Tim Palmer: Non-Locality, Universe on a Fractal, Quantum Mechanics

Okay, we're here at the Royal Society of London with Tim Palmer. Professor Tim Palmer, welcome.

Thank you. Nice to be here. Nice to talk to you. Nice to meet you. Nice to go out for lunch earlier, and you showed me around. I appreciate that.

Tell us about the state of fundamental physics today.

Well, physics, I mean, in general, has been a phenomenal success story over the last, uh, well, since my career. So, I mean, I started, uh, researching, you know, uh, doing my PhD back in the 1970s, and, um, you know, physics has gone from strength to strength. Um, and in certain aspects of fundamental physics, you know, it's gone from strength to strength. You know, we have completed the Standard Model. Um, you know, just this morning I've been at a, um, a meeting discussing some of the results from the James Webb telescope and the implications for our understanding of cosmology. All that's fantastic.

However, having said that, some of the problems that were there at the beginning of my research career, you know, have hardly moved forward. And they evolve around things like the fundamentals of quantum mechanics. What do we mean by a measurement? What is the measurement problem? How do we interpret, you know, the entanglement of particles? Is it really telling us that the universe is non-local? Does it really have the kind of spooky action at a distance that Einstein so hated so much? And then, above all, you know, we still really haven't made, uh, we haven't, at least we haven't solved the problem of how to unify all the, the electromagnetic and nuclear forces with gravity. You know, and that, um, some people would argue we haven't. I mean, some people argue String Theory took us a long way forward. Others would say, well, we haven't actually moved forward at all since the 70s. So it's a kind of a mixed bag, I would say. Um, but I still think we're facing very fundamental questions, particularly in this issue of, of, uh, quantum mechanics and gravitational physics, um, where we've sort of got to get back to basics, I think, a bit more and and, uh, and start asking the deep questions rather than just plowing through calculations.

What's the missing link then to solve the issues between quantum mechanics and general relativity?

I mean, I personally think it's to do with, at, at the sort of deepest level, it's to do with, uh, understanding the, uh, if you like, the holistic nature of quantum physics a bit more explicitly. And we, we know that that's the case, or, you know, many of us believe that's the case in, in gravitational physics. And Mach's principle in gravitational physics is, which I think, although it's not something that's been rigorously proved, I think many, most physicists are sort of sympathetic to the idea, which is that the, you know, when you, when you spin round and your arms flail out, or when you watch a Foucault pendulum and its plane of oscillation moves as the Earth rotates, and you say, well, what actually is causing that rotation? Or what is causing my arms to flail out? Mach's principle says it's the distant mass of the universe. It's the mass, it's the totality of the universe that is doing that. And I think we have to, we have to kind of take that notion, uh, and move it more into the quantum regime. Um, and, you know, sometimes, I mean, this is not a completely new idea. People like David Bohm and Basil Hiley, their famous book on quantum mechanics was called *The Undivided Universe*. Uh-huh. So it's a, it's a concept that's sort of been around, but I think we have to take it a bit more seriously and recognize that both in the quantum world and the gravitational world, the local laws of physics are probably determined by the large-scale structure of the universe.

So what is Mach's principle? Can you state it rigorously, or you can't? And that's the reason why it's unproved rigorously?

Well, Mach, as I say, Mach's principle is that the, you see, the question is, when we rotate ourselves, if we spin round, our arms flail out, or if we drive around a curve in the, in the road, we, we kind of get pushed to the side of the car, and we say, you know, that's, that's caused by the centrifugal force or something like that. But that's just a label. And the question is, let's take the rotating case, what actually is it that, um, that determines whether you're rotating or not rotating? And Mach, I mean, this actually goes back to Bishop Berkeley back, you know, in the earlier times, but in the 19th century, Mach and Mach said, well, the reason why we say we're rotating is because when we rotate, we see the distant stars, uh, moving. And he said, well, that's not a coincidence. It's because of the kind of gravitational effect of the distant matter that defines what a, a non-rotating frame is and a rotating frame. And this was a very big, uh, inspiration for, for Einstein in his coming, in his developing his theory of, of general relativity. Um, and indeed, in general relativity, there is an effect called the Lense-Thirring effect, where if you have a, a, a mass, a massive shell, if you like, and you rotate it, that rotating mass drags the local frames of reference around with it. So, in other words, locally, that, that tells you, you know, whether you're rotating or not.

Now, it then becomes a kind of, you know, then you ask the question, does, does general relativity automatically, um, uh, is it, is it internally consistent with Mach's principle? Well, you can have space-times which, where there is no distant matter. So the Schwarzschild solution for an isolated star or black hole, um, doesn't have, doesn't have distant matter. Um, but in some sense, you have to define then what you mean by rotating or non-rotating. In the case of a Schwarzschild, it's a non-rotating solution. You kind of have to impose that non-rotating condition by a boundary condition at infinity. Um, but most, I think most cosmologists believe that, um, in the real world, where, you know, we don't live in, you know, the Earth is not an isolated mass in an otherwise empty universe, it's, it's part of the universe. And, and I think most cosmologists would, um, accept that that Mach's principle probably is the key reason why we experience so-called inertial forces in rotating frames of reference, because these are the ones moving with respect to the distant stars. But it's hard to prove it because we don't yet know, you know, we don't even know whether the universe is infinite or or finite. We don't, don't really know how much matter there is. Uh, so it's become, Mach's, I, I like to think of it like this. Mach's principle has become a little bit like, um, some of these famous sort of conjectures in, in number theory, like Goldbach's conjecture, you know, that every even number is a sum of two prime numbers. I mean, everybody kind of believes it's true, but nobody knows how to prove it. So it's not very high up in the research agenda because nobody knows how to prove it. And in a way, I think Mach's principle is similar. It's difficult to, to know how to prove it rigorously, but, um, but I think most cosmologists would sort of accept that that it's something, there's some truth in it. And I think, as I say, I think that we've got to get to that sort of stage in thinking about quantum mechanics as well.

Okay, fill in this blank for me. Mach's principle is to general relativity as ______ is to quantum mechanics.

Well, okay. I mean, I have my own, you know, you're, you're putting me on the spot. I mean, I have my own, um, you know, ideas about, um, about, uh, quantum mechanics, or quantum physics, let's say. Um, and I've tried to propose, you know, an idea which I call the cosmological invariant set postulate. And this is very much, you know, um, motivated by chaos theory. That there are systems which exhibit this extraordinarily beautiful, uh, geometric structure in their state space, and that they, you know, if you leave, if you leave these, if you start them from an arbitrary initial condition and just leave them for a long time, eventually they just evolve on this, what's called an invariant set, or sometimes called an attractor. Um, but these are fractal attractors. And my, my, uh, kind of principle, which would be consistent with what I'm talking about here, the holistic nature of quantum physics, would be the universe evolving on a cosmological invariant set. So Mach's principle is gravity, as you know. Um, perhaps this cosmological invariant set postulate is to, to quantum physics. And the reason for saying that is that it can explain some of these difficult issues like entanglement and Bell's theorem without having to invoke non-locality or indeterminism, all the sort of things that Einstein hated.

So, Professor, I'm a stickler for words, and I noticed a few times you were going to say that. Okay, so I have three questions here. Einstein's theory is is consistent with Mach's principle, then with Mach's principle, then you corrected yourself and said, is internally consistent with Mach's principle. So one of the questions I have is, well, what's the difference between being consistent and internally consistent?

Okay, so we'll get to that. But just in a moment, so I don't forget. You corrected yourself when you're saying quantum mechanics, you switched it to quantum physics. So I'm curious what the difference is there that you, you see. And then another time is invariant sets, which is the same as a fractal attractor. Okay, if it's the same as a fractal attractor, why did you rename it as invariant set? Okay, so those are three questions. The first one was about internal consistency versus consistency.

Okay, on the first question. Um, there are solutions of Einstein's equations, and I mentioned earlier, the isolated body and the Schwarzschild solution. Um, another one is what's called de Sitter space, you know, which, which has no matter in it, and yet space is curved. It's curved by the cosmological constant. Um, so you can, you can, um, you can come up with space-times which are, which satisfy Einstein's field equations, which are not Machian, you know, because there's no distant matter for that to be. However, what I'm saying is, um, the real world, forget what I actually said, because what I'm trying to say is the real world, which is governed, you know, which is governed by, let's say, to a good approximation, one of the Friedmann-Robertson-Walker solutions, the cosmological solutions of Einstein's equations, um, is consistent with Mach's principle. So not all solutions of Einstein's field equations are consistent with, uh, with Mach's principle, but the, the, the Friedmann type of equations are, and we live in a Friedmann type of universe. That is, we don't live in de Sitter space, and we don't live in an isolated Schwarzschild space. We live in something which, at least on the very, very, very large scale, approximates quite well to the Friedmann-Robertson-Walker cosmology. So from that point of view, that's consistent with Mach's principle.

Um, quantum physics versus mechanics? Yeah. No, quantum mechanics is a very specific theory. It's the theory that Heisenberg first proposed, almost 100 years ago next year, I guess, and then Schrödinger very shortly afterwards with his wave mechanics version. So that's what we mean by quantum mechanics. It's a specific theory of quantum, quantum phenomena. But I use the word quantum physics in a slightly more generic way, which is, you know, the set of all observations of the world, you know, involving atoms and particles and entangled systems, where maybe quantum mechanics isn't the final word. I mean, that's would be my belief, that it's not the final answer to how to describe quantum physics, um, in as accurate a way as possible. I see.

Okay, now the third one was invariant sets versus a fractal attractor. Yeah. Well, the concept of an attractor is that if you start with any old initial condition, um, and you run your equations, your differential equations, forward in time and just leave it for, you know, a very, very, very long time, then the state gets attracted to this special, um, uh, special fractal. But the hypothesis I want to put forward is that the universe has always been evolving on this geometry. So it's never, it's never been attracted towards it. It was, it was always on it, and it always will be on it, potentially, you know, for an infinite time in the past, or a finite time in the future. It might end up being cyclical. It might end up repeating itself, um, at some stage. But the point is, I'm not invoking the concept of it being attracted, the states of the universe being attracted towards this geometry, but it's just this is the geometry that it evolves on. And the, the concept behind this geometry is that it's, it's called an invariant set because if you're on it today, you always will be on it in the future, and you always have been on it in the past. So the set is, in some sense, or the geometry is invariant under the, under time evolution. It's invariant under the propagating forwards in, or indeed backwards in time. So calling it an attractor, I don't like using that word because it implies that I'm thinking about states. Yeah. Think, you see, the point is, if you're outside it, you're violating. And this is an important point of my reason for believing why this is important for Bell's theorem. If you don't lie on the attractor, you're inconsistent with your, this basic postulate. The states which don't lie on the invariant set, uh, by definition, don't satisfy the postulate that, that states of the world evolve on the invariant set. So the concept of being attracted towards it is, is not really a useful idea in this respect.

I see. So, in other words, the invariant set is the attractor, but we just, yes, our states are always on it. Exactly. That's exactly right. The invariant set is the attractor, and states are on it. They always will be, they always have been. And any point, any hypothetical, this is the point, if you, if you imagine a counterfactual world, a hypothetical world in your head, where you slightly changed, you know, you've changed, you've slightly moved the position of that chair, hypothetically, you know, you haven't actually moved it, but you say, well, maybe I might have moved it. You've invented a counterfactual world which doesn't really exist. It's a hypothetical world. Now, if that world, if that hypothetical world, if you've nudged your, the state of the universe off this invariant set, when you, in, in formulating such a hypothetical world, then I would say, uh, uh, uh, that's inconsistent with, uh, the laws of physics as I, as I myself see.

So you may as well have said, what if this, what if this ball lifted up into the air? Like a counterfactual such as moving this chair. A nudge that takes you off the attractor set or the invariant set, right? Is equivalent to saying, what if there was some elephant that just appeared here? But even in that case, there is some quantum mechanical chance that an elephant can appear here. It's just minute.

Well, there's a quantum mechanical chance. I mean, of course, there is a chance an elephant might walk into this room, right? In five seconds, who knows? I don't know. It's always possible. Um, should we wait and see? No, didn't happen anyway. But it could have happened. But that's not quite the point I'm making. The point I'm making is, if we took the world as it was 20 seconds ago, and said, okay, no elephant walked into the room 20 seconds ago, but is a world where everything was the same, you know, you and me talking, uh, the people in London doing their shopping, you know, the Earth going around the sun, the sun going around the Milky Way, everything the same, except that an elephant walks into the room. Is that hypothetical world? Because it is hypothetical. It's not an actual world, because the elephant didn't walk in the room, right? It's just maybe we're hypothesizing the possibility that everything stayed the same, but an elephant walked in the room. What I'm saying is, if that hypothetical state does not lie on this invariant set, and I don't, I'm not saying that it does or it doesn't, but if it didn't, then that would not be consistent with the way I formulate the laws of physics. The point about this is, in quantum, in quantum physics, this notion of counterfactual, counterfactual worlds actually occurs quite a lot of the time. And most of the, um, the so-called no-go theorems, and the classic one is, of course, Bell's theorem. Implicitly, there's an implicit assumption, and it's not, you know, it's in regular proofs, if you like, it's not drawn out terribly explicitly, but there's an implicit proof when you introduce, for example, a hidden variable model, that that hidden variable model has the property that you can, for example, keep your hidden variables fixed, but change the actual measurement or orientations that you actually did the measurement with, and assume that that, that hypothetical counterfactual measurement is consistent with the laws of physics. That's, that's an implicit assumption. And I'm trying to draw out an example, and my examples based on this notion of an invariant set. And therefore, it brings in the, you know, the, the large-scale structure of the universe, where those counterfactuals would be inconsistent with the laws of physics. So, what being the case, you no longer have to, uh, conclude that the world is non-local or indeterministic or anything like that. It's just that certain counterfactual worlds, which might seem plausible, you know, to your, in your head, are actually technically inconsistent with the laws of physics.

So some who shall not be named may call that conspiratorial.

Okay, so I want you to explain your thoughts on that, as well as tie it into when people say that the universe is not locally real. What are your views?

Well, just on the last point, this is, of course, is, was, was the headline, if you like, when, you know, Clauser, Aspect, um, and Zeilinger won the Nobel Prize, um, a year or so ago. Um, the, the headline, you know, for, for showing that Bell's inequalities were violated, the headlines, you know, the world is not locally real. So, what I mean, what they mean by that is that the world is, is, you know, either it's not, there's something kind of inherently indeterministic about the world, or at least the world can't be described with, kind of, with deterministic equations. Um, or that, and possibly, and that, um, the world, the world is non-local. So, what does that mean? The world being non-local means that the result of an experiment which I might do here in my lab, not that we're in a lab, but if we were in a lab, um, uh, the result that I get, if the world is non-local, the result that I get can depend on whether a colleague of mine, who could be on the other side of the world, or in principle on the other side of the galaxy, or indeed on the other side of the universe, um, uh, what, how he set up his measurement. Mhm. So the setup of a measurement, you know, a gazillion miles away, can affect the outcome of, of an experiment here. That, that's what non-locality means. Now, back in the 30s, Einstein, Podolsky, and Rosen, they wrote this famous paper in, in '35 on, you know, introducing this, uh, concept of, of, of entanglement and so on, and they just, they just said, well, that's manifest nonsense. You know, I mean, the world could never be like that. That's just crazy. Um, and yet, you know, um, whatever we are now, 90, yeah, almost 90 years later, we're somehow concluding that, yeah, the world might be non-local. The one thing that they rejected is completely barking mad. So, um, so this is why I, you know, I have been, and, you know, kind of motivated by my background in nonlinear dynamics and chaos theory and chaotic geometry, I, I've been kind of going back with an absolutely, you know, fine-tooth comb through the Bell, Bell, the Bell proof of Bell's theorem and pointing out this kind of implicit assumption, which comes, it's usually introduced when, when they introduce a hidden variable model. There's an implicit assumption that the hidden variable model has this property that you can hold fixed the hidden variables and change the measurement settings, um, as you like. So you say, okay, well, I did an actual experiment with an actual hidden variable and an actual experimental setup, but according to my model, I could have hypothetically, counterfactually, could have changed the measurement setting, keeping the hidden variable fixed, and got a, and get it, and I got a sensible result. And I'm saying, um, there are models, and the one based on this invariant set idea is, is an example of that. And we could talk, perhaps talk about another one, because it also comes out of, you get the same result if you discretize Hilbert space. So we'll talk about that in a minute, and that may be a more direct way of seeing it from, if you're an expert in quantum mechanics. Um, but that's, you know, if you, if you have this property of counterfactual definiteness, then, then you can indeed, um, violate these Bell inequalities without having to, uh, assume indeterminism or non-locality.

Now, is it conspiratorial? Um, no, it's not. Um, the conspiracy argument, um, I mean, it, it arose from a paper which the philosopher Abner Shimony wrote, um, shortly after one of Bell's, um, papers in the 1970s, um, where they introduced, it's kind of a bizarre idea in a way, but it's a, you could technically explain the Bell inequalities if, um, if somehow you could corrupt the minds of the experimenter to perform an experiment, you know, where the experimental settings were, uh, let's put it like this, you know, if you imagine the particle that was being, being, uh, measured somehow sent a weird message out through the ether into the brain of the experimenter, and, you know, the experimenter wasn't aware that they were, their brains were being corrupted, and they, and the message was telling them, make sure you set your measuring system like this, you know, so that would be a conspiracy. And that would apply even in the case that the measurement was made by some random number generator.

Well, yeah. You're right. So Bell, the reason Bell didn't run with that was precisely because, sort of, what you say, it, it, it invokes things you don't really want to get into, like how does the brain work, and what, what do we mean by free will, and all sorts of sort of quite metaphysical stuff. So, so he, in his paper, which was the response to the Shimony paper, he said, look, let's forget about humans, because it's just complicated and it's kind of messy. And just say, let's imagine that the measurement setting is determined by a pseudo-random number generator, you know, which spits out either the number, let's say, zero or one. So zero, you set it, you set your measuring, measuring system one way, and one, you set it in a different way. Um, and this pseudo-random number generator is such that you feed in a kind of an input number, and what it does is it picks up on the millionth digit of the input number. So, you know, you could, you could, you could imagine a computer doing a calculation of a, solving a quadratic equation or something, where the number was definitely an irrational number, and, you know, eventually it would come to that millionth digit, and depending on whether that millionth digit was odd or even, then the output variable would be zero or one. Okay, so that, that's fine. So then Bell said, the crucial issue is, um, is that millionth digit important for any other purpose than setting the output of the, the pseudo-random number generator, and hence setting the measuring apparatus? Does that, that millionth digit serve any other purpose in the world? Okay. Now, he said, you know, he said, my intuition is that it doesn't serve any other purpose. But at the end of his paper, and this is a crucial point, he said, actually, I'm not completely sure that's correct. The correct conclusion, and it may be that one day somebody does come along and explain why that millionth digit could be important for a distinctly different purpose.

So Tim came along. Well, whether I came along or not, but I am claiming that, you see, changing that millionth digit is again an example of a counterfactual. And if, just for the sake of argument, if that changing that millionth digit took you off this invariant set, then that perturbed state of the world, where the millionth digit was different, would be, the world, the whole world would be inconsistent with the laws of physics. So all the galaxies would vanish in a puff of smoke, you know, puff of metaphysical smoke, let's say. Everything in the world would vanish in a puff of metaphysical smoke. So that millionth digit is, is not only important, it's vital for the existence of everything in the world. So that would be, now, right or wrong, don't know. But this is a contra, this, this would contradict Bell's intuition, which, as I, I tried to emphasize, he wasn't 100% sure about, that that millionth digit could actually be an important piece of information for, for other purposes than just setting the apparatus.

There was a philosopher named David Lewis, and he had this construct called possible worlds and impossible worlds. So are you suggesting that what's not on the invariant set is then an impossible world that we thought was possible?

That's exactly right. That's exactly right. So it looks possible, and our brains, you know, which have limited, um, computational capacity, let's say, you know, think of it as possible, but actually it's an impossible world. Um, and I just want to say one other slightly technical thing, because sometimes this is another argument people, uh, raise with me. Sure. Which is that, you know, I'm, I'm seemingly invoking, um, tiny, tiny, tiny, tiny, tiny perturbations, you know, to the millionth digit, or to the billion, could be the billionth digit or something, and saying that has a radical effect on, you know, on this, the ontological status of the world. And isn't that fine? Isn't that very fine-tuning? You know, isn't the world very fine-tuned, then, that you're not even allowing me to change the millionth digit? One of the things I try to emphasize in the geometry of, of fractals, is that real numbers are actually not a very useful tool for, for looking at the geometry of fractals. And there's a whole different type of number system called p-adic numbers, which are completely bread and butter to, to number theorists, which tie much more closely to fractal geometry than real numbers. And these p-adic numbers are associated with a very different type of distance function or metric compared to real numbers. So real numbers have what's called the Euclidean metric, which we're all familiar with. You know, if I, as my fingers approach each other, their Euclidean distance is getting smaller and smaller. But the p-adic distance behaves a bit differently to that. And in particular, if two points are on this invariant set, then, and their p-adic, you know, their p-adic distance can be small, but a point that's in the gap between them actually has a very large p-adic distance to a point on the invariant set. So this is a very robust, uh, scheme from the p-adic perspective. Um, and this is one of my sort of, sort of goals in life is to educate physicists much more with p-adic numbers, because I think they give whole new insights into the way of the world. And we might talk about one of these perhaps a bit later, which is the difference between, um, determinism and predestination.

So explain to this face, explain how is it, how are you supposed to know that what you said was an impossible world versus a possible one, a priori? Like, you can always say, look, what you suggested was impossible. How do you know?

No, you don't know. And that's, I mean, one of the, one of the things about the world is we can't compute. You know, there are certain things we just can't calculate, compute. I, I don't know. And there's no way I could know that something I might say in two minutes will cause you to be so crossed with me that you'll turn down, you'll turn off the camera and storm out. Almost happened two minutes ago already, I bet. You know, I can't prove that. But all I, but what I can compute, what I can compute is that if you, let's, let's put it in terms of EPR, the Einstein-Podolsky-Rosen, you know, normally, you know, we've been talking about spin and things, but, you know, they, they framed it in terms of position and momentum. This is the original Heisenberg thing, you know, you can measure spin, you can measure position, or you can measure momentum. Now, the way I frame this is, if you measured the position of a particle, then your measurement of the counterfactual, which is that what result would I have got had I measured the momentum of that particle, whereas in the real world I measured the position of that particle, I'm claiming that that would lie off the invariant set. The momentum measurement would lie off the invariant set when the position measurement lay on the invariant set, or conversely, if you had measured momentum, then the position measurement would have lay off the invariant set. So I can't predict what you'll measure, but what I can predict is whatever you measure, the other variable, the counterfactual variable, would lie off the invariant set. So as soon as you've, you could say, up to the time where you did the measurement, you were free to to choose position or momentum, choose away as you like, whatever you want. If you want to use the millionth digit of some irrational number, that's good with me. If you want to use your grandmother's birthday, that's good with me. If you want to use the Dow Jones index, that's good with me. But so I can't, I don't know, and you're free to choose that. But having made that choice, then the counterfactual world where you say, what would I have measured on that particular particle had I measured momentum, whereas I actually measure position, then that, that counterfactual state's off the invariant set.

Notice that so far, technically, the discussion on Bell's theorem or Bell's inequalities has lasted maybe 20 minutes. Some people who are physicists may understand it, some people who aren't may not. But why is it the case that it's taken so long to explain something minute about Bell's theorem when in math, if you have a proof of something, it's quite clear? Now, maybe it's difficult because you're not technically proficient, like Andrew Wiles' proof took quite some time to go through. But this isn't at the level of Andrew Wiles' proof in terms of abstraction or mathematical ability or what's required mathematically. So explain to people who are unfamiliar why is it that you said you went through this with a fine-tooth comb? Many others have gone through his theorem or his proof with a fine-tooth comb. Why does it even take through with a fine-tooth comb, tens of decades later?

Well, I can only hypothesize about this, right? I don't know for a fact. But I think, I think most physicists tend to associate, let's say, so one of, one of the, you know, one of the conditions of Bell's theorem is about, you know, is the world deterministic or indeterministic? Now, what do we mean by determinism? And I think most physicists tend to think of it as, and, and indeed, you know, many examples are like this, as initial value problems. You, you give me some initial condition, you know, at some initial time, and I have a, you know, a computer, or I, you know, I can do a calculation in my head or something like that, and I tell you, given that initial condition, what the, what the, what's happening in the future. I mean, weather forecasting, you, you take a gazillion weather measurements and use those to determine an initial state of the atmosphere, the atmosphere today. You stick all that into a big computer, churns away, comes out with a state tomorrow. Now, when you frame it like that, then there's no reason at all why counterfactual, um, worlds aren't consistent with the laws of physics, because I can change, I can change the initial state, you know, as I like. I mean, you, you know, the initial conditions are usually, you know, just given, they're prescribed by you or by somebody, and then you do your time evolution. If you want to have a different initial condition, that's fine. Um, or, or in fact, what you can do is with deterministic equations, you can say, um, you know, let's say, I mean, today we're looking out over central London, it's a reasonably, reasonably sunny day for England. You could imagine, okay, well, let's imagine it's raining. Okay, I'm going to, I'm going to sort of, That's so unlikely in London though. Well, you know, sometimes it rains. So I'm going to, I'm going to take, I'm going to put, I'm going to change all of the isobars, the pressure, so there's a big low-pressure system right over, over the UK. Okay. I could take the laws of the Navier-Stokes equations, the laws of, you know, classical physics, fluid mechanics, and so on, and I could sort of, you know, work them backwards in time to produce an initial state, let's say two days earlier, that would lead to it being raining today. And that, you know, that what would happen is that somewhere over the North Atlantic, the pressure fields and the wind fields would be slightly different to what they were. But, you know, there's nothing, there's nothing in those laws, those classical laws of physics, that would, say, okay, that that slightly different initial state was somehow inconsistent with the laws of physics, right? Just do it. So when you have, when you have like standard, a standard initial value problem, what I'm saying about counterfactuals sometimes being inconsistent with the laws of physics never arises because you can always change, you, the initial state, you can perturb as you like. So this only comes about, and this is what, so this is, this is my argument, this only comes about because I am moving away from that paradigm to where my definition of the, of physics is this geometry of the invariant set. The dynamics is encoded in these sort of p-adic type of equations, which encode the geometry of this invariant set. So I'm moving away from the, you know, the standard initial value problem to saying this is actually a problem in geometry. And when you do that, then, then you have this possibility that states which don't, counterfactual states which don't lie on the invariant set, then become inconsistent with the laws of physics. So my answer to your question is, I think it's because people have thought somewhat narrowly about what a, you know, what a hidden variable model or what a deterministic model of quantum physics would look like. Um, but this in turn then brings me to the point where we started with, which is, you know, that what is this invariant set? It's the invariant set of the whole damn universe. It's, it's the totality of everything. It's, it's thinking of the whole universe as a dynamical system evolving on some, you know, cosmological invariant set. So that's why I'm saying you cannot dissociate the local, uh, laws of physics, you know, which, which govern how a Bell experiment would work in the lab, from the very large-scale structure of the universe. Because that's where the invariant set concept comes in. And that's why I say that is the kind of parallel, if you like, with, with Mach's principle for gravity. Uh-huh.

So, okay, being on the attractor set, evolving it forward, you mentioned that you can continue it at infinity and evolve it backward at infinity, which has inside it infinite and infinitum. So let's talk about infinity. Yeah. Some people think of infinity as just a placeholder, like a heuristic for this is a sufficiently large number beyond our grasp. Right? Okay. Computationalists are fond of that. What do you make of the concept of infinity in physics?

The concept of infinity in physics is really interesting. And let me ask you a question, Kurt, because we have, you know, we have classical physics, you know, the laws of fluid mechanics, um, for example, is a good example of a classical system. And then we have quantum mechanics, which replaced it. Now, you have to, so if you ask about infinity, as you say, there, you have two possibilities. One is that actually, in physics, we know infinity sort of exists as a concept in mathematics, of course, it does. But in physics, is infinity, when we use infinity, or or conversely, one over infinity, an infinitesimal, when we use these concepts, um, do we really mean infinity is absolutely literally infinitely big, bigger than any finite number? Or is it just a placeholder for a very big number, but we don't particularly care exactly how big it is, but it's, it's a very big number, and, you know, the laws, the laws of physics don't depend sensitively on that number, and, you know, all experiments, it doesn't really matter if that number is like a googol or a googolplex or something like that. So do you think there's a difference between how infinity is used in classical physics and quantum physics? And if so, how, how would you see it? Do you think infinity is more of a, you know, as, as an, as a number, let's say, infinity as a number that's bigger than any finite number, so not a placeholder? Do you think that's more of a concept in classical physics or in quantum physics?

In quantum physics, there are infinite-dimensional Hilbert spaces. And then because of that, there's something like the Stone-von Neumann theorem. And that's one of the reasons why QFT is not so trivial compared to quantum mechanics, because you have the conjugate variables of X and P.

Okay, well, look, I, I think I agree with your answer, but I think you're making it too complicated. I think you're, you're right, but let me, let me put it like this. So, if we, you know, even at high school, when we learn Newton's laws of motion, we, you know, they're framed in terms of the calculus. F equals, I mean, I'm assuming that that's, I'm not even sure today whether high school students do the calculus or not, but anyway, certainly first year university, say, force equals mass times acceleration, and acceleration is the second, you know, rate of change of position with respect to time. So we use Newton's calculus, Leibniz's calculus, and the calculus involves, um, you know, infinitesimal numbers like d, d by dt. DT is an infinitesimal number in, in, uh, in calculus. So, so you might think that, you know, infinity or an infinitesimal plays an essential role in classical physics. But, you know, people might think, well, in quantum physics, it's all about discrete jumps. You know, everything is discrete jumps of energy, and therefore it's all somehow finite. Everything is finite. But actually, it's completely the other way around. Um, because in classical physics, I can take a, a differential equation, you know, for, I mean, we do this with weather forecasting, of course, um, where the, you know, the, we have partial differential equations that underlie the movement of air, but they're represented on a computer with finite derivatives. So that we don't have like d by dt, we have delta by delta t, and these deltas are finite things. And, you know, we know that at least for short-range forecasts, the, you know, that that does pretty well. So there's no, there's no kind of essential reason in classical physics why we need to be working with a continuum, the real number continuum. We can just work with discrete numbers, and we get answers. If we want to get a slightly more accurate answer, we'll have the time step or quarter the time step. But you tell me how accurate you want to know it, and I'll tell you the discretization length and the discretization time that will give me, yes, an answer to the accuracy you want. But in quantum mechanics, it's completely different. Because the basic concept behind a quantum state, it's sort of, you, you mentioned this effectively, is it's an element of a Hilbert space. And a Hilbert space is a vector space. In fact, in quantum mechanics, it's a vector space over the complex numbers. Um, but the point about it being a vector space is that you, you want to be, or you need to be able to add together two vectors, in other words, two different quantum states, and the resulting addition is itself a quantum state. So that's a, that's a really important property. Now, if you start discretizing Hilbert space, you will typically lose that property. You'll, you'll, you know, you'll add together two vectors, and the, and the resulting vector will kind of lie in between two of your, um, you know, two of your points in your discretized space.

Wait, that's not so obvious. So let's say you make a grid. Okay, so it's just integer steps. I mean, make a grid and take, take a vec, say you have an origin and you have two vectors which point to, um, two, two of the points on the grid, and add them together. That the sum of the two is going to split the difference, unless you've got a grid point which splits the difference. Then your vector will no longer be, um, will be in the space. Let's say you have something that's unit one in length, and then another that's unit one in length, then you get something that's unit two in length, but along the same axis. But what's wrong with that? No, that, that, that might work. But if you, if you imagine, um, uh, well, if you imagine to, you know, if you have, if you discretize, say, a circle, and you imagine two vectors pointing to, let's say, neighbor grid points, and then add together, you're going to split that difference, and your vector will, um, will typically then not lie on that, on that, um, on on either will.

Okay, so it depends on your discretization. It will depend on the discretization. But the, but the generically, to get, you know, these algebraic properties, you need a continuum space. And actually, this, this, I'm not the first to make this point. This was made by, um, Lucien Hardy from Perimeter some years ago, when he came up with what he called reasonable axioms for, um, for quantum mechanics. And one is this notion that, it's called the continuity notion, that you don't have, you don't have a space where you can get discrete jumps. Even though quantum mechanics is all about discrete jumps in the so-called unitary phase of quantum evolution, you actually need continuity. It's a critical property. Um, now, but then the question is, why is it a critical property? And it's only a critical property if you believe that these Hilbert spaces and Hilbert vectors are the fundamental objects of your theory. In other words, if you say, what is the, you know, if I have my theory of quantum physics, what is at the deepest level? Now, in quantum mechanics, at the deepest level, is Hilbert space. That is the, that doesn't go any deeper than that. That's it. Okay. Then you have to have the continuum, uh, of, of Hilbert space, of Hilbert, um, Hilbert states, rather, to describe quantum mechanics. On the other hand, if you say, well, um, uh, if you say that, which is sort of what I'm trying to suggest by virtue of this cosmological invariant set postulate, that there may well be something deterministic that underpins quantum physics, then the Hilbert states are not really fundamental. All they're doing is, they're the mathematical quantities that you would use when you know that there is some inherent uncertainty in your knowledge of the system, and you want to represent that uncertainty, um, in a kind of statistical way. So Hilbert states coupled with Born's rule, which is about probabilities of outcomes, um, then just becomes, if you like, it's not a fundamental, um, property of your theory. It's just something which is useful to use when you want to describe things in a statistical way. Um, and this is actually again, where p-adic numbers come in, because on a, on a fractal, you can, you know, you can add and multiply using these p-adic numbers, and the result is a p-adic number. So you have this closure under addition and multiplication at this deeper deterministic level. Um, so I, I, I have a, I mean, I have a, a scheme which leads to a particular type of discretization, which I feel might be too, a bit too much technically to talk about here, but it, it, it exactly has all these properties that you add two vectors, and typically the sum doesn't lie, you know, in the discretized space. However, it has this deeper deterministic underpinning. But importantly, it has exactly this property that, that when you look at the count, if you try and estimate the, oh, sorry, if you try to, um, define the quantum states associated with entangled particles, where you do these counterfactual measurements, then the...

Hilbert to describe the Hilbert States, you will need strictly irrational numbers, either for the amplitudes or the phases of the quantum state. And those are things that are forbidden in the discretization. So this captures precisely this notion of moving off the invariant set, but now it's framed in terms of rational versus irrational numbers in the definition of the of the Hilbert state. Um, so I think actually that is a probably a more an easier way to, you know, for a, say, for a practicing quantum theorist to kind of get to the type of model I'm trying to propose here.

So you have two theories: one called rational quantum mechanics and another called invariant set theory. Right? Are those two the same? Yeah, I think they're the same. I think they're the same. But I, I have to confess, they're still, you know, I, I believe they're the same. It's a bit like, why are you, why are you not sure? Well, because there are some technical details which I, which I haven't yet figured out. I need some, you know, I'm kind of reminded of, you know, in the, I'm, I'm rising above my station now considerably, but if you'll forgive me for it for saying this, because I'm, I'm sounding a bit bigheaded to say this, but I'm kind of a little bit reminded of the, you know, in 1946 or 7 or whatever it was, when Feynman came up with his theory of QED, and, uh, and Schwinger had a theory of QED, and everyone said, well, these look completely different. And then Freeman Dyson actually sat down and looked at them carefully, actually, no, they're the same thing. And I think the same is true here, but, but I need a Freeman Dyson to.

So you have hints of this unity? Yeah. And what would those hints be? Um, are they just at the level of intimations or feelings? It's no, no, no. It's at no, it's at the level of, um, as I say, it's at the level of, um, no, it's had a sort of more technical level than that. And it's to do with these, the, so the link with the fractals. First, the first link is p-adic numbers. P-adic numbers, um, you know, have certain representations in terms of digits where the digit takes a number, 0, 1, 2, 3 up to P. Um, and that number P gives you a, um, if you like, the number P describes the discretization of the Hilbert vector. So there are, there are, there are, I can kind of see the pathway to making it completely correct, but I, I, you know, some of the details have yet to be filled.

Has Sabine Hossenfelder worked on this with you? Uh, not. I mean, Sabine and I are both convinced that this is the, broadly speaking, I mean, so my proposal about counterfactuals would, the way it comes into the Bell inequality is through what's called the measurement independence, or sometimes called statistical independence. I've come to the conclusion statistical independence is is not actually a very good phrase, but it's quite measurement independence postulate. And that's the thing that basically says the measurement independence postulate says I can keep my hidden variable fixed and and vary the measurement settings. And this is what, and that, so Sabine and I both agree that is the key assumption that is false in, um, in Bell's theorem. Um, and I think she, she, I think she agrees with me that my ideas about counterfactual definiteness and this invariant set stuff are plausibly correct, but we haven't worked. No, we haven't done any, any technically yet on linking the, the discrete Hilbert space idea to that. I mean, she has her own, she has her own agenda, of course. So I, I don't want to, you know, force, I don't force anybody to work on what I do. When you say agenda, you mean she has her own point of view and her own? No, she has her own interests, you know, she has, she has things she wants to do. Um, and of course, she's got a fantastic, uh, outreach channel as well. So, you know, she, she, um, uh, you know, people, people, people do what they want.

Okay, so in your book, yes, "The Primacy of Doubt," yes, either in the preface or the introducing chapter, the introduction, you say something like, quantum mechanics and general relativity are not merged because of conceptual difficulties, and people don't want to deal with these conceptual difficulties. Now, me and you talked about an hour ago now, or approximately off air, you mentioned some conference where Chris Isham spoke. You were relating this to how when I spoke to Neil deGrasse Tyson, he was more on the "shut up and calculate" side, and I was trying to dissuade him from just being merely on that side. Can you talk about that lecture that happened approximately, no, exactly 50 years ago this year? Yeah. And how that relates to the conceptual difficulties, and what conceptual difficulties even mean? Right. I mean, this, this was literally the first conference I ever went to. So I, I was at that stage, still actually an undergraduate, but I was, I knew I was about to start a PhD program in general relativity at Oxford. Um, and, um, I'd been invited basically by my, uh, physics tutor. I was at an undergraduate at Bristol University, and my, um, actually mathematics tutor, um, had suggested, well, since you're interested in relativity, you should come to this conference. And, um, it was called the First Oxford Quantum Gravity Conference in 1974. So literally 50 years ago. And it was just phenomenal. Um, uh, you know, Stephen Hawking, who hadn't lost his voice then, so he could still speak, he announced his famous, uh, evaporating black hole result at the conference. John Wheeler spoke about, you know, his ideas on, on quantum gravity. Roger Penrose talked about twister theory. Abdus Salam, who was Nobel Prize winner for weak interaction, spoke about his ideas. So, you know, it was just full of, um, amazing people. But Chris Isham, who, um, was a professor at Imperial College, um, at the time, theoretical physics, gave a kind of an opening, um, survey of the, of the field. And, um, you know, there were different approaches to quantum, to quantum gravity, um, but they were all sort of variations of, of quantum field theory, and, and they all were quite, you know, technical, and it all revolved around whether, you know, the theory was renormalizable or whatever, and so on and so forth. And he kind of ended up by saying, well, you know, it's kind of the sexy thing to do are all these complicated calculations, but we shouldn't lose sight of the fact that there are profound, concern, perceptual problems with quantum gravity. Um, and that we kind of, we, if we ignore these conceptual problems, we do so at our peril. And that, you know, people may end up, you know, um, you know, sort of, I'd say, wasting their lives, but they may end up really not making much of an advance because we have, they haven't really solved the, the, the tech, the conceptual problems.

So by conceptual problems, I mean, you know, it, quantum mechanics is basically a linear theory. The Schrödinger equation is a linear theory. Uh, it's not especially geometric. It's not really deterministic. It's about probabilities. That's what comes out of Born rule. General relativity is, is geometric. It's certainly deterministic. It's nonlinear, profoundly nonlinear. It's sort of like almost everything you think about quantum mechanics and, and general relativity are kind of 180 degrees opposite to each other. And, um, uh, all he was pointing out was, you know, you shouldn't just ignore these conceptual problems and just launch into very complicated calculations because you might end up not getting anywhere.

That doesn't sound like a conceptual problem. Sounds like a mathematical problem. So what, what was his definition of a conceptual problem? Well, a conceptual problem could be superposition. I mean, we, we live with superpositions of, um, of, uh, of electrons going through interferometers and things like that quite happily. But when we talk about, you know, a gravitating object and its effect on spacetime, it's something very definite, you know. Um, we don't talk about, we don't even know what it means to talk about a superposition of spacetimes. I mean, it just, you know. Um, so, but that's what I mean. I, I, you know, I, what I, by conceptual and technical, I do mean by conceptual, I mean, I do mean in a sense, what is, you know, it's like how Einstein came to general relativity. It wasn't about doing complicated calculations. It was thinking about what would happen if I was being towed in outer space in an enclosed, you know, box or something by an alien. You know, could I tell the difference between that and being in a gravitational field? That's kind of, and I think it's almost at that level, you know, what does it mean to say gravity is geometric, nonlinear, deterministic, and quantum mechanics is, you know, is, is linear, is not geometric, and is not apparently deterministic? Um, and thinking about maybe this idea of holism. You know, are we, are we, by assuming we can formulate everything, you know, that goes on in the laboratory by laws which only recognize stuff in the laboratory and couldn't care less about what happens in the rest of the world, maybe we're missing, we're missing something.

So explain how there can be a relationship between local laws and some large-scale structure. What does that mean? What does that look like? Well, we spoke earlier about Mach's principle, you see. I mean, you know, um, when I turn, you know, if I rotate round and round, my arms flail out, and I attribute that to some kind of centrifugal force. But what is the origin? What, what determines the fact that when I spin round, I'm in a rotating frame of reference? And, you know, when I don't spin around? I mean, why is the other way around? When is it when I'm? And in fact, to some extent, you know, because the Earth rotates and we see the effect of the Earth's rotation through the structure of weather systems, you know, the Coriolis force is a big, plays a big role in the structure of weather systems. But what is, what determines that the Earth is in a rotating frame of reference? You see, we, we don't, we don't directly perceive the rotation of the Earth. You know, I can't tell the Earth's rotating now. Um, so unless I do careful experiments. So, but what is it that that makes the Coriolis force act in one frame of reference and not another frame of reference? That's a local question. I mean, that's a question which you can address in a laboratory. But the answer to the question, if we accept Mach's principle, which I do myself, is that whether you're in a rotating frame or a non-rotating frame depends on your, uh, depends on whether the, the, the so-called fixed stars, the distant universe, is you're rotating with respect to it or not. And why is that important? Because the distant universe can exert some, um, effectively gravitational force on, on, on you here, here in the, in the laboratory. So, yeah, we can frame things in terms of centrifugal forces and Coriolis forces, but they're just stop, those are stopgap, you know, um, words, if you like, uh, that we, we won't get a deeper understanding of until we, until we understand our position in relation to the bigger universe. And that's sort of the argument that, you know, we can do all these Bell experiments in the lab, but we won't really understand what they're telling us unless we understand the relationship of the lab with respect to the bigger universe.

Is this the holism that you referred to? It is. Yeah, that's absolutely what it, what it means. Yeah. And when people say in the popular press, the word fractal, generally they're referring to something of a self-similar nature. But fractal doesn't always mean that. In fact, generically, it doesn't. So when you're using the word fractal and holism and the local influencing the large and vice versa, are you using it in the self-similar fashion? Well, what I'm really, I'm just ref, what I'm really referring to is this notion of an invariant set, which we, which we referred to earlier, that, um, um, you know, when you have nonlinear dynamical systems and you, you, you start, you can start them from any initial condition and just let them run for, um, a long period of time, um, they will tend to asymptote, if you like, to to one of three different types of invariant sets. One is a fixed point, so that the system just grinds to a halt and stays there as a fixed point in state space. Um, the other is where it actually just evolves cyclically, going round and round in the sort of, um, uh, yeah, sort of cyclical motion, repeating itself like Groundhog Day or something. Every now, would it technically be repeating itself or just sufficiently close to repeating? Precisely. The invariant set is a circle or a topological circle, anyway, and it just repeats itself. Okay, so that's not a fractal. That's not a fractal. No, but the third possibility, which is when you have chaotic dynamics, which is, you know, we live in a world which is chaotic. We have any number of applications or any number of illustrations of that. The most dramatic being billiard balls, which we could talk about if you like. Um, the world is, is chaotic, and the invariant set for these chaotic systems is a, is a fractal. I mean, the fact it's a fractal is kind of doesn't really matter that much. It's just, it's a, it's a, it's a geometry that doesn't isn't described topologically as a point or a, or a circle. It's something more complex.

Okay, let's talk about billiard balls and infinity. Right? So many people think that something being non-computational has in it embedded the notion of infinity. And that's one of the ways that Penrose goes off the rail when he makes a non-computability argument about the mind from Gödel's incompleteness theorem, because Gödel's, Gödel has in it infinity. Talk about Berry's billiard ball thought experiment. Right? Yes, because I, I think there's a very nice example of, you know, of non-computability, which, which Michael Berry, theoretical physicist, incidentally, one of my very early tutors in my undergraduate days at Bristol University, um, very inspirational person. No, he, he, um, you know, he just asked, uh, you know, imagine the game of, of billiards or snooker or pool, whichever you sure, whichever you like. Um, and he asked a question: how many collisions would the snooker ball have to have undergone before its motion is sensitive to the gravitational effect of somebody who is, say, you know, a few hundred yards away or something, just waving their arms around? And it's surprisingly small. It's about, I can't remember the precise, say, 15 or so collisions. Then the whether that person waves their arm or not would influence the motion of the ball after the 15th collision. Well, then Berry goes on to say, well, okay, well, what about, um, how many collisions would it have to make, um, before the motion of the ball was sensitive to the position of an electron, a single electron, at the edge of the visible universe? And the answer actually only goes up to about 50. You know, I can't remember the exact number again. This is all to do with the power of the exponential. If you ever have doubt, you know, exponential growth. This, this is a, this is a consequence of exponential growth, by the way, which is, you know, lies at the heart of chaos theory. The uncertainty, um, grows exponentially. Um, but this has the, you know, the implication that if you, you might imagine, well, I'm going to try and compute, you know, say, just before the billiard ball is set off, you try to do a computation. So you set up your computer and you, you have it running away, you know, and maybe, maybe you don't, you know, you don't do the computation here, you do it in the other side of the world. Does it make any difference? The fact you've set up your computer, you know, where this is again, this is a little bit with counterfactuals. The fact you have set up the computer to do the calculation, that will affect the result of the, of the, um, collisions of the, of the snooker ball, because the orientation of, not only electrons, but atoms and, you know, everything, um, will be different by virtue of the fact you've done that computation. So really, what we're talking about here, this is actually an example of what Stephen Wolfram would call computationally irreducible, because basically we're saying, you know, maybe the whole universe as a whole, as a holistic whole, um, might be computational. But any, but, but what does even what does that mean? I don't know, because any, any attempt to compute part of it, if you like, in other words, you know, you, or compute it with a simpler system than itself, will fail. It won't give you the same result. Um, but I think for all practical purposes, this is a non-computational system, because if you set up your computer to do the calculation, just the very fact you set up your computer to do the calculation will affect the, uh, the, the, the snooker ball or billiard ball after the 50th collision. Um, and again, you know, this is a lovely, beautiful example of, of holism at work. This is, you know, you, you're saying if you want to know precisely what goes on, it does depend on, uh, stuff that's potentially distant, not in a, not in a non-local way. It's not, there's no violation of causality or, you know, things going faster than the speed of light here, but it's just saying you have to take account of, um, uh, you know, what happens in the Andromeda galaxy, you know, will propagate, the, the gravitational waves will propagate, and they will affect the motion of the snooker balls after 50 collisions or so.

Now, is this related to predestination not equaling determinism? Yeah, I think it is. Because this is, I mean, you know, this, these are, this is a difficult, um, issue. But, you see, I, I mean, I know there's been a lot of discussion recently, books have come out about free will and and determinism and, and, you know, mo, mo, I think a lot of scientists seem to think that, um, if the world is deterministic, we can't have free will. But for me, that again, um, is predicated on this notion that a deterministic system is one where you have an initial condition that's somehow given. God gives you the initial condition, or somebody gives you the initial condition, and then you have an evolution equation which takes it forward in time. So that's, that's the, that's the kind of canonical way of thinking about, um, determinism. The problem I have with that, you know, just at a, as a human level, is that I just find it unacceptable. You know, for somebody like Adolf Hitler, let's say, to sort of say, well, I had no choice but to commit genocide because it was all in the initial conditions. You know, I don't blame me, blame whoever set the initial conditions. I mean, that's totally, um, that's totally unacceptable. So the question is, is there an alternative? You could say, okay, random. You know, do we, I mean, random doesn't help. It, it doesn't help the case either, because then he says, oh, well, I didn't really want to kill all those people, but a random, you know, a random flip in my brain made it happen, and that was just, that was again beyond my, uh, ability to control it. So anyway, so all those, so, okay, so how do we deal with this situation? I think the problem is again, this sort of, um, conflation, if you like, of predestination with determinism. And the billiard ball is, is, is an example. But in a way, my whole, in cosmological invariant set thing, is, you see, that is a deterministic system. But the way I, you see, what, let me, let me just, I've tried to say earlier, um, from a mathematical point of view, we've talked a little bit about real numbers and p-adic numbers. And I'm saying p-adic numbers are are kind of the way to, um, uh, to describe fractals. So let me just try and explain how that works, because because with a real number, I mean, typically what will happen is, you know, if you, if you specify the initial conditions to, you know, 10 significant places or something, you could maybe make a, a reasonable prediction a little time ahead with your evolution equations. If you want to make a prediction longer into the future, you have to know that initial condition even more accurately. So in a, some sense, the way you see real numbers, real numbers fit into that paradigm of the initial conditions and the evolution equations quite well. And basically, the more, the more information about the real number in the initial condition you have, the, the more the further ahead you can forecast. So that's, that's, that's the kind of picture. With p-adic numbers, it's quite different. So the way p-adic numbers work is that they describe the totality. I'm just going to say this in words without going into details, but they basically describe the totality of the whole fractal attractor, or the fractal invariant set, as I prefer to call it. Um, but the more digits you specify, the sharper that picture becomes. So you, you know, with with a very few digits in your specification of the p-adic number, you just have a very kind of fuzzy view of the whole. But it's always, you see, it's all at once. It's always that you see it all. Oh, but it's just, you see it with different resolution. You see it all at once. But you see it more and more with more and more. So in other words, if you just have a few digits, you just see what, what the trajectories as, as like thick blobs. And then as you get more num, more digits, you know, those blobs break up into smaller, okay, lines, that sort of thing. So, and that's different to the, actually, that's different to the real numbers, where all, when you specify more of the real numbers, you just know more of the initial conditions, and that allows you to predict a bit further ahead. You never have a, you never have a picture of, of the whole attractor at once. So the point is that these, um, fractal invariant sets, these are deterministic structures. There's nothing random. There's no randomness in it. Everything is deterministic. But the way, the way the, the p, the p-adic picture is that the more information you specify in the p-adic number, the sharper the whole structure becomes. But you're always seeing the whole structure. You just see the whole thing at different levels of, of, of accuracy and, and granularity, if you like. So the point is that you never, it's deterministic, but you never, um, you never, you never frame the problem in terms of initial conditions. And in fact, with the bill, the billiard ball problem, is, is a good example of where that is a futile, it gets you nowhere. If you could specify the initial conditions as accurately as you like, but unless you've got that electron in the last corner of the universe, you're never going to do anything. So I like to see my invariant set postulate as something which is deterministic, but where things are not predestined. There's no predestined nature to it. So, you know, so, uh, I would like to think, you know, when, if Adolf Hitler had been in the, in the, the dock, and he had pled, you know, his innocence because of being predestined by the Big Bang or something, I, if I had been the judge, I would have said, look, I'm sorry, but you don't have to look at it that way. You can look at it from this invariant set way, and that doesn't resolve you, or, you know, you have moral responsibility in that picture, because there is no predestination. Um, you know, um, it's all at once. It's all everything is there, and just the information in the p-adic numbers gives you more and more, uh, structure. Look, I'm not saying that this is an easy thing, and I'm trying to kind of write, write this in a, in a, in a way which is perhaps understandable, but I just feel it comes back to the point that we've discussed, that most people treat as sort of synonymous, this notion of determinism with, uh, the initial value problem. You know, it's, it's a kind of manifestation of what it means to be deterministic. And what I'm saying is that that doesn't have to be like that. The, the invariant set is a completely different perspective on this problem of determinism.

So there's another physicist named Karen Marletto and David Deutsch, who have Constructor Theory. And they don't like the initial boundary approach plus evolution. Right? So firstly, is there a relationship between your theory and Constructor Theory? Um, yeah, I've heard David talk, I've heard Karen talk, and a lot of the words resonate, but I haven't yet, like, okay, at a technical level, I haven't yet found the connection. But, but I think a lot of what they say is consistent. Yeah. So I need to get back to them, actually. I mean, David was, David was, he and I had adjoining offices in Oxford. We did our PhDs exactly at the same time. So I almost wrote a paper with him on causality conditions in general relativity, um, but never quite finished it, to my regret, actually. Was one of the things I would have liked to have done. The door is open if you both want to talk on it. I am down. Okay.

Have I been misnaming invariant set theory, and it should be called cosmological invariant set theory? Well, I, I, well, I wouldn't call it theory, because it's, I mean, that kind of, I always feel that that elevates. You call it theory? Well, I, okay, well, I didn't mean to. I mean, I just call it a postulate. I mean, you know, um, when something becomes a theory, I don't quite know when something becomes a theory. Um, maybe model or something. But yeah, cosmological. I mean, the, I, I use the word cosmological to emphasize this holistic aspect, because it is, it is a, it is a geometry of the whole universe, and it's not decomposable. You know, I can't, like, break it up into, um, three or four chunks and, and say, okay, you know, we're just looking at the, like, direct product of these four chunks. No, that's not the way it works. It's all, it's all, um, like intertwined together. You can't break it up into chunks.

Speaking of cosmology, downstairs right now, there's a conference on the problems with the standard model of cosmology. So earlier, you spoke about the FLRW, I don't recall the initials. What do they stand for? Friedmann, Lemaître, Robertson, Walker. Yeah, their equations. And you said that they're valid at the, at the large scales. But I, I know that there's some controversy there. It's not clear if there's an anisotropy in the universe, or sorry, yes, it's not clear if the universe is is homogeneous. Right? It's not clear if this dust postulate is the, is the correct framing. Right? So I don't know if that was a controversial statement or if it's considered consensus. It's only a few heretics who don't believe in the Lambda CDM or the, or the FLRW. Yes. Well, the thing that, the thing that I, one of the reasons I went to it, um, is less to do with that, but more to do with Lambda. Because my, so yeah, so this is, this is quite an important point. My model of quantum physics, you see, requires, as I was mentioning to you, the universe to be somehow evolving on this, yes, cosmological invariant set. Now, if the universe had started at a Big Bang and expanded and is now accelerating to some sort of heat death, where every atom, you know, is infinitely dilute, somehow, um, then that's the complete antithesis of an invariant set. Then I mean, basically the universe is heading towards, I mean, the invariant set of the universe would then be a fixed point. It, it would just be a static fixed point where everything was infinitely dilute. So this is the absolute, I could almost say, if that really is the way the universe is evolving, then my model is is wrong, or there's something I haven't understood.

All right. So in the Friedmann-Lemaître-Robertson-Walker, there's the other, you know, I mean, Friedmann himself, who, by the way, started off life as a meteorologist. I always think that's rather nice. And did his work on cosmological models in his spare time, almost. So you did the opposite. I did the opposite. Yeah. But you know, I have affinity for people who change like that. Um, um, I mean, he discovered these two solutions, two topologically different types of solution. One is where the universe, um, yeah, kind of expands forever, and the other where it has this cyclical, um, behavior. So, now, up until now, most people, with the particular, with the discovery of dark energy, accelerating the universe, have suggested that, or I mean, it points to this, uh, uh, you know, it points to the universe just expanding forever. By the way, I should say, we don't know whether the universe is infinite in scale or finite, because its spatial curvature is flat, as far as we can tell now. If it's flat minus epsilon, if you like, it then it's a sphere, but a very large one. And if it's flat plus epsilon or something, then it, it's, it's infinite. So we don't know that. But the question is, is it accelerating away now? The interesting thing is that the very, very, very latest sort of, um, cosmological observations from this dark, from from this, yeah, what's it called, Dark Energy Survey, DESI, SGG, just came out a week ago, kind of pointing to the possibility that that dark energy is not constant. Interesting. I don't want to say this too strongly, because I think the statistics are weak, and the answer is we don't know. But it's, there is a, a possibility which I would like to believe is the case. So I'm looking forward to the day when we have more of these observations. But the indications are that that dark energy might be weakening. So the deceleration is weakening. Now, what you would need for a, a cyclical universe is, is something where the dark energy actually eventually changed sign and it became, uh, an attractive force, so it caused the universe to collapse. Um, now, the point is, you wouldn't expect each cycle to be exactly the same. It wouldn't be like Groundhog Day, because the system is chaotic. So what the, what the cyclical model would suggest is very much along the lines of this chaotic invariant set type of concept. But it does require, if, again, this is another falsifiable thing, if, if it really is the case that dark energy is just the cosmological constant and it's constant, um, then, then that puts my ideas in trouble. But I'm sort of, I was pleased to see this morning that that's not the latest results, although the statistical significance is still weak, but the latest results give some hope that that dark energy might be weakening. Now, the Hilbert way of deriving general relativity from varying the action, and you get the cosmological constant, it's a constant. It must be, because if you make it somehow a field or something that varies, then you introduce an extra structure. Yeah. Well, I mean, people have done this. I mean, famously Turk and Steinhardt have, you know, they call it quintessence, which is a field, you know, they introduce it as a scalar field which has its own kind of Lagrangian dynamics and so on, and that gives rise to a, um, a time-varying, um, you know, term which, which, um, has its own energy momentum, you know, equations and everything. So, I, I think it's, uh, it's just, it wouldn't be called a cosmological constant in that case. Um, but, you know, you'd lump it more on the right, right-hand side of the field equations. On the left-hand side, for people who are interested. Neil Turo was just on Theories of Everything talking about the current state of theoretical physics and his minimal model of cosmology. Now, you mentioned the cyclical model, right? So as a closing quotation, I want to bring up Roger Penrose. Yeah. "I think the universe has a purpose. It's not somehow just there by chance. Some people I think take the view that the universe is just there and it just runs along. It just computes and we happen to somehow by accident find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe. I think there is something much deeper about it." That's a quotation from Roger Penrose. And I believe that's also, that's something that you reference. So tell me, what do you think about that?

Okay, um, well, let me say this. I, I've spent, you know, most of my research career, professional career, like, you know, most scientists, um, you know, writing research papers and getting them published in journals and things, and these would be read by my peers and colleagues and stuff like that. When, when COVID, you know, kept us all at home, um, I, I suddenly thought, well, you know, I've had the idea in the back of my mind to write a popular book. Um, you know, if I'm ever going to do it, this is the time to do it. So I did. I wrote this book called "The Primacy of Doubt," which is, let's say, broadly about the science of uncertainty. And I, I try to cover a range of topics from, you know, economics, climate change, and quantum physics, you know, even consciousness, bit of consciousness, all that free will. Yeah. So towards the end, yeah, towards, I mean, there were kind of three parts to the book. Um, the, the third part was was the more speculative. I think the first two were fairly solid, um, you know, scientifically. I was a bit more speculative in the third part. But anyway, the point is, I, I kind of quite enjoyed, um, the experience. And in fact, you know, I met people like you, Kurt, as a result of this book. I don't think we would probably have interacted otherwise. So I, I did a number of, um, you know, podcasts with, with people. Just as a note for people who are interested in your previous Theories of Everything podcast, that's with Tim Modlin, and the link is on screen. Yeah. Also the book "The Primacy of Doubt," the link is in the description, and it's on screen right now. Fantastic. And, um, and I recommend you read it or listen to it. I listened to it. Yeah. I, I even read it. Uh, you even read it yourself? Well, my son, my son said, you can't let somebody else read the book, you've got to read it yourself. So that was quite hard work. Um, but, you know, if you're, if you're interested in, you know, brushing up on your English accent, then you can listen to the book. Um, anyway, it, it got me, uh, thinking about, uh, you know, do, do I have enough in me, in me to, um, write a second book? And that's what I've literally been doing the last few months. And it, and it's been focused on sort of the Penrose quote, you know, is life, you know, cuz if you take the standard model of cosmology, not only are we humans an irrelevance, we're an irrelevance for an infinitely small amount of time. You know, the universe is going through this infinite phase of becoming infinitely dilute, and we're around just for this finite period, which in the length of the, of this universe is an infinitesimal period. Um, is that all there is to it? Now, I know, you know, people would say, well, you know, you should believe in God, you should believe in a creator. Okay. Uh, I don't particularly, um, that, that doesn't appeal to me. I, I have to say, you know, I'm, I'm, what I'm doing here is, um, you know, for, for right or wrong, I, I, I've, I've kind of developed a, a scientific intuition about things, and people will agree or disagree with me about them, which is fine. But my, my sense of scientific intuition is that there is something more, um, to the world and to our existence in the world, than either, you know, a product of some external creator, or as just a complete irrelevance, you know, the product of some random Darwinian mutation that, you know, we, we will have our day and then we'll fade into nothing, and the rest of the universe will carry on without us. Um, so Kurt, I'm not, I'm going to studiously avoid answering your question in detail, but I'm going to put forward some possibilities, let's say, to answer this question, because I think Penrose, Penrose's intuition is probably not that much different to a lot of scientists, that they kind of don't feel comfortable with either of the, it's God or we're in irrelevance. There's something in the middle. And, oh, my impression is that the majority of scientists are extremely comfortable with we're in irrelevance. I don't think that's so. Lawrence Krauss, like, revels in it. Well, Lawrence may be, maybe a unique. I, I suspect. You see, the problem is that there's a bit of a stigma. I mean, if you, if you start giving ground, then you, you, people say, oh, well, he's gone soft, you know, he's halfway to becoming a religious, uh, person. So, um, uh, anyway, look, I, I can, I lay my cards on the table. I'm not a religious person at all. I don't believe in God. However, I have some other ideas which are scientific. They're based on scientific principles, which I think could well give, and Penrose himself has said, you know, he's not a believer in God. So, you know, somebody as clever as him saying that there might be something more to it, I think we have to take that seriously. And I'm going to try and put forward some proposals in the book, so that's what I'm writing now. It'll be at least a year or or two before it comes out. We can have another chat. I'll be very happy to talk in more depth about it. I'm trying to sort out the details at the moment.

So you have a third option that is not meaningless chance, and it's not the other thing is to say it's not like panpsychism or anything. A fourth option. It's a fourth option, exactly. So I go through the three options, which is religion, sort of spiritualism, M, panpsychism, that sort of mysticism types of stuff, or just sort of in irrelevant scientific irrelevance. So I'm trying to put forward a fourth option. That's right. And was there a Freudian slip? The reason why you used your middle finger for the scientific irrelevance? I don't know. Well, the other option of course is is agnosticism. And I, I mean, for years I would have probably called myself an agnostic. Um, but in a way, that's a bit of a cop-out. It's a bit of that wouldn't be a single hero, just be uncertainty between choosing one of these, though. Okay, well, maybe, yeah. Um, I don't know. But anyway, that, that's, um, yeah, we'll see, we'll see how it goes. Professor, it's been a blast. Thank you. It's been great. Yeah, I hope, um, well, I hope we meet again and I hope we have another chat. And, um, you know, I feel, you know, you put me on the spot, and I always feel, um, I don't come over as humbly as I should do, because I'm putting my points of view more forthrightly than perhaps, you know, is is fully justified. But, but I'm glad to have the opportunity anyway to do so. And also, I want to thank the guy who's behind the camera, you can't see him, it's Dougall McQueen of the Royal Society of London, and he has helped set all of this up. So thank you, Dougall. That's it. Thank you. Firstly, thank you for for watching. Thank you for listening. There's now a website, KurtJongle.com, and that has a mailing list. The reason being that large platforms like YouTube, like Patreon, they can disable you for whatever reason, whenever they like. That's just part of the terms of service. Now, a direct mailing list ensures that I have an untrammeled communication with you. Plus, soon I'll be releasing a one-page PDF of my top 10 Toes. It's not as Quentin Tarantino as it sounds. Like secondly, if you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like, helps YouTube push this content to more people like yourself. Plus, it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm, which means that whenever you share on Twitter, say, on Facebook, or even on Reddit, etc., it shows YouTube, hey, people are talking about this content outside of YouTube, which in turn greatly aids the distribution on YouTube. Thirdly, there's a remarkably active Discord and subreddit for Theories of Everything, where people explicate Toes, they disagree respectfully about theories, and build as a community our own Toes. Links to both are in the description. Fourthly, you should know this podcast is on iTunes. It's on Spotify. It's on all of the audio platforms. All you have to do is type in Theories of Everything, and you'll find it. Personally, I gain from re-watching lectures and podcasts. I also read in the comments that, hey, ToE listeners also gain from replaying. So how about instead, you relisten on those platforms like iTunes, Spotify, Google Podcasts, whichever podcast catcher you use. And finally, if you'd like to support more conversations like this, more content like this, then do consider visiting Patreon.com/KurtJongle and donating with whatever you like. There's also PayPal, there's also crypto, there's also just joining on YouTube. Again, keep in mind, it's support from the sponsors and you that allow me to work on ToE full-time. You also get get early access to ad-free episodes, whether it's audio or video. It's audio in the case of Patreon, video in the case of YouTube, for instance. This episode that you're listening to right now was released a few days earlier. Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much.