Building A Theory Of Everything | Stephen Wolfram | Escaped Sapiens #70

We think we understand the very low-level machine code of the universe. But getting to stuff that we recognize, and even that has been analyzed in existing mathematical physics, that's a big, long, you know, effort. Where do the fundamental laws of nature come from? That's what this conversation is about. I speak with mathematician, computer scientist, and theoretical physicist Steven Wolfram about his attempt to derive the laws of nature starting with almost nothing.

Now, these ideas are outside of mainstream physics, and they're not fully fleshed out. But the claim is that some real progress has been made. So that's what this discussion is about. It's an overview of some of the main ideas for a candidate Theory of Everything. I'm Shane Fonsworth, and this is the Escape Sapiens Podcast, supported by the Andrew Fon Brown Foundation. If you enjoy this content, you can support it directly by liking, subscribing, and sharing. And now, here's Steven Wolfram. I, I hope you enjoy Escaped [Music] Sapiens.

So let's start with a few definitions that will pop up again and again in the conversation. What is computational irreducibility, and why should people care about it?

Right. Actually, it turns out it is almost exactly 40 years old as of later today. The very first time I wrote the words "computational irreducibility" happened 40 years ago later today, so to speak. So, what is it? So, when we think about how we figure out how a system behaves, we, we can say we may know the rules by which the system operates. For example, we may know some rules for some program. We know the rules by which it operates. The question is, how do we find out what the system will do? So, let's say we want to know what will happen after a million steps of running these rules. Well, one possibility is that we can just work out a formula and say, "Okay, we know after a million steps, we just fill in the number a million into some formula, we get the answer immediately." That's kind of the, the concept that has been typical in kind of mathematical approaches to science, that it's kind of like you can solve it and get a formula for what will happen.

What computational irreducibility is about is the fact that in many systems, you can't actually jump ahead and see what's going to happen. The only way you can work out what's going to happen is to follow each step and just see, see what will, will happen. So, why is that important? The, essentially, the, if you look at kind of the, the, the big arc of sort of history of science, the, um, the sort of the, the thing that really launched a lot of modern science was the idea of introducing mathematics into, into science. And sort of the concept there was, "We'll just work out a formula for what, what's going to happen. We'll use calculus or something. We'll say this is, you know, this is the formula at, at time T. We can work out the result." And that's been kind of a dominant theme in sort of the exact sciences for the last 300 years. And it's led people to believe that, "Oh, you know, if you're doing epidemiology, you've got these formulas, and you can just work out what will happen." Kind of the assumption is with mathematical science, we can always jump ahead and see what the answer will be.

Computational irreducibility kind of denies that. Computational irreducibility tells one there are systems where there is no way to kind of jump ahead and just say what the answer will be. It's something where to know what the system will do, you essentially have to follow every step, just as the system itself behaves. And so that, that's a, that's a pretty important idea. It turns out because it kind of gives one, from within science, it kind of shows one a fundamental limitation of science. It also has some very positive aspects. For example, you know, we lead our lives, we, you know, we progress through time and so on. If the, if we could always say, "Oh, we know what's going to happen in the end. In the end, the answer is going to be 42 or something," there would be nothing achieved by the leading of our lives and the passage of time. But what computational irreducibility tells us is that actually there is something irreducible that's achieved by the passage of time, by kind of the progression of, of, of things that happen through the course of, of, of time in our lives and so on. So it's kind of a, on the one hand, it tells you that the bad news is it tells you you can't always know what will happen. The good news is it tells you there's something that, that, that you can't always know what will happen, and that there's sort of a thing achieved by actually progressing through that time.

Now, you know, there are many, many places where computational irreducibility shows up. Like, for example, if you're interested in AI and you're interested in, "Can we set it up so that AIs will only do what we want them to do?" Well, if you have computational irreducibility, you'll never succeed in doing that, because there will always be a way that the AI, as it actually progresses and does what it does, will surprise you. And so I think one of the things that, for example, will be a big, kind of societal debate is to what extent do we allow computational irreducibility for AI type systems? If we say, "No, they, they've always got to be predictable," well, fine, then they'll only do what we want them to do, but they won't be able to do a lot of the powerful things that they would computationally be able to do. So it's kind of like, do you want computational irreducibility, or do you want to sort of fit everything into this reducible box where you can know what's going to happen?

So, you know, you might ask, I mean, there are, computational irreducibility has many consequences. And, you know, at, at some level, it has very basic kinds of consequences, like saying, "You know, why are there bugs in programs?" Well, because most, you know, it is hard to foresee what programs will do. That's a, that's a kind of computational irreducibility symptom. The thing when I was inventing computational irreducibility, uh, you know, I thought, I thought of it as kind of a limitation on science. But, for example, one of the things I absolutely never imagined is the idea of proof of work in blockchain, for example, which is essentially a computational irreducibility idea. And, it's, you know, and that because that's a thing where one is, one is taking irreducible effort, and one is kind of to, you know, do Bitcoin mining or whatever else. And that's, so there's again, there's something is being done, and that something is sort of this irreducible computation. Something I certainly didn't imagine when I, when I came up with computational irreducibility.

Another, another aspect of it is, I mean, it, it just has many, many consequences. But another one that I studied, uh, quite a bit, is trying to understand things like the second law of thermodynamics. The fact that when you start systems off in a kind of orderly way, like gas molecules sort of arranged in an orderly fashion, they tend to become more disordered over time. And the question is, why? Why does that happen? And what's happening is that they are effectively, the, the collisions of these gas molecules are implementing an irreducible computation. And that means that when we look at it, because we are bounded in the sort of computational work that we can do in perceiving what's going on, we can't decode kind of this irreducible computation that happened. And so we just say, "Oh, it looks random to us." And that's kind of the essence of the second law. Is that there's this mismatch between the irreducible computation that's happening in these actual molecules computing where they're going to go, and the bounded computation that we are able to do as observers of that system.

And it turns out this, this interplay between the computational irreducibility of underlying systems and our computational boundedness as observers of those systems, that seems to be the key thing that basically leads us to the laws of physics that we have. This is something I absolutely did not see coming. And it's, I think it's a very remarkable thing that the, the, sort of the three key theories of 20th century physics, statistical mechanics, second law of thermodynamics is one of them, general theory of gravity and spacetime, another one, and, uh, quantum mechanics, the third one. It turns out, to my great surprise, I think all three of those theories are derivable, not just things that happen to be the way they are, but they are derivable from sort of the underlying computational, uh, character of the universe, effectively of, of, of sort of an inevitable computational structure combined with the fact that there is computational irreducibility interacting with essentially our boundedness and other characteristics of us as observers. So, in other words, for observers like us, it becomes inevitable that we must perceive the laws of physics that we, that we know, so to speak. And that, that, that the fact that that works is one of the important components of that is computational irreducibility. Computational irreducibility is what leads us to believe in things like the gas laws, to not have to say, "We're going to look at every individual molecule. We have to know what every individual molecule does." We are only able to look at this kind of aggregated view of a gas, and we're only able to do that because what's underneath is computationally irreducible. And the same thing seems to be true for spacetime. That the fact that we believe in the continuity of space, for example, is a consequence of the fact that, sort of underneath, there's computational irreducibility. And because we are computationally bounded, we can't kind of get to see all those details, and we just have to summarize things by saying there's, there's continuous space. And it's the same type of story with quantum mechanics. We can talk about, talk about in more detail.

But so, so that's, that's kind of the, the key idea of computational irreducibility is that even though you may know the rules by which a system operates, that may not tell you what the system will do. You may have to just explicitly follow those rules. In a sense, you have to do the same amount of computational work that the system itself has to do to figure out what it will do. And so the way that, sort of, why is, why, why is there computational irreducibility? Well, the answer, I think, is you can see it as a consequence of this thing I call the principle of computational equivalence. And what the principle of computational equivalence says, if you look at all these different possible systems, then above some very low threshold, all these different possible systems will be equivalent in the sophistication of computations that they can do. So, in other words, we can imagine a microprocessor in a computer, we can imagine a human brain, we can imagine a bunch of molecules bouncing around, we can imagine some simple computational system, some little program that you can, you can specify with black and white squares or something like this. Any one of those things will be able to do the same level of sophistication of computation. And that's closely related to the fact, the idea of universal computation. It's also closely related to things like Gödel's theorem. Um, it's a, it's a slightly tighter version of those kinds of things.

Okay, so if one has this principle of computational equivalence, what does it mean? Well, what does, what does predicting a system consist of? The system is doing what it does. It's computing what it's going to do. And you, as the predictor of the system, are going to be smarter than the system. The system is doing what it does, it goes through a million steps or something. You're some going to be able to do some smarter thing and conclude in five steps, this is what the system is going to do. Well, the principle of computational equivalence says that that isn't going to work, because it says that whatever means you have to do the prediction, whether it's your brain, or mathematics, or a computer, or whatever else it is, it is computationally equivalent to that little system that you're looking at. And so you can't expect to kind of jump ahead and predict what it's going to do. You basically are stuck having to follow step and, and just do the same computation that it does. So it's kind of a, the, the reason why computational irreducibility happens is because there is this one sort of uniform level of computational ability that exists, and at least in, in things like our universe, so to speak. And, and that's, uh, sort of embodied in this principle of computational equivalence.

So that, that's, that's, uh, it's, you know, I have to say, in the, in the 40 years since I originally kind of came up with, I, I actually, the idea of computational irreducibility was a few years older than that, but the term computational irreducibility is almost exactly 40 years ago, um, today, um, but, you know, it's been, it's been interesting the extent to which, sort of, that way of thinking about things has affected, uh, kind of has, has found its way into all these different domains and all these different issues. And I mean, there, you know, they're just, there's so many of these things, like, for example, another one is to ask the question, kind of, things like, "Will there be, will there be an end to science? Will there be an end to mathematics? Uh, will there be, will it come to be the case that, sort of, AIs can do everything and there's sort of no place for the humans?" Type thing. These are all questions that are deeply informed by this idea of computational reducibility. And if you don't, if you don't know about that, you kind of might conclude, for example, that, uh, you know, you'll, you'll finish science and you'll be able to answer every question, so to speak.

And another one that's come up very recently is the question of, you know, with, with AIs, can you just solve all of science? You just find an AI that's, you know, a million times smarter than us humans in some definition of smartness, and it'll just say to any science question you might ask, to say, "I know the answer." Um, and computational reducibility basically tells that that won't happen. Um, there's a small footnote to that that was quite interesting, which is it won't happen if you imagine considering, sort of, all possible science questions. If you restrict that to science questions humans care about, it's a slightly different, there could be a different answer. But, um, in the, so, yeah, I mean, I think this, this notion of, I mean, I, I, in a sense, as I was mentioning, one can see computational irreducibility as kind of the one of the key pieces of, sort of, a different paradigm for thinking about the world and about science that we've had for the last few hundred years. I mean, in, before, it's kind of this question of how much can we understand about the world? And, you know, if you'd gone back 500 years, many people were like, "Well, there's a lot of things we'll never understand about the world. They just happened to be that way." And then there was, "No, we've got all these mathematical ideas. We're going to crush it. We're going to understand everything." And by "understand," what tended to be meant was, "We'll be able to predict everything about what will happen." And that got, sort of, enshrined in various people's ideas of what science fundamentally is about: prediction and so on. It's not applicable to all areas of science, but in an area like physics, for example, that's a, that's a typical kind of, "If you know what you're talking about, then you will be able to predict what will happen in the world," so to speak.

And computational irreducibility is kind of, I think, sort of the next step in that, in that journey, because it tells one, "Yes, there are things you can predict. There are slices of reducibility. There are actually an infinite collection of slices of reducibility. But there's also kind of a, a kind of a background of things that you cannot predict." And, and that's sort of the, the, you know, so that, that gives one a different view of what science is about, what science is capable of doing. And it has, well, both an intellectual implication and also at some level, kind of a societal implication. Because in a sense, there's, you know, there's a, there's a kind of people have often believed, "Oh, you know, if we could solve society, we could make the world a better place," so to speak. What computational reducibility tells you is, you'll never be able to quote, "solve society." The concept of a kind of a controlled system, it just is never going to work, because when you think you have this controlled system, something, because there's computational irreducibility, there will be an infinite collection of surprises, and you'll never be able to get this kind of closed, "we know what every, how everything is going to work," kind of system.

So, one of the main points that you bring up, that you brought up, is that the laws of nature that we develop, they really depend on who the observer is. And so, how do you know what is fundamental? So, you mentioned, for example, uh, um, in your writings, you mentioned that even the three-dimensional space that we live in might be emergent, and there's some underlying computational structure from which it develops, right? So, how do you know what is fundamental if, if everyone is, if we're just observers, right? All we have is the perspective that we have. How do you know that space, for instance, is not fundamental, if that's the thing that we really see?

Well, it's a reasonable question. I mean, it's kind of the, it's, it's kind of, what is the ultimate abstraction? What is the, you know, what is the thing that is, you know, is there an ultimate abstraction? And I think the answer is that, you know, I, I think we can, computation, at some level, the following of rules, is the closest I think I can come to giving you kind of an ultimate notion of abstraction. That if you say, if, if you imagine that things follow definite rules, we're not saying what those rules are, we're just saying things follow definite rules. At, at some level, things follow definite rules. Now, actually, it's more complicated than that, because what I actually believe now is that things follow all possible definite rules. It's not like you can hold one up and say, "This is the rule for the universe." Instead, in this concept that I call the ruad, it's this kind of object that essentially runs all possible rules. And what your, the ruad consists of, kind of the entangled limit of the running of all possible computational rules. So, in a sense, there is still, you know, in even defining that, I'm kind of using as a basis for what I'm talking about, concepts like rules and computation. And there might very well be a different way to formulate the same object using a completely different way of, kind of, presenting it. But the thing that, you know, the thing that that I have believe is kind of the ultimate fundamental thing is this ruad object, which I could, I at least think of as the entangled limit of all possible computations.

And the, you know, by the way, just to make it even more complicated, that's the ruad. There's also, one can imagine the hyper ruad. One can imagine a thing which, in our view of computation, we could say, "Let's figure out what's going to happen after an infinite number of steps of this computation." Okay, we can't determine that in any finite number of steps, but we can imagine, as actually Alan Turing did at one point, we can imagine that we have an oracle which just tells us the answer, and we add that onto our computer. So what we end up then with is a sort of hypercomputation, in addition to the, the standard computation that we do, where we follow every step in the rules. We're also saying, and we're adding on this black box that will just tell us the answer to what happens after an infinite number of steps. And so we can go, in addition to the ruad, which is this thing that's the entangled limit of all possible computations, we can imagine a whole hierarchy of hyper ruads that that embody the, the entangled limit of these hypercomputations. And so you say, "Well, well, why, you know, why did we get the ruad and not one of these hyper ruads?" It turns out that, for example, it's just a matter of, kind of, science and computation theory and so on, that there's sort of an inventor's RIZ between our ruad and any hyper ruad. So there was no, there's no connection. There can be no computation that goes from the ruad into the hyper ruad or vice versa. And in fact, it's probably the case, although this isn't something we've completely figured out, that there is a kind of Rael relativity. That as an observer in any level of hyper ruad, it will look the same to you. In other words, what, what's tricky is you have this ruad, which is supposed to represent, sort of, everything that's possible and everything that, in a some sense, happens. But we, as observers, are embedded inside the ruad. And so it's this, it's this elaborate question of, "How does an observer within the ruad perceive the ruad?" And the answer is, it depends what that observer is like. If we didn't say anything about what that observer is like, we wouldn't be able to make any statements, because the ruad contains all possible computations, and we just say, "We don't know what's, you know, everything is happening." But the fact is that because we are observers of the kind we are, that that forces us to perceive certain things about the ruad.

So, simple example of this, we talked about the second law of thermodynamics. The fact that we are observers who are computationally bounded means that we don't get to follow every individual gas molecule, and we believe in the second law of thermodynamics. We believe that what tends to happen is that we will get this computational irreducible process which, which we can't decode, and it'll look random to us. If we were a different kind of observer, if we were an observer who traced every molecule, who was very small and had lots of computational ability, we wouldn't particularly believe in the second law of thermodynamics. It would be something which somebody would tell us, "Oh, you know, if you looked at all these molecules in the aggregate, this is what you'd see," and we'd say, "Well, okay, we, you know, that's not something we notice." I mean, by the way, it's, it's really, kind of, notable how much the way that we perceive the world depends on our means of perception. So, for example, space, which we think of as a very, you know, it's obvious space exists, we might say. And it's obvious that as we look around us, that we see, kind of, a slice of, at a particular, we see a region of space at a particular time. And then, you know, time passes, and the, we see a region of space at a different time. But that depends a lot on a bunch of scales. So, for example, if we look around, you know, if you're in a room, you might see 10 meters away. Well, it takes light, you know, milliseconds to get to our eyes from that distance away, and it takes us milliseconds to process what we've seen. So, for us, in the, in the environment that we're looking at, it's as if we see instantaneously. We see this lump of space, and then we see that changing over time. We kind of see this, every movie frame, we see a complete movie frame, and then we see it progressing through time. But if we thought a million times faster than we do, you know, digital electronics runs a million times faster than our brains do, um, if we thought a million times faster, we wouldn't have that perception, because the light travel time from the distances we're looking at would be significant, and we would be seeing, you know, this cascade of photons arriving. And if somebody tells us, "Oh, by the way, you can think of that as slices of, of, uh, of space progressing through time," we'd be like, "Well, that's kind of interesting, but it's not the way that we perceive things."

So, you know, there is a, a great, this, it's, it's not surprising that the way we perceive the, the kinds of things that we talk about and the kinds of laws that we ascribe to the universe depend on, kind of, the way in which we're perceiving things. And what, uh, you know, what seems to be the case is that for the three, sort of, key laws of physics that, uh, that we kind of learned in the 20th century, that two attributes of us as observers seem to be critical: one is that we are computationally bounded, that we can't decode all that computational irreducibility, and the other one is that we believe we are persistent in time. So, in other words, even though at every moment we're made of different atoms of space in our models, we still believe that it's the same us, and we have the single thread of experience going through time. And it turns out those two characteristics allow one to derive, M.H., in precise form, these three core, kind of, laws of physics that arose in the 20th century, which is really surprising.

Can, can I ask you about that? When you're building your models, how do you introduce those assumptions? So, how, how do you introduce, so you start with the ruad, in some way, um, which is the space of all computations, or? Yes. And so then you somehow introduce the notion that there's an observer that's completely bound, that's not completely bounded, that is bounded in some sense? And then from that assumption, plus some other assumptions, you can derive, for instance, that space is three-dimensional? Or that? No, we cannot derive that. We don't know how to derive that. We can derive that space satisfies, uh, general, satisfies the Einstein equations, including Lorentzian signature. Yeah. Yeah. Yes, you get the signature as well. Absolutely. Because in our models, so one thing, not uncommon in the history of science, you kind of have to go back before you go forward. The fact that people concluded in 1919 that, uh, that no, actually earlier than that, it was 1909, maybe, that space and time could be bundled together. I mean, that wasn't an original idea of Einstein's. That wasn't in the 1905 kind of special relativity package. That was a subsequent, kind of, mathematical gloss on that, introduced by Minkowski, actually, that, kind of, you could think of space and time as sort of both facets of the same kind of thing. In our models, they are absolutely not facets of the same kind of things. Space and time are completely different. Space is associated with, well, time, I already described. Time is kind of this, this progression of computation. The passage of time is the doing of computation. So, what's space? Well, I mean, that's a slightly longer story.

So, the beginning of that story kind of starts in antiquity, when everybody was arguing about, you know, "Is the world discrete or continuous?" And, you know, "Were there atoms? Was there just a flow? How was the world made of? What was the world made of?" And it wasn't known for a long time. Finally, by the end of the 19th century, it became clear that matter is discrete. There are molecules and atoms and so on. And then pretty close to that time, it became clear one could think of light as being discrete as well, made of photons or whatever. At that time, in the early 20th century, as I've come to really understand this history better in the last couple of years, most people believed that space was discrete. Most people thought that just as matter was discrete, so space would turn out to be discrete. But nobody could make the mathematics work out. In fact, you know, people like Einstein made statements like, you know, "In the end, it will turn out space is discrete, but we don't currently have the tools to see how to make this work." Well, 100 years later, we finally do. And that's sort of an underlying idea in these kind of computational models that, that, that we built. And so the idea is, what is the universe made of? Well, one view of what the universe is made of, probably not the only possible way to think about it, but one that I found convenient, is you just imagine there are these atoms of space. Just these elements that are discrete points. Just a big collection of discrete points. They're not in space, they, they are what makes space. How do they make space? Well, these points have relations to other points. It's as if they have a giant friend network of the atoms of space. And you can represent that friend network as a graph, a network. Actually, formally, it's convenient to have it be a hypergraph, where instead of just saying that that two, that that there are pairs of things that are related by being connected by an edge, that you can have any number of things related by hyperedge. But, so, so what one imagines is, what is the universe made of? The universe is just a giant hypergraph. And every feature of the universe is a feature of that hypergraph. So, it's kind of like, if you were looking at, I don't know, fluid like water, and you say, and you, and you see an eddy going through the water, that eddy is just made of those molecules of the water. It is a, a feature of, kind of, the, the collective motion of those molecules that there is this thing that we identify as an eddy. And so similarly, our belief is that, for example, quarks, electrons, quarks, whatever else they are, they are things like eddies, but in this hypergraph. We actually know how that works for black holes. We kind of know how black holes can, can exist in this hypergraph. And we can, and the, the thing to say is, so there's this hypergraph, and it has features that correspond to, we think, particles. We don't know how that works yet. We, we kind of know how black holes work there. And then what happens to the cyerra? Well, the cyerra is constantly being rewritten. So there are these rules that are applied to the hypergraph, and the rules basically say, "If you have a little tiny piece of hypergraph that looks like this, it will be rewritten to one that looks like this." And the passage of time is the application of all these little rules. And so that, um, and so then the question is, well, if you have all these little rules being applied, what is the kind of aggregate effect of that?

Could I ask, just quickly, is the idea that the, the smallest units here, all about the same rule, or is it that different universes correspond to different rules?

Okay, this is complicated. Okay. So, in the, in the, in the sort of the first way of thinking about this, everything obeys the same rule. By the time you get to the ruad, and that's another conceptual step when is dealing with all possible rules. But let's, let's say there is a way of describing our universe in which every update is by the same rule, or by the same small collection of rules. Um, it's, um, so, so the question is, if, in the case of a fluid, again, you've got these molecules bouncing around, and they obey certain laws and mechanics and so on. And you can ask the question, "What will be the large-scale behavior of the fluid?" At the small scale, you've just got a bunch of billion balls bouncing around. But what does that mean when you have a trillion of those, or a trillion trillion of those? What is the aggregate effect? Well, the mathematics has never been precisely filled in. And in fact, some things I've done recently fill in a bit more of it. But what we know is that the limit, the large-scale limit of molecular dynamics of molecules just bouncing around, is fluid dynamics. The large-scale limit is the, this sort of continuous fluid that obeys certain fluid equations and so on. Well, you can ask for the same kind of derivation of the continuum limit for this hypergraph that's being rewritten. And the first thing that I, I kind of knew in the 1990s, that, um, uh, the first, sort of, big fact is, the continuum limit of all that rewriting, with certain footnotes and so on added in, which we can talk about, but essentially, the continuum limit of all that rewriting is not the equations of fluid mechanics, but Einstein's equations for the structure of spacetime. So that's a, it's a really quite remarkable fact that from this kind of underlying microscopic, kind of, very simple structure, the, the limit of that is, is, is, is the equations that we know govern spacetime.

Now, there are many, many issues that come up. So, for example, you say, "Well, the universe seems three-dimensional to us, but this hypergraph has no notion of dimension." All it has is a bunch of points. And you know, there's the question, "How do we figure out even what effective dimension the thing has?" And the answer to that is, it's a pretty simple kind of mathematical construction. You say, you've got this big graph, and you've got a particular point in the graph, a particular node of, in the graph. You can say, "Well, that node is connected to some number of other nodes, a distance one." You just follow one edge to get to those other nodes. Then you, then you do it again, and you follow two edges, and so on. And if you say, if you go R steps from that given node, you ask the question of how many nodes do you get to after you went R steps. And so, if you, if you just think about a grid, for example, a two-dimensional grid, then as you go out, you know, R steps, you'll get some kind of diamond shape of where you got to. And the number of nodes that you get to goes up like R squared, just the, the area of the, of the, that diamond shape. So, and in three dimensions, it would be R cubed, and so on. And so what you realize is that the effective dimension of this hypergraph is the growth rate, is that, is that exponent of the growth rate of the number of nodes that you get in a ball, effectively, of a given radius. So that's an example of how you deduce from this hypergraph, if you've got a big hypergraph, you can generate one in a computer, um, you get a big hypergraph, you can deduce the effective dimension of that hypergraph.

Now, that, that, and then you go on, and, uh, actually, if you look at that growth rate of that, that volume of that ball, there's a correction term. The correction term is proportional to the curvature of the effective space, the Ricci scalar curvature, um, and that happens to be the thing that shows up in the Einstein equations, interestingly enough. And so that's sort of the beginning of how things, how things develop. Now, actually, the thing I said, it's a bit more complicated than that, because you're dealing with, kind of, growth rates not of balls in purely in space, but of cones in spacetime. And so it's just a little bit more ornate, but the idea is the same. And that's kind of how you start to derive the equations of general relativity, the Einstein equations, from this underlying structure. But it doesn't tell you, it, it simply tells you that, well, you, if you assume that things are finite dimensional, not infinite dimensional, that's one of the footnote assumptions that you have to make to derive the Einstein equations. Um, but one of the things that is a, a very interesting consequence of all of this is because dimension is now a dynamical parameter, you can have dimension change in the universe. So we think that the universe is three-dimensional, but actually, we're pretty sure in our models that the universe started infinite dimensional and just gradually became something that was about three-dimensional. And so one of the things that's of great interest is to try and find dimension fluctuations, and if we're lucky, they will have survived to the time of the cosmic microwave background, and we'll actually be able to see the effect of dimension fluctuations in cosmological measurements and so on. And so that, that's an example of, of, um, but, but, you know, so in our models, there's this, the, the idea that space, sort of, is three-dimensional is an emergent phenomenon. And I think that it gets a little bit more, we don't know how to derive the number three. Um, in the three dimensions of space. And, and, in fact, my guess is maybe I, you, you asked, is it one definite rule? Okay, so I have to explain that a little bit more.

So, the next thing to realize is that when you're updating this hypergraph, there are many different places where updates can occur. And there isn't a single, kind of, thread of time where you say, "Oh, this is the next state of the hypergraph. This is going to be the next state of the hypergraph." Because there are all these different places where updates could occur. And that, that fact is what leads to, kind of, the, the way that relativity works. That fact also leads to quantum mechanics. Because the essence of quantum mechanics is that, that it isn't the case that definite things happen. You know, in classical mechanics, you throw something, it goes in a definite trajectory. In quantum mechanics, kind of the idea is there are many paths that are followed, and we just get to be aware of certain probabilities in the aggregate of these paths. Well, in these models, there's, it's very explicit that there are different paths of history, because there are different ways in which this updating can occur. So, and, and sort of then there's a whole story of quantum mechanics about how we, as observers, who are ourselves branching to many parts of history, observe a universe which is, it, which is also branching through many paths of history. But setting that aside for a second. So, kind of the concept is that there are many different places where a particular rule could be applied, many different orders in which the rule could be applied. But then when we get to this whole ruad object, we're not looking at a particular rule being applied in many different places and ways. We're looking at all possible rules being applied in all possible places and ways. And so there's, and then there's the question of what, you know, from out of this ruad, how does, uh, you know, what, what slice of the ruad are we perceiving as observers like us, so to speak? And any observer who is computationally bounded and believes they're persistent in time will observe, with some footnotes, the Einstein equations for spacetime. But we don't know how many dimensions that'll be in.

So my guess is that there is another aspect of us as observers that's probably something unbelievably obvious to us, um, that we effectively implicitly assume that leads us to have our perception of this ruad object be the one that corresponds to three-dimensional space. So, you know, there are many aspects of, of the way that we perceive the universe that seem totally obvious to us, but actually are not so obvious. Here's an example. One example is, we can do any experiment we choose. We are not constrained. We, we have free will to do experiments. That's not at all obvious. It could be the case that the reason we see what we see is because we are forced to do just this experiment, then this one, then this one, and we're not free to make a choice about what experiments we do. That's an example of another assumption that we make about how we exist in the universe, so to speak. Another one is that we believe in objects. We believe that there are things that have some unchanging character to them. So, for example, we believe in pure motion. We believe you can pick up a thing and move it, and it's still the same thing, which is not obvious. And, for example, even in traditional general relativity, if you're close to a spacetime singularity, things don't stay as things, they get sort of arbitrarily ripped apart. But the fact that we believe in pure motion, we believe that there are objects, is another one of these implicit assumptions about the world that has consequences for the laws of physics that we perceive from this ruad, from the slice that we're taking of this ruad object. I don't know, you know, it's one of my, my current things to think about is kind of, what, what are those implicit assumptions that we're making that end up feeding into our perception of the world, so to speak? And, and what things are, as I say, given even very coarse knowledge about our assumptions, we can already derive quite precise facts about the kinds of laws that we will perceive. But, you know, it's a question of, well, what, what other things are we assuming that lead to other aspects of, of what we perceive in the universe?

So, one thing that I'm rather curious about here, in terms of your setup, is, so, in, in a geometry, in a geometric space, the points are coherent in the sense that I can derive what the distance between various points will be if I know enough information about other points in the system. Let's say, like, if I know two angles of a triangle, I can work out the other angle, and so on. Yep. Um, in your initial setup, is there initially a coherence that you put in, or the coherence is derivable through this computational, uh, development?

Yeah, it has to be derived. I mean, one of my projects right now is what we're calling infra-geometry. Can we build a version of geometry that doesn't start from an assumption of the existence of space? So all of these things that you're talking about, like angles, angles are a pretty non-trivial thing in our, in our setup. Uh, points, we know what they are. Lines are shortest paths in this graph. Geodesics in this, in this graph. We kind of know what lines are. If we say, "Are two lines parallel?" We can, we can actually give a definition of that, which is this kind of operational definition that's based on distances from every point on one of these lines to every point on the other line, and so on. It's a, just a little ladder type thing. You can, you can make. You can start. So the, the goal is, can you build up, you know, this is, this is our goal is to build up differential geometry, basically, from a new basis. Because usually, in, you know, in, kind of, when you, when you think about manifolds, for example, kind of the idea is, when you look with a, a powerful enough microscope, you see Euclidean space. You see standard, standard geometrical space. But in our models, that isn't true. What you see in a, in a, you know, in a good enough microscope, you'll just see a hypergraph down there, which doesn't have any reason to look like Euclidean space. And so the question is, what does, what does, how does, how do you build up geometry from that? And, and this question of, of sort of how, how you, this question of how, why is space coherent? Why is there sort of, why is there a space there at all? That's an interesting question, and it has, I think, deep, sort of, roots.

So, in our models, the hypergraph is effectively knitted together by events that occur. So, actually, when I say there is this hypergraph and it is the universe, that's kind of a simplification. Because what we really think of there being is, we can, we can think of that being, sort of, a substrate of this hypergraph. But the reality of things is mostly the update events that occur in this hypergraph.

Can I just stop you for a second? And for people who don't know what a hypergraph is, is this, do you view the picture of these steps of computation and the different directions that computation can go in? Is that what you mean by the hypergraph? Right. Those series of steps. What do you mean exactly?

Okay, so, so, okay. A graph is something where you have a bunch of points. They might be people, for example. And you have relations between those points. So those relations are, are lines that join those points. So, for example, you have might have a friend network. And the friend network might be, uh, uh, might be just something which is just this graph. It's not laid out in physical space. It's just a graph, a network of relations between these points that correspond to people. So the only, the hyper part of this whole thing is only that instead of there being a relation between two entities, there can be a relation between any number of entities. So, for example, you can have a, a three-way friend, so to speak, rather than just a, as a friend of B, it's like A, B, and C are all friends together, and that's a single hyperedge that the, it's.

Not that important that this is a. I mean, it is technically very important because the kind of the the it's a much more flexible structure that allows all kinds of things that otherwise have to be distinguished to not have to be distinguished. It's, it's technically important, but conceptually it's not that important that we're talking about a hypergraph rather than an ordinary graph.

Just so, so essentially what one's thinking about is that everything about the universe is determined by the relations between the atoms of space. That the only thing we can say about an atom of space is that it is unique. This atom of space is distinct from that atom of space. And the only, and then we're saying that we are defining relations between these atoms of space, which we can draw as this graph. And we can potentially lay that graph out, if we were making a picture of it on a computer screen, for example. We lay that graph out in a certain way, and that, that corresponds to, you know, that, that's how we get sort of a perceivable thing from this underlying kind of thing that is just a bunch of relations between atoms of space.

So, and, and so what, um, uh, what I was saying is that that one can think about this this hypergraph as it is progressively being updated. And every update takes a certain number of hyperedges and kind of grinds them up and produces another set of hyperedges. And that, that there's an event that occurs, which is that updating event M. And in the end, the only thing that we can perceive is the causal network of connections between those events. So what I mean by that is some event happens and the output of that event becomes the input to another event. And so that defines a kind of time-like sequence of one event has to occur before another event in time. So that because it is producing the input that will be used in that other event. And so what you end up with is this, is this giant kind of causal graph that is, you can think about it is coming from the the updating of the hypergraph. It is a, a graph of all the causal relations between every event that happened in the universe.

So, for example, when you get to a black hole, there is an event horizon, which means there is no causal connection between what's happening inside the black hole and what happens outside. There are no causal edges that cross the event horizon in that direction. And that's, that's kind of the, the defining feature of a black hole in, in, in our models. But so this causal graph, in a sense, is more real than the hypergraph. It is the thing that we are really capable of perceiving. To get the hypergraph, what we do is we take this big causal graph and we have to define what counts as a particular moment of time for this causal graph. And there are all these events happening, and they happen all over, you know, all, all over this causal graph. What we have to do is say, this is a set of events that we will say are simultaneous in time. And this is a setup very similar to the the way that special relativity is set up. Um, it's, you are defining a simultaneity surface of the events that you say are happening at a particular time. Then another set of events happen at a later time. And we can know that that there's is's a certain ordering, a partial ordering of these events, where some events have to be after other events, and so on. In some events, we don't know what their relative ordering is. They're space-like separated in the language of relativity. And so, so we build up these space-like slices of, this is what could be the current moment of time. This could be the structure of space at the current moment of time. That thing that we get from slicing the causal graph is the hypergraph.

So we can, we can think of it that the real thing is this causal graph. And the hypergraph is the, is a thing derived by saying, we choose to say that this is what counts as now across all those different events. So the way to think about it in terms of the structure, how where space emerges from, is the fact that space is coherent. Is a consequence of all of these events occurring. In other words, if you, if you stopped the events from occurring, you would, you could, for example, you can, you can, you can break off a piece of space. If there are no events that knit together two pieces of space, those pieces of space are not causally connected. They are effectively independent. So in effect, in, in, you know, in our perception of the universe and of space, the vast majority of everything that's happening in the universe is the knitting together of the structure of space. So probably, we, we don't know, but in some vague estimates of parameter values, there's really only one parameter in our models, but which is the the size of the elementary length, the size of the elementary time. Um, they're all related. Um, but with some very vague estimates of these things, maybe one part in 10 to the 120 of the activity of the universe is is not just the knitting together of the structure of space. Um, so most of the activity of the universe is knitting together space, making space coherent, making it be the case that there is a causal connection, for example, between one part of space and another part of space, making effectively space in some sense come to equilibrium, just like the molecules in a gas will come to equilibrium and have a certain uniformity to them. The fact that space has an apparent uniformity is a consequence of the fact that there's a sort of causal connections between different parts of space. It's effectively sort of this knitting together process in space is what leads to things like the apparent homogeneity of space. And so this, this phenomenon. So it is an incredibly non-trivial thing that space is coherent. In other words, that that there is a, um, and that, um, uh, and, and that breaks down, for example, when you have an event horizon. You no longer have causal connections between, well, in the case of a black hole event horizon, between what's going on inside the black hole and what happens outside. I mean, there are different kinds of event horizons where you can have sort of a, a separation going in both directions and so on.

Do you resolve the singularity at the center of a black hole in your framework?

There is no singularity. It's discrete. So, yeah, it's discrete space. And so, so that means that, for example, in traditional generality, one of the mysteries is there has to be essentially topology change in spacetime because a black hole with a quote singularity ends up being spacetime with a hole punched in it. And how do you get from spacetime without a hole punched in it to spacetime with a hole punched in it? Well, that's not something you can discuss in the context of standard general relativity. In our models, it's, that's easy to get because everything is discrete. And so there is no discontinuity between kind of the thing with, with, you know, with the formation of an event horizon, for example, and the formation of a singularity. It's something that is, it just, you can, you can get there because you're not trying to take a continuous space and change its topology. You've got a, a graph. You can think of a topology of a graph, if you want to, but the, the effective, you know, the, emergent continuum topology can, in the limit, can change discontinuously because you're taking the limit of an infinite sized graph. But when you look at the, in, at the graph itself, you just see, well, it tangled itself up in a particular way. So yes, it resolves that issue, um, as well as many others. I mean, it's, you know, in the simplest kind of black hole in our models, time just stops. I mean, that, that you see that in general relativity as well, that, you know, every, every path, every geodesic is a, is a finite length in a, in a, in a non-rotating black hole, for example. And so in our models, the way you see that is there are these update rules, and at the center of a black hole, it will just turn out there's no update you can apply. So time stops. I mean, it's, it's, it's very interesting how a lot of things that one thought about as mathematical become, in a sense, mechanical.

So, for example, time dilation, standard phenomenon in relativity, that things that are moving quickly, uh, effectively have time running slower for them. In our models, that becomes a very mechanical fact because if you have a thing like a particle, for example, that is sort of, uh, that is at a particular place in space and it is kind of progressing through time, it's being rewritten, its structure is being rewritten, but it's staying having the same overall characteristics. But it's using, it's sort of computation budget is used to rewrite itself through time. If the thing is also moving, it has to do two kinds of computation. It has to, it has to recreate itself at a different place in space, and it has to update its internal structure in time. And so when it's using its computation budget to move in space, it doesn't have as much computation budget left over to change with time. And so effectively time goes slower for it. I mean, that, that all turns into a bunch of mathematics, but that's the intuition behind how time dilation works in our models. And, and there are other things like how gravity works, for example. There's another very lovely thing.

Well, actually, before, before we go into gravity, can I just stop you for a second? Because I, I'm curious about something else, which is when you're doing this update, is the fact that you're looking at all possible updates, is that what gets you around having, for example, a hidden model, uh, hidden variable model? The fact that you're?

Yeah, I mean, the structure of these models is so different from the traditional setup of few-particle quantum mechanics that it's hard to pinpoint exactly what it is that kind of gets you around kind of, you know, violation of Bell's inequality issues and things like this. But the end result is, yes, you get around those things.

But you must be able to show that, right? You must be able to prove that.

So, so, and when you follow that proof, what, what, what are the characteristics of that proof? What does it look like?

Well, okay, so I would say that the, um, this is still, it's still kind of complicated, and it's not, I would say we haven't got the cleanest version of this. We haven't got as clean a version of this as I would like to have.

Okay, I can give you some intuition and I can tell you some mathematics. But the intuition would be great.

Yeah, yeah, they don't quite meet in the middle. Okay. So, so, um, so essentially, one, one question is, might say, is our model a deterministic model?

Yes.

And the answer is yes. The whole, what we call multi-way graph of all possible paths of history, there's just one of it. But the question of what we experience is a question of sort of where we are in that multi-way graph. So to explain that in physical space, we experience the universe in a particular way because we are at a particular place in the physical universe. Oh, we think it's the same laws of physics in another galaxy, but we're not actually seeing what happens in that other galaxy. Well, turns out there's an analog of physical space that is the thing that's relevant for quantum mechanics. We call it branchial space. Here's how it works. So we talked about how kind of you get these multiple threads of history, you're rewriting these hypergraphs, you get these different possible rewriting paths. So what can happen is there's one form of hypergraph, and there there are two possible rewritings that can happen. So you get, get two different hypergraphs. It could turn out that those two different hypergraphs, the rewritings, two different rewritings of those, end up converging to a single hypergraph in the next step. So you end up with this kind of branching, merging set of paths of history. So that you, you think about that whole structure that's branching and merging successively over time. Now, you say, let me take a slice of that at a particular moment in time. What you'll end up with is all these different branches of history that sort of poking through that slice, all the kind of loose ends of this thing that was branching and merging with different paths of history. So you can then ask the question, how are the different paths of history knitted together? And it's actually the same story as with causal connections in spacetime, but played out on a different situation where if you say two different branches of history are next to each other, if they came from a single common ancestor one step before. So you've got these branchial connections, so to speak, these connections between branches. And you can kind of make a map of all these connections between branches. And what that does is it builds up essentially a space that we call branchial space. And if you ask, what is that, what is the structure of that branchial space? We wish we knew more about that. It's some kind of Hilbert space thing, but we don't completely know how that limit works. But so qualitatively, what happens is there, you, you are kind of moving to different parts of different paths in history as you move through branchial space. You're moving to different parts of history, different, different possible parts of history. Now, where it gets tricky is we, as observers of the universe, also exist in branchial space. We're extended objects, actually, in branchial space. Our minds do not experience a single thread of history. Our minds experience a clump of threads of history, just as we're not used to experiencing individual atoms of space. We experience these big lumps of atoms of space that correspond to what we can perceive at the scale that we're at. And so it's the same thing in, in this branchial space that we are sort of experiencing this kind of lump of branchial space. And so then the, there are, well, a bunch of different issues. One is that it is a non-trivial fact that we believe that there is a single thread of history that we experience, because, because actually what, what's happening is that we are conflating all the branches in this lump of branchial space. We're conflating them all to say there is a definite thread of things that happen. And that's, so when, and that's, for example, imagine that we have a quantum computer. That is what a quantum computer is doing is it's following multiple threads of history. But then what has to happen at the end of doing that quantum part of the computation is you have to knit together all those threads of history to, uh, to get an answer that we humans will say, yes, that's the answer. There's no good to say, oh, well, the answer is the superposition of all these different threads of history. It's, we want to know, did the computer say one or zero? And so we have to knit those threads of history together. And that's what in the standard formalism of quantum mechanics, people talk about as a measurement process. But nobody says what happens, what the mechanism for that is. They just say, and then measurement happens, and we get a definite answer. And so in our models, we get to say something about sort of the mechanism by which we knit together those different threads of history. And it's not, not obvious, you know, the amount of effort that it takes to knit together those threads of history might actually be quite large.

But, but now you can ask questions like, okay, so, so one key question is, in this branchial space, what is the interpretation of kind of motion in branchial space? What is, what, what changes as you move around branchial space? And what seems to be true is that motion in branchial space is like a change of quantum phase. So, you know, a key feature of quantum mechanics and the usual formalism is that you describe things in terms of complex numbers, and that you describe quantum amplitudes in terms of complex numbers. That's not actually necessary. There's an alternative formalism of quantum mechanics where you describe everything in terms of positive and negative probabilities and so on. But in the simplest way of doing it, in the usual way that it's treated in quantum mechanics, you have these complex numbers, and they have a magnitude and a phase. Well, what seems to be the case is that the position in branchial space is related to quantum phase. And the reason that's important is, oh, we didn't really talk much about gravity, but in physical space, gravity is all about the deflection of shortest paths. In when there's no gravity, a shortest path is kind of a straight line. When there is gravity, when there is a, a mass somewhere, a lump of mass, that that deforms shortest paths. Do not be straight lines. That's what leads to gravity in general relativity and so on. In our models, what happens is the, the presence of energy. Okay, so what, what is energy in our models? Turns out to be something much simpler than I ever thought it would be. Energy is basically the density of activity in this network. So if you look at how many events are happening per unit region of this hypergraph, that basically is giving you the energy density in that, in that region of the hypergraph. It's a little bit more complicated than that in the, in the mathematics, but that's the, the intuition is the activity in the network corresponds to density of energy in the network. And so what is being said then is that the presence of activity deflects the paths of of shortest paths, deflects shortest paths to no longer be straight lines. And that's what leads to gravity in physical space. And so it turns out the same thing happens in branchial space, that the presence of energy momentum ends up deflecting paths in branchial space. And that deflection of paths corresponds to a change of quantum phase. And there's this lovely thing, the Feynman path integral, which tells you that the way quantum amplitudes work out is according to something which is a change of phase as you have a density of, well, LR in density, essentially an energy density. And so what, what turns out to be the case is that our models, what in physical space is gravity and the Einstein equations and, and, uh, uh, and kind of the deflection of things by by gravity in branchial space, that same thing turns out to be the Feynman path. So in other words, you've got in both cases, the presence of activity in the network, presence of energy, um, deflects paths. In one case, that deflection of paths is a physical deflection of paths where we can just see the thing is turning, going into orbit, whatever else. In the other case, that deflection of paths is a change of quantum phase.

So when, when we're talking about things like, um, it's, you know, it's, it's frustrating because I, I think we're really close to having a very clean, few-sentence version of, here's how Bell's inequality works, but we're not quite there yet. And so I'm kind of circling around kind of explaining pieces of this. And it's, it's sort of frustrating because, because in the end, what we're looking at in quantum mechanics is sort of how does a branching mind perceive a branching universe? And the, the, the question of whether one of the things is, you know, okay, so there's supposed to be probabilities in quantum mechanics. Where do those, where do those different possibilities come from in our models? And the answer, I think, is it's really simple. They just come from the fact that we can be at different places in branchial space. So we, the fact that we observe this happening rather than that happening is because we are at this place in branchial space now. That has little to do with, so the whole sort of entanglement and, you know, we have these multiple states and they're related and they're related in a way that doesn't, uh, correspond to what happens in physical space and so on. That's all rather trivial in our models because there's this whole notion of branchial space. It's easy to have these these, uh, sort of pieces of reality, different threads of reality coexisting and so on. That's a straightforward thing. How that relates to, for example, oh, I don't know, some experiment that you might do, uh, that involves some interference pattern or something like this, um, that's another interesting thing. So again, I can give you some intuition. The mathematics is only partially done for this case. But, you know, one of the mysterious things is, you've got two slits, and you, you, you're, you know, you say, which slit did the photon go through? If we, um, what seems, what we observe happening is that we have destructive interference, for example, where even though, you know, on its own, the photon could have gone through one slit or the other, somehow we end up getting destructive interference, and it's as if it went through neither slit. And how does that happen? Well, in our models, what probably happens is that the photons corresponding to the going through the different slits wind up at different ends of branchial space. They end up far away in branchial space. Those two possibilities, it went through one slit, it went through the other slit, end up being far separated in branchial space. So what, well, if we are an observer who has a limited extent in branchial space, then what will happen as far as we're concerned is because these photons wind up at these different possibilities, wind up at different ends of branchial space, we'll never be able to to have a coherent view of that. We'll never be able to say, yes, we know what happened. And so it's, it's, if we could, if we're only sensitive to this small part of branchial space, so as far as we're concerned, the photons just disappeared because there's no, no place we can be in branchial space where we'd see both of them, given that we're of limited extent. So in a sense, destructive interference, which is a very bizarre phenomenon in some ways, is a consequence, in that picture, of the fact that we have a limited extent in branchial space. And so this, the photons didn't disappear, they're, they're alive and well, but they're sufficiently, the possibilities are sufficiently far separated in branchial space that we can never perceive them. So that's, that's kind of another, another sort of indication of what happens. But, you know, it's a little frustrating because at a mathematical level, we can very beautifully reproduce quantum circuits and things using our formalism. In fact, it's actually now probably the most efficient way anybody knows to optimize quantum circuits is to turn them into these multi-way systems, do a bunch of theorem proving technology at the level of multi-way systems, and turn them back into quantum circuits. So we kind of know that the formalism, you know, reproduces at least quantum circuits. Quantum circuits are not all of quantum field theory, but we have pretty good reason to believe that that we'll get sort of all of quantum field theory. But, you know, when you ask about, um, uh, you know, each of these different things that are typically identified in in quantum mechanics, and you say, how does that, how in detail does that work? We don't quite know yet. I mean, in fact, this is a literally last few weeks, um, somebody I'm working with has been very actively working on trying to untangle this, um, and we seem to be making progress. So in another, in a little while, we may have a nice crisp, this is exactly how it works. Um, it's, it's, you know, one of the things about these about this whole activity is, you know, we think we understand the very low-level machine code of the universe. But getting to stuff that we recognize, and even that has been analyzed in existing mathematical physics, that's a big long, you know, effort. And, you know, it is exciting that a lot of things people have studied in mathematical physics seem to plug in as limits of our models. So, you know, people have studied, you know, some particular mathematical structure, and yes, you can find that structure as a limit of our models. Our models are, in a sense, very flexible, but nevertheless have definite structure. Um, but, you know, various limits of them seem to seem to connect with these other kinds of things. But it's, it's, it's just, um, uh, it's hard work, um, kind of connecting all these things together. And, um, uh, when, when typically, it's very satisfying because when you actually see one of these connections, it's like, oh, wow, that's how that works. You know, I didn't, I didn't understand how that works. In fact, I just had that earlier today about some things to do with machine learning, which one can think about using some of the methods from, uh, from our physics project. And I just, I finally understood something that I wondered about for like decades, um, about, about why machine learning works. Um, and so that's, it's, it's very satisfying, but it's hard work.

Can I ask you, you mentioned, so I understand, because I understand sort of how gravity might emerge in your models, right? Because you start off with the assumption that you have some distances between the sort of smallest units within the computational model. But you mentioned about photons, right? So how does a photon appear in your model? Do you, do you have charge? Is also something you input?

As in, we don't know. We don't know how photons, sharp. I mean, so, so what we do know is how black holes work. Um, and we can simulate, you know, black hole mergers and things like that. And we get nice little ring-downs of of merged black holes and so on. We get little gravitational waves coming out. That's stuff we can, we can look at. And we can see it, it, it follows what the Einstein equations seem to say. Actually, it turns out to be a really good numerical model for doing general relativity. So it's actually something which people have started using as a kind of build-up from the microscopic structure rather than build-down from the continuum equations. But, um, in terms of what a photon is and how it works, we don't know. I mean, in fact, my increasing guess is that particles are very much like black holes. And one of the things that's been kind of a, uh, uh, you know, our expectation, slightly more technically, is that particles are certain kinds of topological defects in this, uh, in this hypergraph. So it's, it's as if, you know, you have a, it's like you have these sort of topologically stable things like an eddy in a fluid, for example. Once you have that fluid going around and you know, having that the spot in the middle, which is, you know, in which things are going around different directions on different sides of that spot, it's, it's hard to untangle that eddy. And so it has a certain stability through through time. And our, our guess is that that's how particles work. But we haven't figured out the details of of what, you know, what are those topological defects. I mean, just to give you a sense of why this is hard. Okay, a few years ago, I was talking to sort of one of the world's sort of leading graph theorists. Okay? And I'm asking about, you know, how do we understand how these topological defects work in graphs and so on? Because there are some sort of elementary versions of this that are understood for planar graphs and such like. He looks, thinks about it for a minute, and he says, come back in a hundred years, we may know a little bit more by then. So, in other words, it's, you know, there's a lot of mathematics that is not known. Um, and we're having to try and build that. And that's part of the reason it's hard work. I mean, you know, when it comes to general relativity, for example, you know, one understands how tensors work in integer dimensions. One understands how calculus works in integer dimensions. You know, people learn univariate calculus, you've got one variable. They learn multivariate calculus, you've got two variables, three variables, whatever. What does calculus look like when you have two and a quarter variables? Nobody knows. But we need to figure that out. And that's what this infrageometry thing that I was was mentioning earlier, that's kind of that's what that story starts to be about, is how do we define the, the whole structure of mathematics starting from things that are very different from space as we normally think of it. But, um, uh, let's see, you were, you were just asking about, um, um, I mean, I, I think I was mostly making an excuse, excuse for why we don't know what photons are. But I'll mention, make another comment, which is one feature of black holes is from the outside, black holes pretty much all look the same. Black holes, they have a certain mass, they have a certain analog of of spin. Um, but, you know, any black hole of a given mass and spin-like thing kind of looks the same, even though different black holes might have different crushed civilizations inside them. But from the outside, they look the same. So one of the longtime mysteries of physics has been, why do all electrons look the same? And the, you know, it is, I think, far from impossible that they only look the same from the outside. And in other words, that there is, in a sense, there's an inside that might not be the same. But like black holes, from the outside, they all look the same. You know, I don't know whether it will turn out to be that way, but my guess is that there is a very close connection between black holes and particles. And that's something that is not nearly as obvious in continuum models of space. In continuum models of space, black holes, I mean, you know, particles generally are thought of as as as infinitesimal geometric points in in continuum theories. You know, the electron is just a point, and quarks are just points. And there isn't really a, a way of, you know, taking them apart. In our models, there is. We don't know exactly what they're like inside.

Now, you asked about things like electromagnetism and so on, um, and the idea of, well, local gauge invariance, the idea that there are, um, that, you know, it's kind of like you have an electric charge, and it produces an electric field, and the electric field, you know, goes out like a, you know, like a hedgehog or something. Actually, real hedgehogs don't really look this way, but it has spines going out in all directions. Um, you know, it has, it has pieces of field going out in all directions. And then always the question is, if you move that electron, how does, if you go far away from the electron, how does the moving the electron affect the electric field? And what we know is that that, what has to happen is that there has to be this wave that goes out that sort of resets the each of these directions of the electric field. And that's what corresponds to an electromagnetic wave. That's what, and, and it's kind of this, this sort of, uh, communicating the information about where the charge was is something for which you need photons, for which you need electromagnetism. And so there's an analog of that in our models where, again, we don't know the details of how this works, but where there's essentially what, um, this, this idea of sort of here's, yeah, let's see how to put this. So, you know, when we think about this hypergraph, space is an emergent feature of this hypergraph. When we look at it in the aggregate, even though we got all these details about all these little pieces of hypergraph, when we look at it, you know, zoomed out, it's, it, it seems like continuum space. But there are potentially aspects of this hypergraph that are not like ordinary physical space. There are sort of details of the hypergraph inside which we can think of as being like some other kind of internal space. The kind of thing that is the standard model of electromagnetism, for example, where there is kind of an internal degree of freedom that is associated with every point in physical space, that is the thing that corresponds to the gauge field for for electromagnetism. Well, this idea of of this notion of sort of a, a way of picking a direction in this kind of internal space, this idea that you have to make a connection between the directions that have been picked in different places in physical space, shows up in a very concrete way in our models. And that, that kind of shows us sort of how the whole structure of of gauge, local gauge invariance, and gauge fields and connections and so on, seems to work. Again, we don't know, as you know, the details, hopefully as time goes by, will be pinned down in in in, uh, more precisely. But the qualitative picture is, as you try and figure out these sort of internal directions, internal degrees of freedom that we, that correspond to aspects of this hypergraph, you have this issue of how do you align what you consider to be the, you know, the zero direction or something in this hypergraph at different places in physical space. And that's, that alignment, the need to have that alignment is what leads to connections and, and the whole structure of of local gauge, gauge theories and so on.

I mean, you know, this, this whole issue of, um, uh, um, of kind of what, um, um, you know, I think you, we were talking earlier about inevitable, about sort of geometry, the emergence of geometry. I make the comment that, um, this R-ad object, one of the different ways it plugs into existing thinking about mathematics is it's closely related to this incredibly abstracting the infinity groupoid of Grothendieck and so on, which I thought I would never care about in my life. I thought it was just one of those things that floated off into the far reaches of pure mathematics and would never be of relevance to me. But actually, it turns out it is mathematically very similar to the the structure of this R-ad thing. And it has, I mean, roughly what happens is, oh, I know, this gets, this gets, I mean, this is, it's basically, yeah, I mean, there's just a very elaborate piece of pure mathematics that, uh, that also has as a hypothesis, the sort of inevitable emergence of geometry. And that hypothesis is is basically closely related to the to the emergence of kind of something like continuum space in our models. It's kind of the same. It has, it has comes from the same mathematical place as this Grothendieck hypothesis about inevitable, well, inevitable topology in that case, but in the end, inevitable geometry from these, um, uh, from these kinds of setups.

You mentioned, so this idea that particles might be defects, right? It's quite interesting because you might ask the question, you know, when, when I produce a defect in my network, how does it arise? Is it, does it pair produce? Right? For instance.

Sure. What does a, what does defect actually mean?

So I, I imagine the picture I have in my mind is that you, you can update at each step in different parts of the network. And over here, you do some local update. And over here, you do some local update. And you go forward a few steps. And at some point, you realize that those were incompatible with one another. The changes that you made down here. Is that what you mean specifically by defect?

Or no, that's yet a different issue. I mean, so in a simple case, imagine that you have a a planar graph, a graph where everything can be laid out, so no two, no two lines cross. Okay? You have such a graph. Now, you put into that graph something that is, oh, particular. There's a theorem in graph theory that says every graph gets to be non-planar by having a particular subgraph present, or two, actually one of two particular subgraphs present in that graph. So there's a graph, if you have, like you have three points and you have another three points, and you're joining each, each point to every one of the three points on the other side, K33 subgraph. Then, then sort of that's a, a, a lump of non-planarity that can can exist. Now, once you have one of those lumps of non-planarity, you never get to get rid of it if you are rewriting this graph, but always preserving planarity. That lump of non-planarity is essentially a defect. You can't get rid of. If you have two such lumps, they can wander around and eventually they can cancel each other out. But if you just have one, it's stuck. So that's, that's kind of the, the junior version of, of how you might have sort of topological defects in a graph. What the more senior version of that is, that's what the graph theorist I mentioned earlier said, come back in 100 years, and we may know more about it. More or less. Um, but that's something for which we have toy models of this. But it's, I mean, it's just one of these things that you have to, you know, by by doing a bunch of computer experiments and by thinking carefully about them, I fully expect that we will know a lot more about this. We don't right now. But that's the, that's the rough picture of what what can happen. Now, in, in the case of, um, uh, well, a black hole, you really have to think not about this hypergraph, but about the causal graph. Because the black hole is kind of this, this, um, uh, lump of of, um, uh, this kind of causally disconnected region inside the causal graph. So it's yet a different level of kind of of defect that you're looking for there. But I think, um, I mean, in terms of of, okay, so how should you think about, for example, pair production? Okay, so in our, our physical space is knitted together by all these little update events happening. And so the usual picture in field theory, which is rather confused, is that in the vacuum, there is an infinite amount of production of virtual particle-antiparticle pairs and so on. Um, and the, um, um, actually, here's an interesting thing I just, just thought about this recently, so first time I mentioned this, um, the, when you think about a particle, an ordinary particle like a photon, electron, whatever, in standard particle physics, it has a certain mass. It can move around. It can move with a certain momentum. It's just, it's a real particle. It just, you know, has its definite mass and it moves around and so on. But there's this notion of virtual particles. That's a feature of quantum field theory that's been thought about since the 1920s, where a particle like a photon can exist for a short period of time. You know, usually photons have zero mass, but a virtual photon can have any mass it wants, but it can exist only for a very short time. And it is the, what one has in in the vacuum of physical space is lots of virtual particle-antiparticle pairs generated for very short amounts of time that then sort of, you know, disappear. The vacuum will produce the particle-antiparticle pair, it will exist for a very short time, then they will annihilate again. So the thing that, um, well, essentially the, um, there's a question in physical space, a particle is a, a lump of reality that can undergo pure motion, that can move around in physical space without changing its character. Okay? That's sort of the defining feature of a particle. So it, it, okay, so the question is, what is a virtual particle? And I think a virtual particle is the direct analog of that in branchial space. That is, that a thing that can move around without change in branchial space is effectively a virtual particle. I, I don't know for sure yet, but I think that's, I think that's how it's going to work out. Um, and so what, why is that? So, so the picture of the vacuum, by the way, in, most of the time, there's all this activity in the vacuum. And that activity is most of the time, in our models, not readily decomposable into saying, oh, there's a virtual photon, anti, you know, there's a virtual electron-positron pair or something like this. It's, it's just, there's activity in the network. And it's not, oh, we can decompose that into, oh, there's this particular lump that corresponds to an electron here, and we can distinguish it from this particular lump that's a positron here. Um, it's something where most of the time, most of that activity is much more jumbled up than that. If you insisted, if you said, I've got to make a decomposition of all these, all this, all these details into something as you do in quantum field theory, into definite modes, like, you know, you've got your string wiggling around, and you say, well, actually, it's a superposition of, you know, the fundamental mode, that's a single, you know, a single sine curve, and, and the, and the first harmonic, that's a thing that wiggles once, and the second one that wiggles twice, and so on. You can describe any shape of of curve, you know, by that Fourier series of of a sum of sine curves. And that's what's done in quantum field theory to describe the vacuum as this infinite sum of different particle excitations. But the fact that you can really identify, say, this really is a particle excitation in our model, you're at a much lower level. And most of the time, you can't sort of, it is not useful to say, oh, actually, this, this just random update here can be thought of as a little piece of this very complicated electron, you know, structure that existed there. So it's kind of, um, you know, what we're, what we're seeing is something below the level of these particle-antiparticle pairs. Um, and as I say, my, my recent guess is that, you know, I've been wondering, what is motion in branchial space? What is it? What corresponds to a particle? What, what is a, what is a thing that maintains its identity through through branchial space? And my guess is that it is effectively a virtual particle. But that's a, that's a future, a piece of of future, future science, so to speak, because I really haven't, haven't figured that out in detail.

So one question I have is, as you mentioned, particles are coherent through time, right? You can update with any rule possible, right? At any point, at any step of the process, right? How does that arrive in your models?

Well, okay, so look, the, the thing is, each, it's complicated because with a given rule, you, you have, you know, if you have a, let's say, you, you're dealing with a given rule, then there will be certain kinds of, for example, let's say it's a rule that preserves planarity of graphs. Um, then that rule will never be able to destroy the K33 subgraph. It's, it's stuck with that. A different rule would be able to destroy that. But in our perception of the universe, we are localized in branchial space, just as we are localized in physical space. So for us, we're mostly looking at this bundle of rules that are sort of nearby in branchial space. And so for us, there are certain things that will, you know, that will be coherently the case because we are localized in branchial space. Now, that's because we're observers like we are. If we were just like in physical space, if we were observers who were, you know, where the parts of our minds were scattered over the surface of the Earth, for example, we would have a different perception of the world than we do. I mean, you know, if we were, and that's a, you know, the fact that we perceive the world, we perceive definite objects in the world and so on, is a consequence of the of the fact that we are localized perceivers of the world.

By that, you mean there's like a weighting on which rules can be taken in each update, sort of like in quantum mechanics?

No, I mean, in, you know, it's like saying we are at a particular place in branchial space. We might be at a different place in branchial space. But just like we might live in a different galaxy, but we happen to live in the galaxy in which we live. And given that, there's a certain perception of what happens in the universe that we get. You know, we, because we live in the galaxy we live, you know, its central black hole is of a certain size, and so certain things happen in our galaxy. If we lived in a different galaxy, we, you know, even though the laws of physics are the same, we would perceive different things to be going on. And so.

I think it is in in Ral space that because we are at a particular place in roal Space the universe we perceive the universe to operate in a particular way. In fact, my my view of that is every mind we can think of as being at a different place in roal space. And minds that are closely aligned are nearby in Ral space. You know, the the humans with similar educational backgrounds are nearby in Ral space.

And by the way, you can ask the question, what is the analog of particles in Ral space? And that one, I think I know. I think the analog of particles in Ral space is a very bizarre thing, is concepts. So, for example, in one brain, there are all these neuron firings that happen. But somehow we package up those neuron firing ings into this thing that we can transmit to another brain. You know, as I try and yak about things, you know, maybe I am communicating some of what's going on in those neural firings in my brain. And I'm communicating them by packaging them up into something that can be transported to a different part of Ral space without much change. A concept, a word, whatever else that you can then unpack in your brain and get something like the same concept, so to speak. Get, you know, get a corresponding set of neuron firings.

And as we, you know, as we look sort of further away in Ral space, you know, we get to the cats and dogs. They have only certain kinds of things that we can successfully communicate. We get further away to other kinds of systems. It becomes very hard to understand the alignment between the way we think about things and the way other systems think about things. I suppose one distinction would be, so in the real world, there's sort of nothing that we see in between in Ron and a muon, right? There's not sort of some. I know that in your writing you talk about this sort of inter-concept space, right? And so if if concepts in the space are particles, this sort of that's a good question. Yeah. Right. There's there's uh, I suppose the idea would be that you would want these to be discreet things where you can't through some continuous process go from one to a different concept. That's an interesting issue.

I mean, so this whole idea of inter-concept space is something that uh, I think sort of maps into this ideas about the ruad. But the place where I kind of came to that is in thinking about generative AI and thinking about if you train up an AI on kind of a bunch of a billion human images and you say, inside the AI, all the kind of the the the kind of the essence of those billion images has been ground up into this underlying model. And then you say, well, what's in that underlying model other than those those existing images? And what you can say is you can identify in this kind of space of possible meanings, so to speak, you can identify this island that corresponds to cats, another island that corresponds to dogs. And you say, well, what's in between those? Well, if you have a generative AI that's producing images, it produces images happily in the space between cats and dogs. And they look like things you don't recognize. They look like kinds of things that there could be, you know, you could have a name for that kind of pattern, but we don't right now. And some of those patterns, maybe in the future, there'll be some art movement that's very keen on this particular kind of pattern, and then there'll be a name for it. But right now, there isn't. So right now, we just have these little islands in the space of possible Concepts, space of possible images. We have these little islands that we have names for, and then we have these vast spaces where we don't have names.

So I think the thing you're you're asking, which is interesting, is is um, uh, yeah, in in so far as when we think about that more dynamically, when we think about kind of, let me think about this for a second. Um, let's see. Um, the question is, the things that we identify as transportable concepts, like cats and dogs, as compared to the things in inter-concept space for which we have not formed kind of a coherent way of describing them. How does that relate to? Yeah, yeah. I think it's the, I think it's the following thing. Okay. So when we have a particle like an electron, it has a definite momentum. There's a thing we can measure about it that's pretty definite, or it has a definite mass. But the vast amount of stuff that's happening in the vacuum in physical space, we don't have that way of identifying. We don't have a way of picking it up and saying, I can talk about this. It has a certain mass. And I think that's the analog. I think that the analog is that the things we consider as particles in physical space are the things for which a certain measurement that we can make on them, a certain way of characterizing them, is definite. Like we say it has a certain mass, which I think is similar to saying we can call those cluster of images a cat. Whereas, you know, so in other words, that that's our, you know, what is it? Well, it has a word associated with it. What is that lump of activity in the network in in space? Well, it has a definite mass. It has a definite momentum. So we can characterize it as a particle. I hadn't thought about it that way before. Thank you for for asking about that.

I mean, that that's a, I think that the, um, um, so, you know, I think I think that's right. I think that the the question of of what, um, sort of we call it a particle because we have a definite thing we can measure about it. See, see, to know that that thing was the same as it moved through space, we have to say, well, what aspect of it is the same as it moves through space? Because if we say, let's identify what atoms of space it's made of, it's going to lose every time. It's going to be different as it moves. But we have to say, you know, the thing that moves, you know, the the glass that moves from here to there, you know, it's made of different atoms of space. But yet we say, look, it's the same glass. But imagine this was really close to a Spacetime singularity. This glass would not maintain its shape as it as it moves through space. The fact that it maintains its shape, I mean, in in actuality, the Earth's gravitational field will be just a little tiny bit different here versus there. And this glass that we think of as circular here, it won't really be circular in a sense when we, if it was perfectly circular in one place, as we change the gravitational field, we change the curvature of space, it won't be perfectly circular anymore. But the fact that we say, yes, it's the same glass as we move it from here here to there, that's a consequence of the fact that the measurements we're doing on the thing make agree when it's in one place versus another. And I think that's it's the same, that's that same idea of the thing we are, the thing we're identifying is, it's cat-like, so to speak. And that's, and and, you know, that's the aspect of this of this in in concepts based, so to speak, that we're identifying. I think that's kind of the analog of this. I think that that's a, um, so the fact that we think it's the same electron is because the things we're measuring about that electron are things that will be the same, even though if we were to get inside that electron and ask what atoms of space is it made of or some other detail, then it would be different.

This is a little bit out of left field, but do you think that it was in some, in hindsight, obvious that LLMs were going to work? Absolutely not. Absolutely not. No, it was not obvious at all. It was, and and the, you know, the fact that LLMs work is a scientific fact about humans and human language. I mean, in other words, nobody knew what, uh, I should just say, I mean, large language models for people. Yes, yes, we're talking AI. G seg from from physics into, uh, um, and, uh, no, I mean, you know, I had followed these things for years. I mean, I started working on, I worked on neuron nets back around 1981, and I could never get them to do anything interesting. And I sort of gave up. And it was the big surprise happened in 2011 when it turned out that if you kept training a neuronet long enough, if you kept giving it enough examples and saying, you know, adjust yourself to follow these examples, and you kept on doing that for long enough, the neuronet would learn things. It was completely not obvious that that would work. Nobody knew. In fact, many people, including myself, sort of guessed that the right thing to do was to try using simple neural nets to do simple things, and that turns out to be much harder than to use complicated neural nets to do complicated things. And I finally have understood this. In fact, it's something I've been working on recently, kind of the fundamental reason that machine learning works. It's not at all obvious it will work. It's not at all obvious that you can. And and by the way, the reason that something like ChatGPT works is that, well, okay, so what is ChatGPT doing? It's basically trying to continue sentences, trying to continue pieces of text. And it's trying to do that by using the example of what's on the web. So we have a sentence like, I don't know, uh, I don't know why I always use this one. You know, the cat sat on the. And you know, if you look on the web, you'll find a bunch of web pages where the next word is mat. And so it's, it's a pretty good guess if you saw the cat sat on the, that the next word should be mat. But what's happening is the fact that you can go on and make a big long essay, you won't find explicit examples on the web of this 20-word sequence that you have to give the 21st word for. You won't find that. So what has to happen is somehow the neuronet has to take the examples from the web and make a model whose extrapolation agrees with what a human brain does. And the fact that that can be done using these neuronet models is completely non-obvious because it's, you know. And the fact that it works, I think, as a consequence of the fact that those models are similar enough to the way that brains are set up that they generalize in more or less the same way that brains generalize. But the fact that that even can be done for language is not obvious. And in fact, I think what it really reveals is that there are regularities in language that we didn't know were there. So, for example, we're very familiar with the fact that, you know, in a language like English, you know, sentences tend to go noun, verb, noun. But there are many sentences that go noun, verb, noun that we would say that sentence is just nonsense. But it turns out that there is more structure. There's kind of a semantic grammar, I think, of what sentences can make sense. And effectively, what was done in these nets is they learned that semantic grammar. They learned the construction kit of how you can put sentences together. And they've done it in a very roundabout and complicated way. But the end result is a kind of scientific fact about how you put sentences together. And that's that's what, you know, that's the, that's the sort of miraculous seeming thing is they're successfully able to spin out lots of text that could have made sense because it's following this construction kit that they have effectively correctly extrapolated from a trillion words deduced from the web or something. So it, it's, but now, you know, I I just recently was working on the problem of why does biological evolution work? You know, why is it the case that you can, why, why do you, when you change the genome just a little bit, why do you not just always get stuck? Why are you not always in a situation where you can never get better, where you can never get, you know, the longer neck giraffe or whatever else? And this is, it's quite an interesting thing because it's actually relates to our friend computational irreducibility that we started out with. If it wasn't for computational irreducibility, neither biological evolution nor machine learning would work. And and the reason is that you're in this, in biological evolution, you've got some genome, it produces an organism, so to speak, that has some particular form. And the question is, as you make the small changes, as you make a bunch of small changes to that genome, what happens? How, how, you know, there are all these different directions you can go in this kind of space of possible organisms. And the issue is, do you get stuck? And the point is, because of computational irreducibility, there's enough effective randomness in what happens in all those different directions that with overwhelming probability, there is a way to improve. It could be the case that what you'd see in all these different directions was similar enough that if there, if one direction didn't let you improve, no other direction would let you improve either. But because of computational irreducibility, there's there's this that you will tend to be in a situation, you, with overwhelming probability, you'll be in a situation where there's enough difference between these different directions that one of them will be a way to get better, so to speak. And it's kind of an interesting thing because I found this very minimal model for biological evolution in which you're generating these computational patterns. And what you see happening is, you say, I want a pattern that lives as long as possible, for example. And what you'll see happen is it has some underlying rule which lets it live for 10 steps, let's say. And then you say, well, mutate it a bit, mutate it a bit. It will have a way to get better because some, it won't, there'll be enough effective randomness produced by computational irreducibility that it will have a a path to improvement. And as it gets better, you get these bigger and bigger patterns, and they get more and more complicated, more and more ornate. And it's kind of looks like the fossil record because it's kind of like, first it had one little idea, then it built on that idea, then it kept building on that idea and eventually made this very complicated thing. And by the way, there was a different original idea it could have had that would have been sort of a different branch in the tree of life. And so what you see, but what you see is these very ornate solutions to the problem of living for a long time, let's say, or generating a particular output. And I think the same thing is happening in in neuron nets. And I think what's happening is that the, what you're doing by doing all this training and so on is you're producing these incredibly ornate patterns of what happens inside. And so when you say, well, what's it really doing? Why does it work? Well, it's, it, it is not, there isn't just some simple mechanical kind of engineering explainable type thing that's going on. Instead, and and you have to see the pictures to to really appreciate this, that, you know, instead it's this very elaborate kind of pattern of behavior that happens to achieve the some particular objective. And that's my guess about what, you know, when we, nobody's been able to really unpeel neural nets, they're pretty complicated mathematical structures. And it's been very hard to kind of get to the essence of what's really going on in these systems. You know, I tried back in the beginning of the '80s and I, I, I, I got some distance and then I decided neuron nets were too complicated and I simplified them to get to these cellular automat and things that I've studied a lot. And, you know, then for the next 40 years, that's what I've been using as my sort of go-to model for very simple systems. And now I, I finally got some kind of merger of those two kinds of things which I think will allow one to kind of get a more intuitive sense of what's happening inside a neuron net and why why that actually works.

It's funny, when we started our conversation, you talked about how we we are now entering a new era of physics, right? For the past 300, 400 years, analytics and calculus and, you know, this sort of thing were really where where you'd go to if you wanted to make progress in physics. But, uh, now in the last few years, we have AI coming on board as well. Not just computation, but can AI really be creative in in the sense that can it really extrapolate or can it only interpolate? And within that context, will AI be able to help us solve science, solve physics at the end of the day?

Okay, so several different things. I mean, first of all, in the solve science, you know, we run right into computational irreducibility here because if you are asking the question, can a neuronet, you know, what we can't figure out by, you know, we haven't been able to find a formula or something for such and such a thing, will the AI be able to do that? Well, the problem is there just isn't any way to find such a formula. We're stuck. So the AI is not going to do any better. And in fact, what we will be best off doing is just doing the computation, so to speak. So I think that's the, that's the, the clear answer to that. Now, in the question of sort of the creativity of AI, being creative is easy. The question is whether what you get is is something you care about. In other words, you can get some random collection of pixels. It's very easy to get an an never seen before random collection of pixels. The question is, will that random collection of pixels, or what you might think of as a random collection of pixels, will that collection of pixels be something that we humans care about? And that's, I think, a, you know, the way that AIs have been trained is on things that we humans have decided we care about. And so this question of whether whether you'll be able to find things that, whether they sort of putting just sort of randomly choosing things, will we get to things we care about? You know, I've spent a lot of my life studying the kind of computational universe of what all possible programs do. And what you discover there is there are these programs that do these amazing things you would never have thought of. And when you first see them, you say, that's really neat. I don't know why I care, other than I can see it's a very intricate pattern and it's very neat. But various times in my life, at least, I realized, oh my gosh, that pattern that I just thought was neat turns out has some big technological consequence or some big scientific consequence. And it wasn't just neat, so to speak, it connected to something that I care about. We see the same thing in mathematics. You know, you can go off and you can enumerate all possible theorems, but most of the theorems you can enumerate will be ones that, you know, we don't have a reason to care about. We build up this kind of sort of social structure of the paths and mathematics that we care about. And it's the same thing, I think, in in, uh, you know, in this this question about, will we be able to find sort of theories that we didn't think of by by sort of rattling around in a sense? My whole physics project is about doing that. I mean, it's about starting from, kind of not starting from the universe as we know it and trying to reverse engineer what's happening. It's instead about starting from all possibilities, sort of the the the the creative effort of of coming up with all possibilities and then seeing what the consequences of that are. So I think, you know, as a practical matter, uh, you know, a lot of the technology I've spent a lot of time building is the things that human brains are not very good at doing. It's building these big towers of computation that are something that we can do with computers that human brains don't do very well. What what sort of the current round of AI and neuron nets have given us is a way of doing things that human brains do do fairly well, but now we can do them in some some kind of automatic way that's a very useful kind of linguistic interface to these kind of deep computational kinds of things. It isn't, it isn't removing the deep computational kinds of things and it never will because of this phenomenon of computational irreducibility. But it is still, and in terms of knowing, for example, what, you know, being able to make sort of analogies between different human things that we've thought about, yes, that can be very useful things done done kind of with with the current round of of sort of neural net AI. But I think it is, it is an interesting point that in a sense, the the whole enterprise that we've been talking about about our physics project and so on is precisely making use of the kind of creativity of computation to say, we're starting off from the infinitely creative thing that is the rou ad, and we're kind of deciding which slices of it we humans care about because because of the fact that we are observers of the kind that we are. And that and that sort of slicing of the set of infinite creative things is giving us the physics that we are currently familiar with. Were we to change the way that we are, we would perceive different physics and we would end up with kind of a different being caring about a different part of that kind of creative space.

Well, Stephen, it's been an absolute pleasure. Thank you for coming on the podcast. Thank you. You asked some very interesting questions and you made me think about some things I hadn't thought about before. So thank you very much.

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