in terms of thinking the scope of some kind of large formations. But nevertheless, they have themselves

both methodological and metaphysical biases. I mean, bias is not in a positive sense, in a completely negative sense. The one that I read was with James Grissimer, called Reproducing Entrenchments to Scaffold Culture, the Central World of Cultural Revolution. Yes, he has also, I think, a collection of essays. He edited a collection of essays. It's called The Scaffolding Something. It's actually quite a very interesting book. It's really expensive, though.

If you can find it somewhere online, that would be great. OK, I'll have a look. See what I can do. I have a scanned version of that. I just need to upload it. It's probably like 200 megabytes. Oh yeah, the MIT Press one. Yes, that's the one. Okay. So if language... Okay, language is one example here of a platform or a scuffle, but would there be another sort of ready example like that of a platform within a hierarchy of complex systems? I think, well, nervous system, neural relation processes can also be taken as platforms.

And this is a really good book on this, is René Thoms' Structural Stability and Morphogenesis. It's like, at the middle of the book, he starts talks about basically neural relation processes and the emergence of nervous system. And this is actually really interesting that basically, yes, the nervous system is really like this platform, not only for cognition, but also for more organogenesis, basically, of the entire body. Once the nervous system generates, we have these new relation processes, it's basically,

As I said, it reduces the diversity of the organogenesis. Nevertheless, it basically adds to the complexity of the organs. Basically, organs become highly specialized. But their diversity starts to flatten out. And this is, you know, we see this basically in kind of a transition of rudimentary, for for example, nervous system to kind of complex nervous system that is not simply a kind of parochial differential responsiveness to stimuli. But it actually plays the role of anticipation, forecasting, a statistical inference of environmental

data, so on and so forth. Also, another great thing is that Tom makes this speculation somewhere toward the end of the book that basically language starts, basically. RASHAD AL- Reza? What book? Are you talking about? TOM BAKERDALA- It's Structural Stability and Morphogenesis by René Tom. RASHAD AL- Thank you. So, yes, so he talks about this when basically the capacity of the nervous system as a platform starts to diminish in terms of basically scaffolding these complex additions.

That's where exactly language starts. And he makes a really fantastic kind of like a neuroscientific speculation about the origin of language and how basically language needs to be understood as a platform that tops basically the nervous system. Precisely because the nervous system at that point in terms of basically the organogenesis It's a scaffolding function for organogenesis. Basically, its capacity bottoms out. And language is basically the one platform that covers this and kind of increases its capacities that itself cannot

afford. There are other examples. I mean, we can see a scaffolding in a lot of, you know, of phenomena. And... Biologically, I was wondering about the role of something like the cell. You know, once you get the cell wall, I was wondering if that's something... The interesting thing about the nervous system is the way that it connects different functions and then allows differentiation. I'm not sure... And a specialization of functions, yes. I think with the cell, we can talk about this. But I think scaffolding is usually, as you said, it's not a unit.

It's an interaction between some scaffolding processes that basically create a stabilized structure and canalization of basically functions, canalization of different, basically, differential structures. Probably, I mean, I need to think about this, whether cell can be called, I mean, if that's the case, then we can say that a gene also is a scaffold. Right, but it's sort of a different scale, right? So driving it now, like maybe you could Maybe you could treat the cell as a platform,

but only at a sort of cell level of . Yes, yes, that's what I was going to say. To the eternal memory of the cell. Probably. I mean, I think that genes are platforms, but that's the whole point, that the scaffolds needs to be understood within their own hierarchy. We can't extend that. And that's precisely, you know, that's so many people like Dawkins overextend the platform of the gene. I guess the other thing is, sorry, there's the nervous system which is a really interesting example and there's also the skin where that's very much like a shared, I don't know, surface

for all the sort of systems within the body. Like, is that just another organ that's specialized, or is there some...? It's usually considered to be an organ, yes. And it is, and this is one of the things that actually... Are you familiar with Alan Berthoud's Brain Sense of Movement? You should definitely check, everyone should read this book. It's one of the best books, really, on brain and nervous system. Brain. Yes, it talks about this in terms of muscular threshold and also skin as basically, and

this is also some theory that they are specialized by nervous system. way that we see them in human function or in a kind of mammalian complex functions. In fact, the skin, I mean, we can see it. So many of our, you know, basically one of the primary motivations, computational motivations behind evolution of the nervous system was simply a spatial differentiation. Basically, the organism should tell the difference between itself, its food, and the predator. And the thing is that as basically this nervous system emerged, it supplements, basically,

it supplies the organism with more sophisticated means of a spatial differentiation, not only in terms of simple responsiveness, differentiation of stimuli and self-contrastation of organism with the environment, but simply proactively explore a space, kind of in a very Kantian sense of a spatial awareness, the outer sense, what Kant calls outer sense. The thing is that a skin actually plays a huge role in a spatial differentiation. In terms of heat dynamics, for example, if you get something before even your eyes can

detect it, if heat comes to your forehead, you can detect it. These are extremely sensitive. Basically, a skin is extremely sensitive to a spatial coordination. actually has a very strong coupling with a sense of gravity, in fact, like inner ear. Inner ear and the skin is very, basically, adapted to detection of, for example, continuity versus inertia versus discontinuity by way of a sense of gravity, hence creating different, basically, orientational cues for the organism. Thanks.

The other thing that came out of the scaffolds thing for me was, I think, maybe a particular class of scaffolds or something, where at least he talks about bundling or sequestration, I think you deliver inside this bundle a set of materials, but also in the organization. And has to do with modularity as well. Yes. We will talk about this today. Yes. It's different. It's different. Sited. Yes. It's different. Basically, the whole notion of bundling is basically different types of modularity,

vertical, horizontal, spatial modularities, yes. It seems that Estefan has the Berto's book. If he can share, that would be great. Unfortunately, I need to spend time to scan that book. And that book is really, for some reason, is out of publication for a long time. And the price has gone up on the use like $2,000. Which one? Alan Berthaud, a brain sense of movement. Berthaud, he's not a neuroscience philosopher.

He is actually a neuroscientist. Another person, you know, if you want to look into more into these kinds of, you know, biological coupling with complexity sciences and stuff, Stanislav Duhan. He's also a great writer. also a neuroscientist, has written really great books in terms of specialization, functional specialization in nervous system, particularly in relation to basically primates, arithmetic

capacities, basically, like dead reckoning, counting, tying knots, so on and so forth. Okay, so, sorry, I'm a little bit coughing. Okay, so, if you remember, we had this list of basically, you know, main features of complex systems. Now, before moving to the measures of complexity, which

I'm going not to talk about all measures of complexity. I'm going to talk about only three, one of them today at the end, the concept of logical depth. And then I observed the two others, the statistical complexity and basically algorithmic computational complexity for next session. And hopefully we will have time also to discuss some model examples, as I promised. If we don't have, if basically just extends, it's OK. We will just use it in the second module and discuss them.

I don't think that we have any problem with regard to time. So I'm going to talk about some of the main features of complexities that I listed. And I very, very briefly talked about, I think, in the first session. I'm going to share the slides. And also, I have not forgotten that I promised to talk about Boltzmann's conception of time

and how it radically, almost rapidly, can basically overthrow all of this, basically, stuff that we have been arguing, the majority of them. Not only are we basically complexity theories, but also really at the fundamental level ways, methodologies of modern physics, observation, theories of causality, so on and so forth. I will try to either today or next session talk about Boltzmann a little bit. So can you see the screen?

Yes. Yeah. Okay. So now complex systems are systems that constitute basically as I said, basically any of these features. But the whole thing is that basically nonlinear dynamic systems don't essentially manifest

all of these properties at the same time. Within this list, roughly properties on the lower of this list are particularly exclusive to basically a living biological phenomenon. We probably can't really observe them within other kinds of physical complex systems. So what I'm going to talk about, I'm going to list these, I mean talk about these.

After talking about them, I will move to basically to kind of, as I said, to kind of like the The discussion, why is that these basically features are important and via what exactly what kind of measure or characterization are we able to detect these features in systems? And that's basically our point of entry to theories about measures of complexity, starting

with fundamental or Boltzmann informational entropic content of a system and then moving to basically Bennett's logical depth. So the diversity and domain specificity of these properties explains the diversity of the notions of complexity, self-organization, et cetera. And the challenges to understanding that these properties continue to pose reduces basically any hope for a unified account of complexity domain in the near future.

Just as there is no canonical list of complex system properties, many of these terms have no canonical definition either. Nevertheless, I'm going to give a brief survey of these properties. We kind of detailed about nonlinear interactions and non-additivity in the first session, so I'm going to skip this one. For irreversibility, a process that is reversible can also be run backwards while still satisfying the same loss. Classical dynamics is time reversible in this sense.

Every dynamically possible process running forward is equally possible running backwards. But virtually, all processes are dissipated. The energy they degrade in quality and shed as waste during the process cannot be retrieved. And this is one of the basic characteristics of complex systems. Because it is converted to heat and distributed randomly throughout the universe. So that they cannot be run in reverse. They also will not persist unless a supply of sufficiently high-quality energy, typically in requisite material forms, is available to continually renew their dissipative processes.

Therefore, they are inherently open systems. only very small brief changes may be approximately reversible, a condition especially obtaining mere dynamical equilibrium. Many, but of course not all, examples of complex dynamics, but all those concerned with living systems are, you know, can be characterized as irreversible systems. Now, the third one is constraints. We are going to basically, when we are talking about complex systems, this concept of constraint is extremely important.

And we need to be very careful to distinguish different contexts of constraints. This is one of them, and then I have listed more down the line. The first one are holonomic and non-holonomic constraints. Constraints on a dynamical process are those limitations on the relationships among its variables that arise from the imposed physical conditions in which the process takes place. Now, this is the definition of the holonomic. A marble rolling in a bowl is confined to the surface of the bowl, whereas a small spacecraft

has no such constraints. You can think of like a particle stuck to the surface of a sphere in contrast to a particle that falls under, you know, or suspends in gravity without the constraint of the surface of the sphere. Whereas a small spacecraft has no constraints, though both move under local gravitational forces. A system's effective degree of freedom are those provided by its inherent variability, dynamical variables minus those removed through constraints.

A dynamical explanation consists in deriving the behavior in question from a model of dynamical variable interrelations, constraints and initial conditions. As the moral ruling without and with friction respectively shows, constraints may apply to both reversible and irreversible processes and are typically required to characterize the specific processes, for example, in a cell. Currently, we only form general analytic dynamics, the Lagrangian-Hamiltonian dynamics, for systems that do not work on their constraints.

The core of these, in turn, is formed by systems also having holonomic constraints, roughly constraints that are purely a matter of space-time geometry and independent of the system's dynamical states and are energy-conserving. There are various simple extensions of this core theory into the holonomic, non-conserving domain, but they remain special cases. Now, it's precisely these features that complex nonlinear irreversible systems lack. For example, a river running between sand and gravel banks has non-holonomic constraints

and what are these non-holonomic constraints? The river banks, which it alters through doing work on them and thus dissipating, i.e. not conserving its energy. Similarly, a group of closed self-reproducing processes, for example, cell metabolism, must do work under many constraints in order to recreate them, while many dynamical bifurications, for example, a boiling phenomena, are initiated in this way. Does anyone have a question? I'd love to hear a little more about holonomic.

I'm not quite getting that. Okay. Holonomic, as I said, is simply the constraints of spatiotemporal geometry on systems, basically. As I said, for example, the constraints regarding the position, basically a spatiotemporal position, quite parochial way, but also you can think about this in terms of, you know, basically the field of space on systems, regardless of basically their internal dynamical states. For example, a particle, a fly on our skin is holonomically constrained by our skin,

Namely, it's field of positioning on a spatial field represented by our skin. It's basically very simple. It's just a jargon for spatial localization constraints. Is there a way to apply that to language? No. OK. You can, yes. I mean, well, not language, but I mean, Basically, there is, I think, a theory. We can develop a theory and talk about the idea that our spatial awareness, prospective spatial awareness, is objectively under the influence of polynomial constraints.

And this is basically, again, one of the fundamental insights of Kant, his theories of intuition of space and time. But not with regard to language such. Yes, basically the way that in fact we talk about objects has this kind of implicit, basically, is implicitly talks about these holonomic constraints.

We can see this in our spatial propositions. For example, if you look at spatial propositions of any language, we have, for example, propositions like on, under, in. These are basically talks about holonomic constraints. But we don't want to talk about that, because that's also a different kind of, it's not really holonomic constraints, but nevertheless, yes, we can, I think we can, we can talk about these spatial prepositions in terms of spatial constraints. But this has something to do with also with our particularly perspectival consciousness.

Would that include hearing? Would this be like the aspect of sound related to language? Would that be a holonomic constraint, or is that just kind of...? No, no, no, no. Yes, when we talk specifically about physical bodies, basically, body systems. Okay, thanks. There is a great book, I mean, if you want to look into a little bit more into these kinds of stuff in terms of how language basically... Language has... When we look at natural language,

it has an element of a naive physics which has been completely unexplored. Very much you can see this naive physics in the spatial propositions of ordinary language. A great book that really explores deeply this problem is Claude Vandeloy, Spatial Propositions, a case of studying French. It's a very great book. I can type it here. That's his... This is the spelling of his name.

OK. Back to our list. Can you see the screen, the slides? Yes. OK. So the next one is Equilibrium and Stabilities. And qualitatively, some aspect A of a dynamical system is in equilibrium if and only if there is no net force acting on the A aspect of the system.

Its A forces are in balance, and there are thus no net second order rates of change, namely accelerations in A. Across the range of possible A choices, an important distinction is then drawn between static and dynamic equilibrium. That is, between cases where the time invariance concerns state parameters and variables. A is basically just system states. And cases where it concerns process parameters and rate variables, A as system processes. The static equilibria require no energy input or output to persist, for example, a crystal

at rest. Dynamical equilibria typically require an irreversible ordered energy, namely negentropy. This is, I will talk about the concept of negentropy later on. So dynamic equilibria typically require an irreversible ordered energy flow to sustain them. For example, water flow to sustain the wave structure of river rapids together with appropriate waste, degraded or entropic outputs. For example, again, turbulent water. For living systems, there is water, food, and hydrogen-oxygen input flow to sustain

them, and heat and chemicals as waste outputs. For these and other dynamically stable systems, energy material flows act to stabilize the system processes so that for small disturbances that do not affect the underlying flows, these systems will behave as if they have a statistic equilibria in these flow variables. In other respects, however, such as river flows acting on river banks and metabolism producing cellular aging, the system may do work that alters and eventually undermines its dynamic equilibria.

Now, equilibria of any sort are stable, metastable, or unstable. And when we talk about complex systems, we usually talk about metastability, but I want to talk about all of these categorizations of equilibria. An equilibria, in some aspect, A is a stable with respect to some class of disturbances, namely perturbations D. If and only if its response to any disturbance from B, the perturbance,

is to soon return to its original A condition under its own dynamical processes and remain there. And equilibrium is unstable to a class D of disturbances if it does not return near to its original A condition, and it is meta-stable to D if it is stable for some disturbances from D and unstable for others. Now, this is an important one. The closed set of states a system repeatedly traverses when at equilibrium is its attractor.

For example, the marble's rest point is a point attractor. If it's circled frictionlessly around that point, but up to a basin wall, it would be a cyclic attractor. And the wider set of states it can pass through while will returning to its attractor is its attractor basin. Basically the bowl provides a literal attractor basin. And this is a diagram of this is also people who are familiar with GlassBeat Collective. That's basically the picture on their website. And it's basically a picture of this, a diagram of this has been provided by Renath Thon.

A complex dynamics may generate several different equilibria of various stabilities, attractor basins of various shapes, either directly intersecting or connected by transient paths, where small disturbances may change its destination, you know, rolling on a horizontal surface in our bold example, plus other transient paths that do not end. This attractor landscape is a system's dynamical signature, basically expressing its dynamical form. A system that remains within a single attractor landscape is structurally stable, meaning it's autonomous dynamics

in mathematical, basically, language, mathematical theory. And otherwise, it's structurally meta or unstable. While mathematical dynamics typically assume structural stability, many complex systems are structurally unstable, i.e., they bifurricate in mathematical parlance, exhibiting phase changes. That's the whole idea of basically phase change. I think the best way of understanding this, how equilibria and instabilities work in terms

of, you know, stable, metal-stable, unstable, is really looking into a couple of phenomena. For example, fluvial dynamics, as I said, the river flow in banks and landscape formation. The idea that basically rivers under the gravity move towards sea, and as they move, basically according to these attractors and various schemas of equilibria, they basically form

different kinds of landscapes. And this whole landscape, again, constrains the river flow and then creates more differentiation. And you can see this same kind of processes via these equilibria, stabilities, holonomic constraints, basically in organogenesis, differentiation of organs, emergence of different organs in the body of an organism. And the best book, as I said about this, is really Thumb's Structural Stability. The next one is amplification. Transient paths aside, a disturbance to a systematic equilibrium will have one of three

consequences. Leave it within its current attractor basin. it into another attractor basin or transform the attractor landscape, leaving the system in an attractor basin in the new landscape. Now, in the first case, the disturbance will be suppressed, negatively amplified, namely amplified down, as the system returns near to its original state. In the other two cases, the disturbance will be augmented, positively amplified, as the system departs from its original state. Now the thing is that amplification is really the norm of almost all nonlinear systems.

We talked about sensitivity to initial conditions again in detail, so I'm going to ask him over this one. But I think it's good to talk about this also a little bit about sensitivity to initial conditions in relation to this idea of attractor and amplification. So just very Very briefly, as we talked about, nonlinearities in dynamical systems permit small differences

in system state to be amplified into large differences in subsequent system trajectory. Now this was basically the whole idea of sensitivity to initial conditions. For example, a system may be poised near the threshold of changing either a tractor basin within the same landscape or changing landscapes so that a small disturbance is amplified to produce a larger change. Again, you can think of this very intuitively in terms of the river flow and how it basically makes landscapes around itself. And while systems left by disturbances inside the same attractor basin are insensitive to

initial conditions in respect of their ultimate destination, they may still be locally sensitive to the path taken back to the equilibrium attractor. Sensitivity to initial conditions is as common as amplification, but under certain conditions, It takes a special form where a strange attractor is formed in which motion is said to be chaotic because it occurs at random. However, the motion remains deterministic and far from being more disordered than a normal attractor is best viewed as super-ordered since every point within it may manifest sensitivity

to initial conditions. Now we talked about this randomness is confined to trajectory locations sample across the attractor and the like. We talked about this a little bit in terms of, you know, when we were talking about the positive global Lipanov exponent. The next feature of complex systems is finite deterministic unpredictability.

systems manifesting sensitivity to initial conditions presents the problem that the small uncertainties, including errors, and we talked about this, errors can be basically measurement observational errors, in initial conditions may be amplified into large subsequent uncertainties in system location and trajectory. That is, system predictability is limited by knowledge of initial conditions. And that was one of the main characteristics that we attributed to complex systems. How severe a limitation this is in practice, and in what respect, depends basically on the amplification processes involved.

In particular, while prediction that a system's state will remain within a strange attractor is often legitimate, knowledge of location within the attractor can be quickly lost, but not always. Conversely, since systems showing sensitivity to initial conditions can be significantly influenced using only small signals or disturbances, then so long as the relevant state conditions can be distinguished, these conditions can be used to sensitively guide or control them. Symmetry breaking. I remember that I suggested a good book on symmetry breaking as Francois

Abai and Giuseppe Longo's Mathematics in Natural Sciences, Physical Singularity of Life. I think he has also another book, which is really great on symmetry breaking, which I read another person. I have forgotten his name. I think it's Perspectives in Biology is the title of the book. I have forgotten. But you can easily find it. And they are both online. So a symmetry is invariant under an operation. For example, the molecular structure of a cubic crystal is invariant under a spatial shift along any axis.

Classical dynamics is invariant under reversal of time. Symmetry breaking occurs when an existing symmetry is disrupted. If water or another fluid is heated from below in a pan, then its previous complete kinetic symmetry, basically same random motion profile of molecules throughout the pan, is broken vertically as layers near the bottom eat up while those at the top remain cooler, passing the applied heat upward by conduction. This already changes the dynamical form of the system from a stable dynamic equilibrium

maintained by internal molecular collisions, producing no net macro force, to a stabilized dynamic equilibrium maintained by irreversible vertical transmission of heat, or in this case, as a kinetic energy. If the applied heat is increased, there comes a point where rolling boiling, basically this formation, sets in. Conduction is replaced by convection, and the fluid breaks up horizontally and vertically into convection cells, each matched to its neighbor

along its boundary. You can basically, as a basic experiment, you can use this basically like a chemical class, experiment it with different liquids, boiling them, or add a little bit of color and see basically how this whole system reforms and basically changes its shape. This change corresponds to the breakdown of previous horizontal symmetry and is again maintained by increased heat flow. In each symmetry breaking, the orderedness and complexity of the system behavior increased. And basically, this is typical in complex systems.

Symmetry breaking may be spontaneous, brought about by the system's own dynamics, or imposed, as in the heating example that we just talked about. Now spontaneous symmetry breaking transitions are assumed to account for all the emergence of order and complexity in the universe since the supersymmetry of the Big Bang. For people who want to look into this whole idea of symmetry and symmetry breaking more in detail, Herman Weil's book, classic book called Symmetry, is a really great introduction

to the concept of symmetry and also symmetry breaking. The thing is that when we are talking about symmetry and invariances, the actual definition of symmetry is much more subtle than what I just gave. There are very actually subtle, you know, basically subtly different concepts of symmetry detection and or basically concepts of invariance. Not all of these concepts involve basically this spatial shift or basically preservation

of invariance under some spatiotemporal translation. Again, Haramon-Weill provides a kind of a wide survey of these different concepts of invariance and forms of symmetry. Bifurrication. A bifurrication occurs when a structural instability in a system leads to a change in its dynamical form. That is a change in the structure of its attractor landscape. Now, there are many dynamically different ways in which this can occur. Broadly classified as either local, where the form changes continuously as some dynamical parameter or parameters

continuously vary, or global changes that involve more complex shifts. Now, among basically the global shifts are phase transitions. For example, gas to liquid, liquid to solid, rivers, including critical point transitions, like simultaneous transitions among gas, liquid, and solid estates, so on and so forth, where changes can be discontinuous and uncomputable, essentially because fluctuations on every scale up to that of the whole system are simultaneously possible. Now, while we can study mathematically the conditions under which a biurication occurs, Beyond the simplest cases, we typically have no dynamical mathematical analysis of the

process of the change itself. Rather, the conditions of occurrence are deduced where possible by matching of characterizations of the antecedent and subsequent dynamical states in terms of parameter changes across the bifurcation threshold. I think basically, you know, thinking about these main so far the listed features of complex systems, these are all intuitive in fact.

we can, as I said, you can think about these and, you know, how basically they work and whether we can really think about them as such or actually are basically they are simply can be talked about within a deductive framework in relation to other properties. You can think about them in, you know, examples of real life quite, you know, intuitively. River change, boiling of water, the convection flow in your oven, and so on and so forth. But there are, as we go to those, you know, a little bit more, as I said, are more peculiar to life biological phenomena, they become less intuitive.

And so we are moving more toward those features of complex systems. Self-organization. Self-organization happens when a system bifurricates sufficiently under its own dynamics to a form exhibiting more ordered and more complex behavior. The molecular motion in a heated pan of water shifting from conduction through random collisions to cellular convecting provides a core intuitive example of self-organizing phenomena.

By contrast, the reverse bifurrications, as heat is reduced to the water, Equally dynamically transforming would not normally be considered self-organizations. They might be considered self-disorganizations. I will talk about this a little bit once we are done with this, you know, at the end of our session about Hegel, the notion of guys, and because we had very briefly talked about in the classroom in terms of this idea of disorganization and self-organization, which is usually associated with, you know, Haeckel's Geist as a complex system.

So please remind me to talk about this at the end. Now, since the condensing of molten iron to form a solid iron crystal is also considered self-organization, it is clear that self-organization has little to do with organization proper, since an iron crystal is too ordered to be significantly organized. Many self-organized states could also be brought about through external manipulation. For example, it's possible to build up an iron crystal lattice by spraying iron ions

a few at a time on a template. While the outcome is the same, here the active self is missing. It's basically the whole idea of self-organization. All things considered, it is probably most useful to consider self-organization to occur where and only where a system bifurricates, sufficiently under its own dynamics, so as to bear an additional system-wide constraint, or at any rate, an additional multi-component that is relatively macro-constrained. The formation of a new relatively macro-constrained, however brought about, creates a new level proper in the system

since the constraint now filters out more microscopic relation detail incompatible with it. The iron crystal, in our example, filters out thermal fluctuations and many external perturbations, dissipating the energy as lattice vibrations. Otherwise, the constraint would not be stable against microscopic originated perturbations and similar external disturbances. So basically, self-organization has its role in this kind of autonomous stabilization process. We talked about the emergence, so I'm

going to skip this one as well. Sorry. Now, as I said, constraints and different conceptions of constraints are very important in complex systems. And this one is especially very important. And it's usually typically Usually when we are talking about constraints, people usually think about it as a kind of a, basically a negative limitation.

But constraints in complex systems usually, often, but not always, play a role of enablement. Basically they enable the system. So the term constraint implies limitation. Most generally, in the context of complex system, it refers to limited access to dynamical states. Equivalently, it means reducing degrees of freedom by limiting dynamical trajectories to subsets of state space.

This is the common disabling sense of the term, basically, constraint. But constraints can at the same time also be enabling. They can provide access to new states unavailable to the unconstrained system. Equivalently, by coordinately decreasing degrees of freedom, they provide access to dynamical trajectories inaccessible to the unconstrained system. Thus, for example, a skeleton is a disabling constraint. For example, limiting the movements of limbs. You can think of this in an octopus.

But by providing a jointed frame of rigid components for muscular attachments, it also acts to enable a huge range of articulated motions and leverages, transforming an organism's accessible niche, initiating armor and predator-prey races, and so on and so forth. Sorry. Each of the great transitions in the evolutionary history, the emergence of multicellular organisms,

marks a new coordination of constraints. This is, again, a very precise discussion on enabling rule of constraint is, again, in the work of Baye and Longo, and also Alan Berthold's brain sense of movement. Basically this whole idea that the coordination of constraint is basically the whole idea of this enablement. By permitting reliable cooperation instead of competition and reliable inheritance of the fruits of cooperation, the new coordinations created new complexity and opened up vast

new possibilities. Coordinated constraints can work their way around physical laws, basically. For example, while no single pump can lift water higher than 10 meters, trees lift it many times this by physically linking together, coordinating many cellular pumps. Now it's possible to obtain complex dynamics in simple systems, for example, like logistic reproduction. However, plausibly the only way in which the complex properties to follow can be obtained is through complex coordination of constraints of the kind neural, muscular, and skeletal

coordinations exemplify. And one of the things that we talked about, you know, basically this whole idea of new relation processes and the emergence of the nervous system is really one basic idea of how that this nervous system as a platform has its role of enablement for the organism is through this massive basically coordination of these small constraints. Now these have their origin in the complex coordination of biochemical products and gradients that allow intercellular chemistry to support cellular maintenance.

We are here far from holonomic constraints, and as cellular regeneration shows, basically the new work and constraints conditions of the standard analytical mechanics and deep into the domain of multiple state-dependent, intracting, non-holonomic constraints." Which as actually Jessica brought up, this whole idea that the nervous system and a spatial locomotive dynamics that the brain allows for complex organisms to initiate, is basically

pushes the organism into the domain of non-holonomic constraints rather than purely parochial holonomic constraints of being basically constrained by rigid spatiotemporal positioning. Questions, discussions before we move forward? I have a question about self-organization. Sure. And a bit confused as to how we would then see something like selection pressure. I may be confused about the kind of sense in which self-organization is quote-unquote

internally or externally driven. And so adaptation relative to a set of selection pressures from the environment, do those contribute to self-organization or are they not? It's a very interesting question. I think the selection, the thing is that, and I might be totally wrong, and I would appreciate like if other people also share their ideas, but to me, you know, the selection criteria I think it would be better for us to dissociate it from self-organization processes because

it is an external factor, an external constraint. Yes, there is an adaptation of self-organization, but self-organization in a physical precise sense is usually that internal physical sense. When we talk about selective adaptation, selection pressure, we are talking about process of construction. Self-organization is different really from construction. It's basically the idea, as I said, it's the autonomous internal, basically self-organizing dynamics, whereas construction, again, technically construed, is a different thing, precisely

because it brings the role of this interaction. And this is one of the things that Darwin shows that basically, and also I think that argument can also be used against these kinds can also be used against these kinds of wild, speculative problems of self-organization dynamics in biology. That Darwin shows that basically this whole idea of interaction, which basically subscribes the organism to these adaptive pressures, selective pressures, can be in fact interpreted as in terms of basically computational constraints of the system to its environment. Where the constraint, yes, play a constructive role in determining, in basically guiding

the self-organizing dynamic of the system. But I think the subtleties of these notions require us to kind of dissociate basically this guiding construction via selective pressure and computational interaction with the environment from that precise notion of internal self-organizing dynamism. Cool. So yeah, there's a bracketing off of the environment in this. Yes. Yes. And actually, the thing is that there is this essay, I will try to find it, I have completely forgotten the title and also the name of the author. He comes up with

this really fantastic kind of philosophical commentary in terms of basically understanding of the difference between construction, as I said, and self-organization. So self-organization basically, in a precise sense, assumes no necessary reliance on this intervention of extrinsic factors.

Now, the thing with the idea of construction is that it can have both self-organization and also something that negates this self-organization, namely an extrinsic external factor. And he talks about this in terms of basically Kantorian understanding the philosophical intuition behind Cantor's idea of self-diagonalization, the idea for construction as different from this idea of self-organization in that sense, is the idea that basically you have the self-referential mechanisms within the self-organizing regime plus a negative regime that should negate

or simply randomize this self-refreshal dynamics in order to be able to reorient it. Thoughts, comments, discussions? So on genetics there? I mean, as an example? Sorry? So an example of this kind of counter system of random selection would be then mutation and genetics, right? Is that a good example? Yes, we talk about mutation, yes.

But also mutation can also be seen in terms of, yes, I think yes, yes, it can. I'm going to talk about this a little bit later, yes. But I think we can, for people who are interested in philosophy, think of Hegel's Geist. Hegel's Geist is not just a self-organizing system. People always, when they talk about, especially in the content of philosophy, they don't take Zizek's Less Than Nothing. refers to Hegel's Geist as a self-organizing complex system.

Now the thing is that once you look at it, it's not really a self-organizing system in that precise technical sense, precisely because Geist develops by negating itself, basically always drawing on the intelligibility of the objective world, using it as the power of negation to basically to what, as I said, disorganize the self-organizational tendencies of the Geist. Because this whole idea of that there is this idea that's, you know, the thing is that if Geist is completely self-organizational for Hegel, then it's actually,

it can never gain traction on the absolute. It basically, it simply, just simply expands the order of itself, which is the order of appearances, the parochial idea of the Geist. But the whole idea of the Geist is that each epoch of the Geist, or each stage of the progress in the Odyssey of the Spirit, is that it's this move between how it organizes itself from what it takes itself to be, namely the order of its appearances, but also at the same time by negating it by the power of reason and using the objective intelligibility of the world. And that's basically the movement of sciences.

Sciences create catastrophes basically for the self-organizational tendency of the spirit within the order of appearances. Is that a kind of heuristics? It's not really heuristics because, you know, it's really the power of judgment. Judgment is very different from heuristics in the Kantian, Hegelian understanding. Judgment is the power of the concept, is the power of the rule that has objective traction, basically, on the domain of intelligibility. My question to build off that, is self-organizing, like this category of self-organization or

calling something self-organizing, is that the kind of heuristic? I guess the majority, actually one of the things is that it's quite strange that so many of these properties of complex systems that we talked about, I mentioned that there There are no canonical definitions. And a lot of them have heuristic basis of basically detecting them or talking about them. Yes. Yeah. I was going to save this for later, but it's sort of building off of our discussion from sort of Hegel, the basic definition of complexity as the amount of history embedded in a system or sort of like historical data.

and I was sort of trying to backtrack from that to sort of the Kantian definition of history and the Kantian idea of technique of nature, specifically, as sort of like, in order to make sort of something that seems random, intelligible, imposing these kind of constraints of purposiveness on it, where you're kind of, yeah, bracketing sort of the outside and looking at a system in terms of what it sees its goal as or what constraints it puts on itself. Uh-huh. And sort of human beings doing this in sort of history with reason, but in order to make nature intelligible, sort of the technique of nature is to treat natural phenomena as if they're goal-directed when you know they're not. Yes, well this is, I mean, this is one of the things that is very, really, I think,

I kind of mentioned about this very briefly in the last session, the idea that, you know, The majority of the stuff that we talk about in complexity sciences, in terms of functions and stuff, are analogically posited on our linguistic reasoning, theoretical and practical. Functions, especially you need to be very careful with this idea of functions in nature, because all of the talks about functions in nature are analogically posited with regard to our models of practical reasoning. Just as, for example, the ideas of representation, and for example, when we talk about organism representation,

computation representation, all of this stuff, three dots, and so fly three dots in this space, all of these talks about representation at any kind of empirical based vocabulary, using any kind of empirical based vocabulary. They are also analogically posited on our theories of truth, theoretical reasoning. Basically, they are linguistically posited. They are analogical. You need to be very careful. That's why we need to be very careful with this stuff. Because if we do not apply these analogical posits carefully, we fall into all sorts of bad metaphysical assumptions about the nature of functions, nature of representation,

and so on and so forth. And this is where I guess sort of emphasizing primacy of the practical or what you were talking about before with looking at sort of like basic assumptions about physics in like prepositions and like metaphor, like in concepts embedded in metaphorical language. Yes, but also analysis. Sure, but I mean, you can think about this. I mean, you can see this, in fact, among all these scientists, or even computer scientists, biologists, neuro scientists, who usually talk about intentionality. Or, for example, the functional architecture of cognition. You see that there is a tendency

that to conflate massive amounts of basically different levels of analogy into just one basically analogical scheme. Representation, basically intentionality, and that's basically what allows them to make that kind of massive reductionist or inflationary leaps from kind of modular representation at the level of modular processes to representation of the conceptual representation that we talk about, namely semantic representation. or reducing, for example, the semantic complexity of linguistic representation or intentionality in our things, again, to that kind of extremely parochial

sense of representation that, for example, you see it in any kind of semi-complex, basically, computational system is capable of simply. You can see it in basically any kind of philosophy of neuroscience. In fact, Mettinger is also, I think, at fault in these things. When he talks about consciousness, he basically drops the, he basically, he assumes an analogical point of view. But then he attributes this analogical point of view as some sort of kind of metaphysical certainty, to the point that basically the entirety conflates

ultimately representation and intentionality at the level of discursive, aperceptive self-consciousness, which is a completely linguistic phenomenon, thought as a picture of language as such, with basically phenomenal consciousness, which is an empirical consciousness. And then he basically tries to come up with this philosophical understandings of philosophical commentaries on basically that our self-consciousness, conceptual self-consciousness, is basically simply an extension of empirical consciousness. Or, you know, then he also, again, does the same kind of move in reverse, inflating the

kind of claims that he has derived from his investigation of phenomenal empirical consciousness again to conceptual self-consciousness and intentionality at the level of conceptual self-consciousness. That creates a kind of really this almost Gnostic, dark Gnostic, basically, worldview. You can see this massively if you look at the kind of the horror side of vlogging speculative realism. You can see it in the work of Scott Baker. This basically massive amount of conflation between the stuff that's going on at different levels. We need to be very careful with this analogical position. Nothing wrong with analogical position.

But we need to know that, first of all, our theories of representation and functions are analogically positive. Analogically positive with regard to what are theories of truth, namely coherent accounts of theoretical reasoning and practical reasoning, accounts of function, malfunction, according to basically how we see things function linguistically. And then we'll talk about this in terms of this whole idea of different levels of vocabulary that people usually deploy to describe different phenomena at different level. For example, the moral vocabularies, normative

vocabularies, logical vocabularies. And some people always try to reduce all of these vocabularies to an empirical base set of vocabularies. Namely, and those empirical vocabularies are the vocabularies of the special sciences. But the whole point is that empirical vocabularies cannot be positioned unless there are different normative modal vocabularies. And we will talk about this at length in the third module, about so many of these kinds of conflations going around in these kinds of looking at complex systems, looking at cognition as a complex system, and then trying to reduce it, or inflating the claims that you have derived from some empirical observation to basically conceptual self-consciousness,

semantic consciousness. Yeah, that's great. I'll definitely want to return to those examples at some point. It's all really helpful. And yeah, I'm still trying to think about this from a Kantian perspective of respecting the separation between the empirical and the transcendental. Yes, yes. No, I think, in fact, I wanted to ask you guys, but I mean, we can talk about this. I think we need to have some sort of extra session somewhere so we can talk about some of this stuff more casual way. It would be great. I mean, I'm not sure how many of you have read Ken's Critique of Pure Reason.

I think it's one of the most astonishing works ever produced. And it would be actually a superb kind of introduction for our third module when we talk about AGI and construction of artificial intelligence and human-level AI. I think these are absolutely, I think, really, because these kinds of stuff that, as you address, is not only methodologically important in terms of not making these kinds of conflations, actually gives a roadmap of how you can analogically posit in a controlled manner a base level,

basically AI, and then analogically bootstrap it to where the AI is no longer analogically posited precisely because it is, at that point, a conceptual intelligence. It has a semantic complexity of cognition. So the representation is no longer analogically presented precisely because it is now embedded in language, because it has autonomous conceptual capacities. Does that bring us back to self-organization, or is this a bad conflation? No, no, no, no, it's not really part of self-organization. We will talk about it simply, it's a construction. Okay. And yeah, do you know of anything that, just to sort of close, that uses this framework

from the third critique of sort of technique of nature in talking about complex systems or trying to sort of draw pattern or structure out of chaos, because they seem sort of very... Hmm, not really, no. Yeah, I guess just by way of the way we talked about history, or like as in a Hegelian or Kantian framework, what someone who's talking about complexity means by history, they would refer to as nature. Yes. And that history, you can only talk about history in that sense as an analogy to a sort of human practical sense of history. Yes, but also... As a way of understanding it.

Sure, but also, in fact, one of the most troubling things behind this is that, as I said in the previous session, there is a time bias inside this. It's the whole idea that consciousness has an inbuilt, prospective temporal awareness, a temporal bias simply. basically sees time in terms of temporal instantiation, temporal asymmetry, past, present, future, with past being always overprivileged, present being the dominant framework.

And then the future is like this kind of nebulous framework that basically is all determined by basically the histories of the system. And the thing is that this absolutely, this kind of temporal perspective that we use, I don't think that we need to, we should inflate it into some sort of objective conception of time. In fact, this is basically the whole idea of Magdegart as a No-Hegelian to show that we absolutely have no capacity. We cannot draw an inference about the objectivity or the nature of time from our temporal perspective or awareness. And in fact, if we do that, in fact, we end up having the unreality of time.

That time, the conception of time as being instantiated in basically within a temporal series, past, present, future, is essentially a vicious circle. It's a regress argument. Basically, all of your past, present instantiation, in order for them to be able to coherently define them, you need to, again, have all of the past, present instantiation, namely the entire temporal concept of time. I wanted to suggest that really I think it would be great if you can look at MacTagart's essay on reality of time.

It's absolutely one of the most phenomenal articles written on time. And so many people usually dismiss this as, oh, well, he argues about the reality of time. No, what MacTagart discusses really is not really the reality of time or showing that there is no such thing as an objective concept of time. where he tries to show that if you try to draw inference from your temporal perspective or awareness, namely a temporal series, and talk about time in that sense, you will end up, your argument basically amounts to nothing but unreality of time, that time becomes simply an ideality, basically, that's exclusive to conceptual consciousness.

And this is something that I will, as I said, will talk about this in terms of Boltzmann and how he basically reinterprets this, this idea that the difference between the block view of time and the temporal or the asymmetric view of time. And basically, one task, I think, would be to show that there is no income and zero-ability between our pragmatic view of time, namely temporalities, past, present, future, and the block view of time. A view of time in which, basically, there is no privileging of any asymmetrical point

within a block of time. You don't call them any more past, present, future, or to be able to talk about the history of the system. And you see that once you can't really talk about the system anymore, then you have to develop a new, basically, conceptual tools in order to explain, basically, cause . You need to be able to develop new tools to talk about system history, in fact. In fact, I would say that system history needs to go away if an objective of time shows that we cannot impose our temporal, basically, conception of time

on physical systems. In any case, I will talk about this much more So, yes. Can you please go back a bit to talk about Geist and why it's not a self-organization system, whatever. Yes. What you started talking about. Sure. You see, the thing is that in Phenomenology of the Spirit, Hegel makes different descriptions

about Geist, and some of them are actually quite self-organizing. They give you the sense that Geist is a self-organizing entity. But toward the end, he talks about this in terms of this self-disintegration, that Geist does not have a self-disorganizing, self-disintegrating mechanism, basically it cannot initiate new epochs. Hence, it ceases to be a geist. It loses its intelligibility of being a spirit as such. Now, the thing is that what is exactly this mechanism of disintegration is the power of negation, implicit within the power of judgment. Determinate negation, not abstract

negation, determined negation, namely resolving incompatibilities in material inference. And this power of determinant negation, for Hegel, is sharpened by, basically, sciences as different but also in connection from manifest rationality. Okay. I have a really quick question on this.

Oh, sure, sure. Would I be right in thinking that this distinction between disabling and enabling constraints, that's well embodied in the notion of generative entrenchment? Yes. Yes, basically the whole idea of enabling constraints is really, which also they play also a disabling role. But once coordinated, they become enablement. Yes, that's precisely the idea of the platform, of generative entrenchment. Yes.

I mean, I kind of like you made a very brief reference to that with regard to our nervous system example in terms of coordination of constraints. And we talked about the nervous system as a platform. And you see that basically once constraints are coordinated, a canalization process occurs. And once this canalization occurs, you have a specialization of functions. Basically enables, this canalization is that enabling process.

So if there's a decrease of degrees of freedom on one level and then an increase on another? Yes, yes. Increase basically, I think one of the things is that when we are talking about enabling the role of constraints, we need to understand this enabling role in terms of functional organization. Basically, they play enablement for functional organization. At the level of the structure, there is that kind of a disabling import, whereas at the level of the functional organization and the functional architecture of the system, there

is an enablement, a global enablement. Cool. Thank you. OK. So let's go. Oh, my goodness. We are kind of behind. It's OK. We had a nice talk. We can always make a make-up session. OK. Where was I? OK. Let me just share the screen. Okay.

So we said that coordinated constraints can work their way around physical laws. For instance, with no single pump, actually I talked about this. Yes, sorry, I completely confused. I think we are actually on the next item, which is intrinsically global coherence. So in analytic Lagrangian dynamics, the globalness or otherwise of constraints is not directly A holonomic constraint provides an inherently global geometrical constraint on motion in

the sense of being specified everywhere, but not in the sense of demanding internal global coordination of variables. Some holonomic constraint may force component motions to be globally correlated, others will not. The same applies to non-holonomic constraints. Moreover, these can be partial rather than global, with a dynamic network of constraints structuring system dynamics, as in the cell. But if a system is to perform a global function, for example, metabolic regeneration, then And this will force a global organization of its components to achieve it.

Therefore underlying a global functionality must be a global constraint on dynamics that ensures realization of the function. And that was just the point that I made. This must be so even when this constraint is realized through a network of a state-dependent, attracting non-holonomic constraints. For example, a network of work constraint cycles in the cell. Multiple global functionalities, characteristics of living systems, require multiple underlying global constraints. And these will normally be significantly but subtly interrelated in order to allow multiplexing,

namely many component roles combining to realize a single function, and multitasking, namely the one component playing roles in realizing many functions. Now multiplexing and multitasking are attractive because they reduce the number of required components, while increasing system functionality and adaptability, and possibly even evolvability, albeit at the expense of increasing system complexity and possibly also increasing system in stability and or rigidity. Another main feature which is important

is this order and organization. The reason it's important is because of this, as I exactly like the concept of constraints, because they have an intuitive appeal when we are talking about constraints. People usually think it's about limitation. There's so much confusion going around these kinds of concepts and complex in sciences. And the same thing about order and organization. Precisely because of their intuitive usage, people usually conflate different conceptions of order and organization. The constraints underlying global functionality

require global organization as distinct from global order. A high degree of orderliness means internal uniformity, a crystal, while functional organization requires inter-articulation of distinct components, For example, in a motor vehicle engine. Now, the root notion of order is that derived from algorithmic complexity theory, the orderedness of a pattern is the inverse of the length of its shortest, most compressed, complete description.

Hence, gases being internally random are disordered, and regular crystals being internally uniform are highly ordered. But neither displays any functional organization. I will talk about this algorithmic complexity theory in the second module, Theory of Solomano, around this concept. Sometimes complexity is taken to be measured by the inverse of algorithmic orderedness. but this leaves gases the most complex systems. In short, it ignores organization. The key to living, basically, complexity. Machines and living things are organized

because their parts are relatively unique, and each part plays distinct and essential roles in the whole. Now, the kolokoya use of organization is broad and vague, though its core examples are functionally organized. engines, firms, rescue teams, so on and so forth. So the use of organization that we are basically address here is basically functional organization. And whenever I talk about organization, unless otherwise I say it as a structural organization, I mean it as a functional organization, organization functionally interpreted.

Now, another use of order we can talk of high order features refers to features characterized by high order relations. That is to say relations among relations among relations among relations. Then organization is a particular kind of ordering in this sense involving relatively high order relations have characterized many nestings of correlations within correlations. Think of correlations within and between the motor, electrical management, and drive chain modules of a car, for example.

That is, an organized system displays a non-redundant global ordering relation of relatively high order, including global subrelations. characterizing global functions. For this reason, organized systems must be less highly ordered than our crystals, but are obviously more highly ordered than a gas. A system's organizational depth is measured by degree of nesting of subordering relation within its global ordering relation. For example, cells within organs, within bodies, within communities.

Living systems are deeply organized. However, organizational depth also does not fully capture complexity, basically, in non-linear dynamic systems. Now, the thing is that as we will talk about this idea of deep objects. As I said, this idea of computational depth or logical depth basically is a very useful way of measuring these various notions of depth. nestings, nestings in hierarchies, time scale, deep, basically, depth of a hierarchy, causal

history, so on and so forth. So the key important thing with this, basically, idea of order in terms of interrelations, organizational depth is when, and also when we are talking about functional organization and organization complex system, is the concept of nesting, nested organization, nested hierarchy. And this is, I think, you need to, you know, have it in order to be able to coherently

differentiate it, this concept of organization and hierarchy, from kind of like the more intuitive idea of vertical hierarchies and basically basic forms of organization. And the great thing about the ideas of measures of computational complexities is that computation as we will talk about in the second modules and the way that computational complexity is defined allows us to basically develop different methods and come up using different computational tool boxes to basically identify and distinguish nestings,

basically, even construct nesting, models nestings, using iterations, recursion, long distance rules, short distance rule, so on and so forth. So this is something that we'll get back extensively in the second module. Now, the next one, which I said was modularity, a system contains a module if and only if to a sufficiently good approximation to capture essential systems functionality, Its dynamics can be expressed as an intractive product, the dynamical product of its intramodular dynamics

and its intermodular dynamics, intra and intermodular dynamics. Three kinds of modularity can be distinguished, a spatial or horizontal, level or vertical, and process modularity, labeled respectively S, L, and P modularity. Now, S modularity obtains when there is a principal division of a system into contemporaneous spatial modules, such that the system dynamics is expressible as a product of the individual module dynamics and their interactions. This is how we currently design and model buildings and machines of all kinds, from homes

to hotels, typewriters, television sets, so on and so forth, and how we usually attempt to model both biological populations, modules being the phenotypes, and often their individual members, where the module is being basically the internal organs or cells. Now L modularity in contrast distinction obtains when a system's dynamics may be decomposed into the interactive products of its dynamics at different system constraint levels. This is often how, for example, business management functionally analyzes a firm.

Often that organization will express a management hierarchy and be graphically represented vertically, often realizing the functional rules vertically in a building, hence the alternative vertical label. It is also often an important part of a machine design. For example, motor vehicle, electrical regulation, and drive chain modules. And of the scale analysis of organisms, cells, organs, organisms, et cetera. Now, P-modularity, process modularity, obtains when a system's dynamics may be decomposed into the intractive product of its process dynamics and is characterized of the analysis of organisms

and complex machines into mechanisms, such as cellular respiration, pulp mill regulation. As motor vehicle design illustrates, all three modularities may be combined, at least to significant extent, in current simple engineering design. Now, S and L modularity will create the constraints to enable corresponding functions in simple, reliable ways through the process, disabling many other, basically, processes. The earlier viable system movements that sprang from cybernetics and general system theory relied on, basically, such designs.

But modular processes may also cut across levels and spread throughout the system, and basically, to that extent, exclude S and L modularity. So we can only have, basically, as a dominant framework, a P modularity, process modularity. Now modularity of any kind reduces system complexity by decreasing dynamical degrees of freedom, while increasing functional and possibly developmental reliability and ease of repair. So we need to understand that, you know, basically the modularity is one of those topics where

the concept of complexity starts to kind of diverge, bifurcates on one side a structural complexity and the other side functional complexity. Modularity usually leads to functional complexity rather than structural complexity. So now, and you know, obviously when we have the modularity and organization and order that we talked about, we need to also take into account basically the property

of the hierarchy. Hierarchy proper is asymmetry of level, basically vertical control in a sufficiently L modular system. While common in machine design and as underlying principle in organism and in situational design, pure hierarchy is in fact the exception and is rarely more than partial in living systems. The higher level constraints, lower level dynamics, as in crystal lattice constraints the behavior of its atomic constituents, will often regulate it through feedback. For example, coherence of crystal vibrations.

And sometimes it will also control the lower levels in important respects. Think of top-down control, you know, brain control of basically muscular thresholds. But it will also typically be true that lower level dynamics will constrain higher levels. example, electron orbital dynamics constrains crystal angles. May regulate them through feedback. In example, in catalysis of chemical reactions. And might control certain aspects of the higher level, bottom-up control, indirect control of volume,

basically in Hebbian learning. And very briefly, Hebbian learning is just this idea of basic model of synoptic plasticity. The basic formula can be rudimentary put as fire together, wire together. This idea that basically the adjustancy, the neighborhood of two, for example, an axon and basically another neuron. Once one is repeatedly, basically, stimulated by the other, then the plasticity of, basically, these neurons starts to structurally shift. They basically structurally change.

And basically, this is one of the things that, basically, the idea of the Hebbian learning or Hebbian rule also can be understood as this kind of variations, these effects of basically shift in a structure between these different levels of hierarchies and how they basically impose a structural change on one another. So, now, path dependence, I remember that we talked about, I think in the previous session,

very briefly, path dependence occurs whenever there is a positive amplification for then Then initially, nearby dynamic trajectories subsequently diverge as a function of small differences in their initial conditions. So the path taken depends on precisely where the first step began. Now a notable subclass of path dependencies are those where once begun development along a certain path itself becomes increasingly entrenched. This applies where an initial fluctuation is amplified and entrenched, especially where that entrenchment involves a bifurcation that reinforces the irreversibility of the development.

Now examples of this include a particular impurity site of first freezing or rolling boiling phenomena that I talk about, irreversible processes. You can think about this as a first oil discovery, basically the opening of a shop in a new suburb that transforms a local economy. Now the thing is that these cases exhibit a clear sense of historical possibilities exploited and correlatively of others foregone, and the resulting paths are often set to fix

their initial historical conditions. By contrast, for stable systems in an abstract or basin basing, there is no overall path dependence, since the same outcome occurs for all beginning points. Basically, for beginning points, by beginning points, I mean initial conditions. The third and the final type of constraints are constraint duality and their relation to supersystem formation. Now coordinated constraints that enable while disabling, the disabling movement constraints

by a skeleton and its enabling of locomotion and leverage exhibit general constraint duality. So this idea of enabling while also disabling. So this is basically the idea of general constraint duality. The notion has a specific application to forming system into super systems through mutual interaction. System constraints may contribute to enabling supersystem capacities. For example, you can think of this rule of mitochondria and eukaryote energy production. Conversely, supersystem constraints

may free up system constraints. For example, whenever multicellular capacities permit member cells to specialize. Well, it has a wider application in considering, for example, social community formation. We can gain a crude measure of the importance of socialization to a species by considering the ratio of usable individual parametric plasticity, i.e., adaptiveness between isolate and communal states. For simpler creatures of lesser neural capacities and more rigid social organization, such as of insects, The relation is typically negative. Individual capacities are sacrificed

to communal cohesion and function. Oppositely, for example, humans increase their coherently usable individual capacities enormously through collective, basically, culture, through this basically mobilizing the constraint duality in order to generate super-system formation. And this happens even while contributing to communal capacities in terms of humans. Unless humans have a sophisticated, high-quality cultural environment in which you develop, there will be vast reaches of our somatic, especially neural organizational space that

we cannot use because it is not accessible to us. Thus, for humans, there is a positive relationship between individual and communal capacities. And that can be expressed through these constraints duality and supersystem formation. Coupled constructively, each enables the other. And together, they largely, but not wholly, dominate those coupled competitively. You know, we can speculatively picture this effect increasing through the mammalian line as brain size, intense socialization, and intentional action increase together. This makes all the difference to the power of basically, you know, a super system formation

like culture, to its significance in adaptive evolution and to the intricacy and globalness of its organized dynamics. So the last ones, the last features, and it seems that we are running low, but I will try to speed it up so we can get at least the idea of the Boltzmann informational content of the system. So the other property of complex systems that needs to be taken into account is this coordinated

spatial and temporal differentiation with functional organization. Multiplexed, multitasked functions, as I mentioned then, cannot all be realized simultaneously at every location. resulting interference would render reliable performance impossible. It is necessary then to distribute the realizing dynamical activities, spatially and temporally, so that each local area over each process cycle is restricted to coherent sets of concurrent activities. Moreover, these distributions have to be subtly organized so that each function is realized at convenient locations and times for receiving its inputs and also useful locations and times

to contribute its output. Similarly, for example, similar metabolism requires a group of close self-reproducing processes to recreate the constraints for each process from the products of other processes. And this to require subtle spatial and temporal differentiation to achieve reliability and effectiveness. So it is kind of this particular feature is very tied to the notion and the concept of scheduling. And we will get to this not particularly under this heading, but we'll talk about this idea of scheduling and its computational importance

when we talk about concurrent systems and the concept of scheduling in basically computer science in terms of this kind of resource sensitive distribution of processes. Questions, stuff, thoughts, commentaries? Any comments on resource sensitivity? Okay, sorry, go on. I was just going to ask again about this distinction within hierarchy and was it between

a structural and functional hierarchy? You want to know what's exactly the distinction between a function, a functional hierarchy and a structural hierarchy. I just wanted to clarify that that was the distinction. Oh, sorry. I totally blanked out. Okay, you need to repeat this. I just wanted to make sure that I have this distinction right in my mind within the notion of hierarchy that one sense of it is the structural hierarchy and the other one is the functional hierarchy? Yes. Just because this is something to be implicated also in the modularity bit as well as...

Absolutely. Yes. Yes. Absolutely. Okay, cool. Yeah. And then as we... Basically, this is the whole... You know, the course as it evolves, it's just what we are... All we are doing is just simply introducing this underlying massive different things that makes a complex system a complex thing. And then we are, as we are moving to the computational module, we are trying to basically start bifurcate, distinguish this structural complexity from functional complexity. And then in the third module, we are trying to integrate them. But the whole idea is that if we are not able to differentiate them properly, we get massive

amounts of conflation, aligning different distinctions at different levels, basically. And this is really, really, I think, an important topic when people talk about in cognitive science, really, picture of the mind, and also, respectively, construction of human level AI. This conflation between function and the structure is important, and also respectively complexity at the level of functional hierarchy and complexity at the level of structural hierarchy. Cool. Yeah. Yeah, that makes sense to me. And another kind of terminology question was, one of the senses of order I had conflated

with level. But I think in your slides you actually at one point you say multi-level and multi-order. And I'm a bit confused between the distinction between level and order in that context. Well, it was about the interrelation of components. Yes. It's about interrelations. You have order, you can think it intuitively in terms of the hierarchical ordering, deep ordering, basically, but also order at the level of intra-level, interrelations among stuff, basically, among components, among generative mechanisms at that level.

So it's inter-level, order can be thought inter-level and intra-level. And you need especially epistemic mapping when we are trying to talk about this order. When actually we look at complex phenomena, we need to come up with models. And that was the plurality model argument that we had last session. In order to be able to map and recognize both interlevel depth, order in that sense, and intralevel order. And this is basically a good book, really, into this different why it's really important epistemically

to differentiate them and develop different models, basically different epistemic maps, in order to be able to map these different variations of orders in physical complex phenomena. A good book on this topic is Mark Wilson's essay on conceptual behaviors. It's a really meandering book, but nevertheless, it's a really classic, superb book on development of mapping and modeling at these different levels in order for you to be able to picture level behaviors, order at the level of intra-level behaviors, and order at the level of inter-level

behaviors. Usually, when typical classical models are basically just try to align the distinction between them, so all you get is either a very classical one-time relationship the level order at the basically in terms of intra-level and your model or model that simply gives a complete basically intra-level without this differentiating these inter-level inter-relations, namely order at the level of intra-level behaviors. Right, because yeah, these distinctions shape into different kinds of production.

Yes, and once you have this kind of elision of different concepts and sense of order and behaviors, not only you get greedy reductionism, but also you get greedy inflationism. Whenever there is a greedy reduction, you need to expect greedy inflation as well. In fact, the majority of people who are that kind of reductionist always comes with inflationary accounts of, for example, as I talked about in terms of empirical consciousness, once you try to basically ally this distinction between different levels of consciousness

and really reduce it to that base empirical consciousness, then you are also tempted to inflate conclusions driven from empirical consciousness to level of basically conceptual consciousness to different levels, other grades of consciousness. Any more comments, thoughts? Ressa, I have a question Sure You've gestured to some of this already but earlier on you distinguished some of these characteristics we've been going through as being associated primarily with living phenomena

which ones would you say are most or might be most directly applicable to understanding social systems and which ones should we be especially skeptical about being applied in those contexts Here I'm thinking about something like war where the notion of sensitivity to initial conditions would be compared to like Baswitz's notion of friction in order to describe how small causes are amplified to produce like macro effects which can't be predicted because of problems from distortion or information overload. Uh-huh. Well, I think sensitivity to initial condition is really one of those base features of all

complex systems. And in fact, it spans across social phenomena, life phenomena, so on and so forth. But for example, when we're talking about self-organization, irreversibility, these kinds of stuff, That becomes very context sensitive. And that's where you need to be careful not to apply, but be careful in applying these concepts like irreversibility especially, like self-organizing from life phenomena to social phenomena, to cultural phenomena. But sensitivity to initial condition, I think,

is probably the most fundamental. It's basically, as we talked about, it's just the consequence of the loss of linear superposition principle, which is the main basic feature of any kind of nonlinear complex dynamic system. So you'd say that functions for that minimal definition? Yes, yes. But as I was trying to argue, it's so minimal, that becomes so prevalent, that it almost becomes trivial. So you need to top it with other features, but those features need to be context sensitive in order for you to be able to effectively talk

about these sensitivity to initial conditions within the framework that you are talking about. Yeah, that makes absolute sense. Thank you. Welcome. OK, can we go for 20 more minutes, or you guys are tired? No, keep going. Yeah, sounds good. Keep going. OK. Fonely? Sorry, I lost my slides here. OK.

My god. This is OK. I'm going to skip over the autonomy thing, because the autonomy, I mean, I will talk about it later on. It's the last in our, basically, and basically, characteristics of complex systems. So in order for me to be able to talk about the Boltzmann stuff. But before that, moving to Boltzmann informational content of a system, I want to talk a little

bit about condition-dependent laws. I briefly talked about laws, determinism, and causality in complex systems in the last session. Condition-dependent laws, compared to the universal, invariant laws of physics, the local idiosyncratic behavioral patterns exhibited by many complex systems don't seem to qualify as laws. Of course, biology and cognitive sciences dealing with complex systems do use universal laws in constructing their models. For example, if their elemental laws of chemistry did not operate the same everywhere. Biochemistry and hence biology would be much harder

than it already is. Even so, the complication arises from the fact that the gnomic invariance largely occurs at, for example, union interaction level. But how n-body, k-component, yon systems operate is often a sensitive function of the initial and constraints condition, especially organizational conditions. The biochemistry, for example, of carbon-hydrogen chains provides eloquent illustration for instance, where folding history can alter subsequent interaction dynamics. That's why no simple set of laws can be deduced in advance in defining complex systems.

However, the phenomenon is universal within science. It's not just restricted to complex sciences. For example, consider that electric circuit dynamics is the outcome of many universally lawful component interaction. But it is the material circuit conditions that determine the outcome circuit law, whether it is oscillation, exponential decay, or other. These circuit design-dependent dynamics are properly considered law-like despite arising from a specific material conditions. Now, moreover, the point extends to law conditioned on self-organization, a distinctively complex

system circumstance. Pertinently, self-organization precisely occurs because of the sensitivity of dynamical form to dynamical initial and constraints conditions. But since self-organization involves a new dynamical form, it is reasonable to say that it obeys new dynamical laws characteristic of that form. For instance, consider this condition. a cooling mold of liquid iron in contact with a heat reservoir of lower temperature. It leads to a new emergent loss, rigid body, not fluid dynamics, crystalline, not fluid

conduction of electricity, heat and sound. The universality requirement drops out, basically, in this schema. Again we are free to see biology and cognitive domains as replete with real laws, just as is physics. It is at most that they will be condition dependent on highly intricate, perhaps idiosyncratic, and typically organizational conditions. They will be a special, as some philosophers say, loss, basically in dealing with complex systems.

Also within this framework, their form will be hard to predict, but so far at least this distinguishes them from the situation in physics by at most a matter of degree, not kind. So when we are talking about condition dependency of law in complex systems, there is this condition dependency, a specialized form of laws, and these specialized laws of complex dynamics and complex systems need to be understood in terms of variation in degrees, not kind, with regard to fundamental laws of physics.

So starting with one of the main figures of complexity sciences, and I absolutely recommend You guys looking deep into his work, is Ludwig Boltzmann, revolutionary, absolutely excellent kind of insights into systems, into how systems work. Basically how things in the broadest possible sense hang together in the most broadest possible

sense. That was basically Boltzmann's idea of how to see the world in light of this formula, how things hang together. And as we talked about in the first session and the second session, this idea of how things hang together, this idea of interaction, is coupled with the idea of behavior of the system. And interaction and behavior in terms of how things hang together in the broadest possible sense is basically the most fundamental idea in complexity sciences, which basically the

whole is not, of course, it's not exclusive to complex sciences, but we saw that basically this idea of how things hang together nonlinearly can lead to massive, basically, new ramifications in how we should approach a system. Now of course, you know, beside this idea of how things hang together as, you know, basic definition behind this whole idea of complexity in sciences, Boltzmann also came up with so many other revolutionary ideas. So many of them have only been revisited in

the last couple of decades. And one of them was the idea of time that I definitely plan to talk about, if not this session, at some point. Now, this idea that how things hang together is a key to understand complex systems brings us to the idea of the information content of a system. Basically, the information content of the system is really answering that how the information content of the system can be described is to answer how can we model and

describe how things hang together in a system. The information content of the system gives us a measure of the ordering present in a system as well as the amount of usable energy contained within the system. In its origin, this parameter can be traced back to, as I said, the revolutionary work of Boltzmann on statistical thermodynamics dating from 1866. His basic idea was that any macroscopic system could be envisioned as consisting of a large number of particles, typically on the order of 10 power 23 particles.

The macroscopic properties of the system would result from a statistical averaging of the mechanical motions of the system constituent particles. It was also assumed that any ensemble of the microstate of a system that yielded a given value of a macroscopic property would be equally likely to occur. On this basis, the probability of finding the system with its known macroscopic properties is proportional to the number of microscopic states available to it. And this number is denoted by the symbol W. Boltzmann defined the macroscopic entropy

of a system by this formula. S equals k natural logarithm w. This is the logarithmic connection between entropy and probability. In this formula, k is the Boltzmann constant. In this formula, we can say that if each of the i a microscopic configuration, has a different probability, p i, of occurring, the equation then becomes this, summation by the i, p i, national logarithm p i, and the k being the

Boltzmann constant. Now in the 20th century, there developed a growing consensus regarding the existence of an intimate relationship between the concepts of entropy and information. And this entropy, as I talked, is basically this idea of how things hang together. And I will talk about this later on again, when I'm going to talk about logical depth. So basically, the consensus, the emerging consensus in the 20th century was this intimate relationship between the concept of entropy and information, the entropy content of a system and the informational content of the system.

Of the key importance in this regard was the work of Shannon, who conjectured that the the information contained within a message must be related to its internal organization. The measure of missing information or uncertainty in messages that he proposed took the same form as the entropy defined in a statistical thermodynamics. He argued that for a system that could exist in n possible states I want to end, each of which had an associated probability of PI, the uncertainty in the

system, U would be given then as this formula, which is basically the definition of the the Boltzmannian entropic states. The negative value of the parameter U is described as the information content, or negentropy, of the system and is usually given the symbol U. Now negentropy is reverse entropy, or basically the entropy deficit of a dynamically ordered subsystem relative to its surrounding chaos. So negative value of the parameter U,

as described as the information content or negentropy, is basically, is really our index of how things hang together in the broadest possible sense in a system. Now in the wake of 20th century system and information theories, it has been widely recognized that systems operate by virtue of the fact that they contain information, or in other words, are ordered in particular ways. And this information content in the sense of how things hang together is precisely what is exploited to characterize and measure complexity of systems.

So ultimately, as we are going to talk about, the measures of complexity are different specifications of how things hang together in the system. In contrast to randomness, which can be measured by the entropy in a system, complexity cannot be determined by one single measure that is applicable in all situations. Different measures are needed to describe such features as the extent to which the parts of a complex system interact and the manner in which the action of the parts are coordinated. It is in this sense that we can see and talk about the measures of complexity in terms of the hermeneutics of complex systems or deep objects.

Basically why I'm using hermeneutics, because hermeneutics traditionally in philosophy is a part of our relationship really. How parts hang together in the broadest possible sense gives us a hermeneutic interpretation of a deep object, a whole. And what we are going to, you know, examine basically in the next session, also in the beginning of the second module, is the computational hermeneutics of deep objects. You can think of life, intelligence, ultimately cognition and language. But before

that we need to introduce one key measure of complexity before moving forward to the other two measures of complexity that I mentioned. And so this measure, this key measure of complexity that basically allows us to talk about the computational hermeneutics of how things hang together as a measure of complexity is Charles Bennett's concept or complexity measure known as logical depth. Now, there have always been, you know, in three major directions along

which the frontiers of physics have advanced toward the very large, you know, toward the very small and toward the complex. A central role in statistical mechanics, you know, the field of physics dealing traditionally with complex systems is played by entropy as I talked about. This role has many facets. First of all, entropy is a thermodynamic concept closely related to temperature and not needing any microscopic interpretation. Secondly, as shown by Boltzmann, it is a measure of disorder or randomness. Finally, according

to Shannon, it measures an amount of information. What information this is depends on the circumstances. In dynamical system theory, it is this final aspect of entropy, third aspect of entropy, which is the most important one. For a chaotic system, the entropy is the most direct measure of non-determinacy, as it measures the amount of information which one needs in order to specify a long trajectory, independently of the particular coding used to describe the trajectory, provided only it is sufficiently fine. This information increases by ht.

T, where T is time and H is Kolmogorov Sinai or metric entropy. It is due to this linear increase with time that chaotic systems are impossible to forecast on the long run. Even if we know the initial state extremely precisely, there will come a time when this This information alone is no longer enough to allow any forecast. And that's basically this whole idea of logical depth basically comes into the picture. But this information aspect is also valid.

in equilibrium statistical mechanics. There, the entropy is the missing information needed to specify the micro state if the macro state is given. While the entropy is still the central concept in chaotic system theory, it has been realized more and more during the last few decades that it does not tell the whole story. There is a widespread feeling that besides entropy or randomness, there exists something which we have so far examined under the rubric complexity. So it seems that we have to get a better understanding of what this is, what this idea of complexity is at the deepest level.

Now in mathematics and computer science, there exists a quite elaborated theory of this informational complexity, a la Boltzmann, Shannon, and Kolmogorov. And it might seem at first straightforward to apply the concept developed there to physics, but this is not quite true. Now the most popular definition of complexity of a string of symbols, the algorithmic or Kolmogorov-Chaitin complexity leads in the cases we are interested in just back to entropy. The Kolmogorov complexity of a string S, let me bring this so we can, the Kolmogorov

complexity of a string s of n symbols with n finally tending to infinity is defined via the shortest string of bits which can produce s as an output on a general purpose computer. This definition was made for strings such as the bits in computer program computing some well-defined function, or as the digits, 3, 1, 4, as in Brighton of pi number. First of all, neither the bits of a well-written program nor the digits of a small n can be random, as they refer to something very specific. Moreover, although the digits of n look perfectly random

to a statistician, the required computer programs are surprisingly short of length proportional to logarithm capital N only. Therefore, randomness in a statistical sense and complexity via a program length are very different here. The reason for this difference is that it is much easier to write a program which gives the first n digits, capital N, of a small n than one which gives n consecutive digits starting at some random position. It is essentially the latter which is tested by statistical tests. This difference does not exist for symbol sequences obtained from dynamical systems,

provided the system was time invariant under translation, and the initial configuration was randomly chosen from a stationary distribution. There it is obvious that coding of a sequence can be made efficient only by using correlations which exists between the symbols, and hence, both concepts are the same. This applies, by the way, also to two or three dimensional patterns generated by spontaneous pattern forming processes. Also here, Kolmogorov complexity is identical to Shannon entropy, and its use is, though not wrong, misleading. More precisely, Shannon entropy is the average of the complexity.

But in these cases, one is only interested in averages anyway. If you want, therefore, a complexity which is not equivalent to entropy in the cases that we are interested in, and as it seems, many take it intuitively granted that such concepts exist, we have to look for something else. The direction where to look is suggested by computer science. In an admittedly vague sense, we can define The complexity of an object, now this object can be a pattern, a string, machine algorithm, so on and so forth, is the difficulty of the most important task associated with this object. For instance, the space complexity of an algorithm is the amount of storage on a general purpose

computer which it needs, i.e., the difficulty to implement it, while its time complexity is the time it requires to basically execute this task. The Kolmogorov complexity of a sequence is in particular the difficulty of uniquely specifying the entire sequence. Accordingly, it seems at first sight to agree perfectly with this definition. But specifying a sequence or a pattern is not necessarily the most important task related to it. Much important might be to understand it, i.e., to describe its meaning, basically the content

of the information, descriptive content of the information. Now the problem with making the latter into something which a physicist can work with is of course meaning and understanding are not well defined concepts in physics. A measure of complexity in this spirit is Charles Bennett's logical depth. The logical depth of a string s is essentially the time needed for a general-purpose computer to actually run the shortest program which generates s. For a random string, the time needed to generate s consists essentially of the time needed

to read in the specification and is accordingly proportional to its length. In contrast to this, a string with great logical depth might have a very short program coding for it. But decoding the program takes a very, very long, much longer than the length of s itself. The prime example of a pattern with great logical depth is presumably life, biologically defined. As far as we know, life emerges spontaneously, i.e., with a program which was assembled randomly and which therefore had to be very, very short.

But it has taken some 10 power 9 years to work with this program until life, biologically assumed its present form, basically its biological form. A more formal example with presumably large logical depth is the central vertical column in the following figure. Can you see the picture? Yep. OK. Now, this is an example from Wolfram.

This figure is obtained with one Wolfram's elementary cellular automator, rule number 86. In the cellular automaton, one starts basically with an infinite horizontal row of zero and one. And it rates by adding in each time step another row under and one, and it rates by adding in each time step another row under the previous one, according to a fixed local rule. Now in rule number 86, one writes one under each of the triples, 1, 1, 0, 1, 0, 0, 0, 1, 0, and 0, 0, 1, while one writes 0 under every other triple.

This figure is obtained by starting with the row 0, 0, 0, 1, 0, 0, 0, and by drawing a black square for each one. Since both the initial configuration and the rule are very easy to describe, the center The central column has zero column over of complexity. From very long simulation, it seems, however, that it has maximal entropy, just as the digits of number pi. Moreover, it is believed that there exists no other way of getting this column than by direct simulation, since it takes approximately n squared operations to iterate n time steps.

We find that the logical depth is large indeed. Now a very brief thing is that Bennett himself defines logical depth as the running time to generate the object in question by a near incompressible program. He states that this is intended as a measure of the value of information within the system. He says, logically deep objects contain internal evidence of having been the result of a long computation or a slow to simulate dynamical process and could not plausibly have originated otherwise.

End quote. Now, the plausibility of its origin of this concept of logical depth comes from the assumption that the most likely program to produce an output would be the shortest one. This idea comes from, and I will talk about extensively this in the second module, comes from Henry Solomone of work, basically, on information algorithmic complexity. He justified this as a physically plausible measure of complexity by its obedience to a slow-growth law of complexity. A slow-growth law of complexity can also be seen in terms of causal history of basically a system. This informational

law states that complexity can only arise slowly through stochastic processes, as presumably in evolution in the guise of causal histories, causal histories of life, causal histories of cognition, so on and so forth. By its construction, one cannot produce a deep object from a shallow one by a deterministic process and only improbably by a stochastic one. Thus, random strings and very simple ones both have a low logical depth. A random string is incompressible, and hence the minimal program that produces it is a simple copying program, which is quick.

A simple pattern can be produced by a simple program, and so will also be fairly quick. Now, one problem which the logical depth, you know, shares with Kolmogorov measure of complexity is that both are not effectively computable. In neither case, one can ever be sure to have found the most efficient coding of what may look like a random pattern. For this reason, one can only get an upper estimate for the Kolmogorov complexity of some not yet understood pattern found in nature.

For the logical depth, this problem is even worse. There one cannot even be sure whether one's estimate is an upper or lower bound, as one could find shorter programs which need either less or more time to execute. Now, it's clear that once the notion of the concept of logical depth as one of the basic and very prevalent measures of complexity is very tight to concept of hierarchies, concept of causal history and basically interlevel correlations.

But I think also we need to be very careful the way that Bennett talks about this, and I have seen it so many people actually very nondiscriminately apply logical depth to any kind of hierarchy. The way that Benet defines this measure of complexity is essentially, basically, assumes a kind of, you know, a kind of basically a, how to say it, a kind of global, basically,

in computability of the hierarchical picture. But when we are looking into basically the nature of hierarchies, what we are really interested in is really basically the computational cost of one hierarchy basically functioning on top of the other. So people usually, when they basically, for example, like I said, like Nick Jobo, he tries to interpret, for example, things like basically tradition as a logically deep object, the concept

of logical depth, and also attributing the concept of computational cost in terms of, for example, what would be the cost of us changing this part of tradition with something new. The whole thing is that this kind of interpretation is shady basically at the base, Precisely because Bennett's idea of logical deep object is globally incomputable. Whereas the computational cost interpretation essentially assumes a computability of basically

the intro-level hierarchical ordering. I'm going to talk about this in the second module much more extensively in terms of these kinds of interpretations that what arises when we apply a measure of computational complexity that has in-computability assumptions to basically to properties of complex systems or measures of complex systems that basically

are based on basically effective computability. Basically, you can't even talk about things like the computational costs and hierarchical systems in that kind of inter-level scenario without having a paradigm of computability, effective computability. Yes, you need to have a general, basically, computability assumption while having a special in computability cases. But you cannot have a global in computability, general in computability assumption, like as implicit in logical depth measure of complexity.

Another thing that before, this was just a very, very basic introduction on this idea of logical depth and deep objects in terms of entropy, informational content, and computational complexity. It was a basic introduction for our next session, which I will talk about this much more in details in relation to the other two main measures of complexity. But one of the things before finishing the session would be just a kind of like as a kind of in order for you to be able to think around problems and ideas

implicit in this idea of logically deep objects, would be think of this question as an exercise. Think of an axiomatic system. And this axiomatic system can be, for example, a geometry book, basically an axiomatic theory based on sets of certain axioms, for example, Euclid's axioms. or any kind of other axiomatic system that you can think of, arithmetic axioms, geometric axioms, philosophical axioms, so on and so forth.

Then think about this, whether an axiomatic system can be thought as a logically deep object or not. And there is a reason that I'm asking this question as an exercise. And the reason will only be clear when we talk about this, kind of address this problem of axiomatization in computer science and the axiomatic concept of program in the second module. But nevertheless, as a kind of a philosophical exercise, thinking exercise, it would be great if you can think about this throughout this week.

Think whether an axiomatic system or an axiomatic theory can be considered a logically deep object. If yes, why yes? If no, why no? And also make clear what you mean by an axiomatic system, because there are different forms of axiom and axiomatic system. But nevertheless, I want to hear your answers to this question, because we deal with this problem quite significantly in the second module. So any questions, answers before we conclude this session? and move to the more serious stuff in our final session

of complexity in the computer module. I was curious about the logically deep object being something that could it express a static form? Does it have to be a process or could it be something... Well, a static form is... These are, I think, idealizations, you know, the idea of the static form. I mean, the idea is that, you know, basically the concept of the logical deep

or logical depth is a processual computational object. But so as everything else, basically everything can be understood processually. Well, you need to basically all you need, and that's the whole. I will talk about this in second module. One of the main difficulties currently in computer science is that there is no canonical definition of processes. And when it comes to processes, there are 700 algebras of processes, but there is no canonical definition. Still we do not know what exactly a process is. As long as you cannot define process and make it context sensitive, then yes, there are

basically the notion of processuality becomes almost trivial. But yes, everything can be said to be a process wall, basically, in that sense of logical depth. So the whole idea of the aesthetic versus process is more of an idealized distinction, rather than something that has an objective reality to it, because everything is really process wall. That's a... I was trying to see it in terms of distinguishing the structure and the function, and the function being more connected to the process and the structure being something that could express stasis? Is that helpful at all? Not really. No. Process is, in fact, more on the side of the structure, interestingly.

functions come, you know, again, this is something that I will talk about. Functions come in different varieties. There are some functions that have a strong coupling with a structure, basically are kind of like tethered to their structural substrate, which also means that the kind of processes drive those structures. But there are also functions that are capable of constituting their own processes, basically giving rise to new forms of structure. And basically a lot of functions in complex systems are of those kinds, are the ones that

capable of weakening their relationship with their specific structural substrate. That's the destabilization? That's a weakening. That's not the destabilization. That's just basically an idea of basically what we talked about in terms of scaffolding and canalization. That's one of the things that happens when functions becomes extremely complex, becomes extremely specialized. They weaken basically.

They give rise to new structures. They introduce constraints, enabling constraints, basically, in the hierarchical depth, hierarchical structural depth. And then they create revision, basically. They initiate structural revisions in the hierarchy. The thing is that stability always needs to be in place, I mean technically understood the word stability, within a structural hierarchy. Otherwise, you don't get anything like hierarchy, because hierarchy is an index of stability,

and that's what we are going to talk about in terms of statistical stability, a statistical complexity and structural instability as one of the most fundamental measures of complexity. That's great. How much are we going to talk about stuff like adaptation, the impact of cycles of contingency, that kind of thing? I will talk about them in the second module, but more under the rubric of this computational

picture of Darwinian dynamics. I think when we are talking about contingency, I hope you do not mean it in terms of absolute ontologized contingency, because we won't talk about that. But we will talk about, yes, contingency proper in terms of, for example, Darwinian selection is a good example of contingency. Yes, we will talk about this in terms of the computational picture of this kind of interaction and the kind of constraints that we talked very briefly this session, the kind of constraints that it imposes and basically starts to guide the system dynamics.

Yes, we will talk about this, but not specifically under the complexity modules. I think because there is a reason for it. Because when we are talking about complexity phenomena, as we talk about, there is no such thing as a complexity phenomena, as a unified global complexity. So we need to kind of fine-grain the discussion a little bit, bring it to that computational level, so we'll be able to have a kind of a fine-grained understanding of how this works, really, rather than just kind of like a kind of microscopic complex phenomenon talk about this adaptation and contingent, basically, positionings.

Cool. By the way, I'm not, I mean, this is one that I ask. How many of you have read Kant and Hegel. I mean, yes, yes, yes, okay. I'm really behind with that. Every time I read Hegel, I just think it's so racist. I just can't even take it. Well, hey, yes, no, I understand. Well,

Germans. That's the problem with Germans. But yes, but I think we can actually, you know, I think his racism is, when you really look at, for example, phenomenology of his spirit as in contrast to like, you know, the more explicitly political stuff, which he stuff, which he talks about the nation state and really puts out his religious dogmas out there, is that so many of these racism and stuff are essentially like time-driven biases. They shouldn't be taken that much seriously. In fact, I think you can extract the main core of the Hegelian system from some of these

racist agendas. But I would say that it is very difficult to rescue Hegelian system in its entirety from his religious doctrines. A good book, if you want to read Kant and Hegel, and if you find it very difficult, because they are very difficult, regardless of what background you are. For the most Kant and Hegel geeks, even though these books are very difficult, a very good accessible book to pre-Kantian themes, Kant and Hegel, is a book written by Jay Rosenberg, simply called Accessing

can't. Hegel, what is that thing? Hegel, I can't read it. Oh, what I typed, it's unrelated to. Oh no, there is, I can see another book, what's that? Hegel, okay. Okay. I don't know about this book. Oh, this is a recommendation for Jessica It's a off topic from this, but on the topic of race. Are you talking about this one? Yes, that one. This is about, I haven't read it yet, but it's about universality and universalism in general. Okay, excellent.

Can you type it in the comment box? It's by a Gellian scholar called Susan Bach Morse. I think it's the one you mentioned, right? Yes. I just bought it. She's really interesting. Yeah, I like her a lot as well. Jay Rosenberg is called Accessing Kant. And Jay Rosenberg is a monumental philosopher, one of the most criminally undervalued, obscure philosophers. He was a student of cellars, and he's really systematically unraveled Kant and Hegel,

really makes them comprehensible and really makes them distinguish why they are such important philosophers and why basically all of this stuff that we're talking about have already been implied by Kant and Hegel, possible pitfalls, possible points of development. But most importantly, when you're talking about AI and constructing human level AI, it's basically the stuff that, for example, people like RL and good AGI experts talk about are essentially Kantian programs. Admin question.

Are this slides anywhere uploaded yet? Tony and or Reza? Oh, no. I had a bit of a catastrophe. I have this apparently very automatic, well-formed system of archiving, but for one reason, I saved these new slides on my previous slides. So what I need to do, I need to get from Tony the password to my presentations and back sessions and then I will, I know how to rewrite them. It's not that much work.

And then as soon as I do that I will upload them. Sorry, yes. It happens all the time with my writings. I save versions on previous versions and I lose a massive amount of work all the time. I'll email it to you, Reza. OK, excellent. So guys, if you don't have any questions, I will let you go so I can have my lunch finally. Thanks, Reza. Thank you very much. Thank you. Thank you. OK, thank you. bye bye