The Theorem That Proves Science Can't Know the Universe

You have all these models around, and a lot of them seem maybe a bit crazy. Some of them might allow for time travel. But things start getting really crazy once you start looking at these Heraclitus-type models. One can use science to show that science has limits.

This is unlike any conversation I've had on this channel, and I've had hundreds of conversations with physicists, mathematicians, philosophers, etc. Professor J.B. Manchak is the only person that I know who connects general relativity to unknowability to the self to Zen Buddhism to time travel. You're in for a treat.

Einstein's theory is replete with models that permit time travel, have exposed points where space-time ends, and even violate determinism. This means it's technically false to say classical physics is deterministic. Most physicists dismiss these as pathological mathematical artifacts and move on. Professor J.B. Manchak of UC Irvine doesn't dismiss them. He proves theorems about them. And what he's found should unsettle anyone who thinks science can tell us what the universe looks like.

On this channel, I interview researchers regarding their theories of reality with rigor and technical depth. Today, we discuss new space-times found by Manchak called Heraclitus space-times. We also discuss how some space-times solve the halting problem. And we close with Hume, Zen Buddhism, and underdetermination in science, where Manchak finds parallels between his unknowability theorems and the Buddhist teaching of non-self. It's quite odd as someone who isn't a guru. You're not a guru. I've never seen anyone connect general relativity to questions about the self, the limits of knowledge, the limits of science even, potentially even spirituality, which we're going to talk about later on in this conversation, in the way that you do. Help me understand your journey.

Oh, yeah. I mean, I think it might be the particular question that I got hooked in on concerning unknowability of the universe. So what that does, of course, it's going to open things up when one is confronted with the possibility that you can collect evidence, all the evidence you want. You can collect evidence forever, any kind of empirical evidence you could ever imagine. And the idea of that not being enough to pin down what the universe is like, that's something that naturally pops up in theories of space and time. So I don't think GR is special here. I think you can prove a similar kind of theorem pretty much in any space-time theory that's modeled on a manifold with geometric structures on it. So a Newtonian version of space-time physics, you'd have similar results there. So I think it's something about space and time.

And what makes a manifold a physically reasonable manifold? Yeah, that's a really great question. I mean, I've been thinking about this question for probably 20 years. I think it's fascinating that we have a situation where a theory like general relativity is so permissive in the sense that it lets so many models into the theory. And then you have all these models around, and a lot of them seem maybe a bit crazy. Some of them might allow for time travel. Some of them might allow for other types of pathological behaviors. And so then it's kind of like, you know, this game where most physicists are going to look at a lot of those things and say, well, those are just mathematical artifacts. They're not really part of the theory. Whereas, you know, a philosopher like me, I'm going to want to really think about this deeply and ask, why exactly is this particular model physically unreasonable? And it turns out that a lot of the usual justifications that are bandied about, when you really analyze them, they kind of fall apart.

So I can give you an example. One thing that I've been studying for some time now is this idea of space-time maximality. Space-time maximality says that the universe is as big as it can be. And I think what happens is you have models around that are extendable. They can become bigger. And so physicists are going to want to say, hey, that's not a physically reasonable space-time. That's not a physically reasonable model of GR. And you ask, well, what's your reason? Why not? And the original justifications given by — it was originally Penrose, Geroch, and a few others back in the 60s in the golden era. You look at the justifications there, and it's very much metaphysics. It's Leibnizian metaphysics. It's the idea that, oh, well, nature — why would nature stop when nature could keep building? So she has to keep building. And so these extendable models are unreasonable because that's just not how nature works.

What's interesting is to look at various subclasses of GR where, you know, if it really is the case that some of these models are physically unreasonable, well, then you've got to kick them out. You can't have them be a part of your collection of models anymore. And when you do that, you open up a Pandora's box. Now, all of your modal structure is completely different because you're working with a completely different collection of models. And so, in some instances, you can actually show that nature doesn't have the option of building, say, a maximal universe. And so then, you know, a theorem like Bob Geroch's theorem that every spacetime has a maximal extension, well, yeah, that's true with the standard collection. But once you kick out those unreasonable guys, now you've got to re-ask the question with this new variant of GR, and it's just not clear whether a theorem like that holds or not. So, I think examining the justifications for certain — I would say almost like dogma — that this type of space-time has to be physically reasonable, you notice that the justifications really aren't what they might appear to be at first.

Hmm. Okay. People are watching and people are probably wondering, well, what is meant by this space-time is pathological, this one is a reasonable space-time, and all of that. I have an article about how GR is indeterministic, technically speaking, that it's not a deterministic theory. And in it, I had to outline, well, what the heck is general relativity? So, firstly, I know you read that article, and I want to know what you think about if GR is technically speaking indeterministic. And then also, what is the definition of GR?

Yeah, yeah. It's a great question. I did check out that article. I mean, I think it gets to the heart of the issue, which is, well, we don't have just one GR. What we have is a lot of variants of GR. GR usually is considered to be a standard collection of models. And so oftentimes when theorems are proved, they're proved relative to that background standard collection. What you're doing in that article is you're pointing to the fact that on questions such as determinism, whether or not GR is deterministic or not, that's going to depend crucially on what that background collection of models is like. And your point was basically like, GR is not deterministic unless you kind of get rid of all those pathological models. And so why are you getting rid of those? I think at one point you said, well, it's just kind of like maybe you don't like it for whatever reason. And that's certainly what happens, right? It's like this little game that's being played where you can say GR is deterministic if you kick out all the models that show that it's indeterministic. I mean, sure. But if you're looking at the entire class of models, there's a sense in which it's certainly not.

I think the game is to have a pluralistic kind of view about these things. So instead of thinking that GR is just one thing, no, we're not in a position really to know what GR is. And so GR is a bunch of different things. And so you can ask the question, is GR deterministic relative to this collection of models, relative to that one, relative to that one, relative to that one? And you'll get different answers depending on which one you're looking at. I think that's all interesting. And if you want to go a step further and say, but which is the real one? I just want to say that's a little bit misguided. We're never going to be in a position to get at that question. Now, we'll never be in a position to get to that question.

Are you referring to some theorem of yours, or are you referring to just the general problem in science of underdetermination? Well, I think I'm referring to, I guess, a few different things. Certainly, it's the case that everyone agrees that GR has unphysical models in it. I mean, I don't think you'll find a physicist around who says that all of the standard models of GR are physically reasonable. That's absurd. And so everyone agrees that the real GR, whatever it is, is some subclass of models. And so the question is, well, which one? And you're never going to find agreement among physicists about which one is the right subclass. Now, that's one side of things. Of course, I'm talking about other things too, including some of my work which says that we can't know what a physically reasonable space-time is because we have this underdetermination problem where we can't rule out, for example, that we in our actual universe have these unphysical properties here with us now. I mean, it very well may be, for example, that our universe allows for something like time travel or indeterminism, but that's just not something that we'll be able to figure out based on any type of empirical evidence and even any type of local induction on top of that empirical evidence. So that's, I think, what I mean by we're never going to be in a position to know what the true collection of models of GR is.

When you say that GR, general relativity, may allow time travel, what's the common misunderstanding of what you're saying? And then what's also the most sophisticated misunderstanding of what you're saying? Well, GR allows for a certain type of time travel. So one common thing that I encounter is that folks want to think of time travel as maybe something that they've seen in a movie or something like that. Going back and changing the past is something that you'll see in a movie like Back to the Future. I grew up with that movie. That's just not something that's going to be compatible with models of GR. What you do find in GR are models that have a peculiar causal structure in the sense that you'll have world lines of particles. This is a trajectory of some massive body that can go forward in time, but the structure of the space-time allows that curve to wrap back on itself, and so the event can be revisited. And that's what's permissible in GR.

Kurt Gödel in 1949, he presents Einstein with this Gödel solution which allows for time travel. Einstein's response was, you know, that's very interesting. We have to see if there are physical reasons to exclude such a model. And I think here we are, many, many decades later, and that's still the position we're in. We're still looking for physical reasons for why we could exclude models like that. And so all sorts of people have proved all sorts of theorems about all sorts of ways in which you can exclude them. But I don't think the situation is one where it's a knockdown argument at any point. There still is a hope, at least by some folks, not necessarily by me, but by some folks, that time travel may be possible. My position is just that we don't know. And I find it fascinating that even a position like that, which is very noncommittal and it's very modest, somehow is seen as almost radical or something.

If the time traveler isn't a point particle, let's imagine it's a person, then how does that work? Because wouldn't entropy take over and then they would just constantly be increasing in their entropy, or does the entropy reset, or what? Yeah, I mean, so the model that you'd be looking at is going to be a structure that has the geometry on it, and then via Einstein's equation, you have some matter fields there. And so if you're a person returning to an event that you've previously visited, then yeah, your mass, whatever's going on with your body, that's going to be exactly the same. So, you know, I don't necessarily think in terms of entropy in the sense that it's not some physical thing that I'm modeling there in the space-time, but it certainly is the case that whatever is going on with the matter, it's in a periodic sort of situation. So the matter is coming back around to itself in the same way that a point particle might return back to itself. So, yeah.

Okay, speaking of coming back around, earlier you said something about collaborating with Malament, or learning from him, from that legendary — and he still is existing. Yeah, yeah, he lives just down the road here, so I still see him every few weeks. It's lovely. We go on hikes together and things. Wow, great, great. There are many people whose names you encounter so often in the literature that, at least to me, I think they must no longer be around because they're legends. Roy Baumeister is one of those in psychology. Anyhow, what is it that you learned from Malament about how to do philosophy of physics? And was it something he taught you, or did you just pick it up by osmosis, or you inferred it, or what? Yeah, I think maybe it was a little bit of both. I mean, I was just watching him, but also he was teaching classes and he was in the classroom trying to convey the way he thought about space and time. And I think his distinctive style is one that's very simple in the sense that he's not one who's drawn to fancy stuff. Really, he is going to focus on making what might be a confusing question, making that precise using the formalism of GR, and then going to work. Going to work trying to see if that conjecture is true or false. So, the real genius of someone like David would be to find a way to ask a philosophical question in a very rigorous type way, and then prove a theorem that essentially gives you an answer for that philosophical question.

Give me an example. Yeah, so I think one that is probably the one he may be most known for was one that happened early on in his career when he was a graduate student. Yeah. Hawking and others had proven that if you give me two models and they have a very rich causal structure associated with them, and you tell me that their causal structures are the same in a sense, that there's an isomorphism of the causal structures, that actually the topology is also going to be the same. So you can determine the topology of space-time via the causal structure of space-time. So you tell me which points are causally related to which others, and then I will tell you what the shape of the universe is like. Now, Hawking and others, they had proven this for extremely rich, causally well-behaved space-times. What David did was to ask the question, how low can you go in terms of the scarcity of causal structure? So I'll give you an example. If you're in a time travel universe where every point is causally related to every other point, you're never going to get a theorem like this to go because you'll have one model which allows for time travel over here where every point is related to every other. So that's some causal structure. And you can have another over here where maybe it's the same model, but then I yank out some points. So now the topology is all messed up. These are going to have the same causal structure because every point is causally related to every other in both of them. So there's a causal isomorphism. They have the same causal structure, but they have very different topologies. And the beauty of what David did is he said, how weak can we go here in terms of what's the minimal level of causal structure that we need to show that causal structure determines the shape of the universe? And so he went to work on that. That was a paper that he published in 1977. Now it's the foundation for a whole program of quantum gravity, the causal set approach to quantum gravity, where the causal set approach says you sprinkle causal little events in space-time, and there's a causal structure there, and then from that emerges a kind of shape. And so they depend heavily on a result like David's for that. But that's just one example. He's done it many, many times over the course of his career.

Later on in his career, he looked at something like rotation. Rotation in standard Newtonian mechanics is very well understood. You ask, well, what does rotation mean in GR? How do you know when object X is rotating around object Y? It turns out there are several things that you might mean by that. And what David did was — this is another example of his style — he would say, you know, let me formalize each one of these criteria of rotation and see if I can't prove either a no-go theorem that says like, hey, these don't really mesh well with one another. You can find spacetimes where an object is rotating around another one according to criterion A, but not according to criterion B. Or try to prove a uniqueness theorem that shows that these all come together. And it turned out that after he really looked and investigated this problem, they came apart. And so now we know that in GR, rotation is a very tricky, subtle business. And it was David who has been able to identify both the problem there and then show how going to work and formalizing this philosophical question, this philosophical problem, leads to a clear answer. So there's clarity there, light — now we understand it better. And so those are a couple of examples.

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Most of my best ideas don't happen during interviews. They come spontaneously, most of the time in the shower, actually, or while I'm walking. Until I had Plaud, I would frequently lose them because by the time I write down half of it, it's gone. I tried voice capture before, like Google Home, and it just cuts me off in the middle. It's so frustrating. Most of my ideas aren't these 10-second soundbites. They're ponderous. They're long-winded. And I wind around. They're discursive. They're five minutes long. Apple Notes, even Google Keep, the transcription there is horrible. But Plaud lets me talk for as long as I want, and there's no interruptions. It's accurate capture. It organizes everything into clear summaries, key takeaways, action items. I can even come back later and say, hey, what was that thread I was talking about regarding consciousness and information? In fact, this episode itself has a Plaud summary below, and I'm using it right now over here. My personal workflow is that I have their auto flow feature enabled, so it sends me an email anytime I take a note. Look, the fact that I can just press it and it turns on instantly, like right now it's starting to record, without a delay, is extremely underrated. This, by the way, is the Note Pro, and then this is the Note Pin. I have both. Over 1.5 million people use Plaud around the world. If your work depends on conversations or the ideas that come after them, it's worth checking out. That's plaud.ai/TOE. Use code TOE for 10% off at checkout.

Yes, and David collaborated with Hogarth to come up with the Malament-Hogarth, if I'm pronouncing that correctly, space-times. I have an article about that as well. I don't know how to pronounce it. I think it sounds like a Yu-Gi-Oh card. Yeah, what's really interesting about that is David never published anything about Malament-Hogarth space-times. I don't know the whole story there, but David was in correspondence with John Earman and some others about this possibility. So it's David's idea, but the first person to publish a paper about this was Hogarth. And so as a kind of compromise, I think John Earman said, let's call these things Malament-Hogarth space-times, but it's David's idea. And the main theorems, including the theorem that Malament-Hogarth space-times are necessarily not globally hyperbolic, that's also David.

Okay, so three terms are likely to come up repeatedly throughout the rest of this conversation. One just came up, global hyperbolicity. So what does it mean for a spacetime to be globally hyperbolic? What is a Cauchy surface? And what is a Cauchy horizon? So why don't you define those? Yeah, yeah, good. Yeah, let's start with a Cauchy surface. A Cauchy surface is a spacelike surface in your space such that every causal curve in the space, if you extend it — maximally extend this curve — it's going to hit this Cauchy surface somewhere. So the physical significance of a Cauchy surface is supposed to be that if you have a Cauchy surface, then everything that's happening in your space-time is registering there. And so there are some deep theorems proved by Choquet-Bruhat and Geroch about how you can take data on a Cauchy surface and use it to evolve that data in a unique way. So it's a type of result that shows that GR is deterministic in some sense. So not all spacetimes have Cauchy surfaces. If a spacetime does have a Cauchy surface, then we call it globally hyperbolic. If it doesn't, then it's not.

If a spacetime doesn't have a Cauchy surface, what can happen is you can have some sort of surface that can be evolved up to some point, and then it can no longer be evolved after that point. And so that point at which it can no longer be evolved, that's the Cauchy horizon. And so one way to formalize, say, Penrose's cosmic censorship hypothesis is that there isn't any Cauchy horizon. So when you take that data surface and you evolve it, what you're going to be left with is a space-time that is globally hyperbolic and can't be extended any more than that.

So to be clear, most of the time when people are thinking of Newtonian mechanics or classical mechanics, they think in terms of, oh, it's laws plus initial conditions. And if someone was intelligent and knowledgeable enough to have access to all the information of the universe and intelligent enough to move it forward, then they can calculate the future state. So you're saying, in this analogy, the initial condition here would be the Cauchy surface. The evolving forward indefinitely would be the presence of zero Cauchy horizons. Is that correct? Or is it the case that if you ever have a Cauchy horizon, a place from which you can't evolve further, that that means you do not ever have a Cauchy surface? Yeah, I'm getting kind of tangled up here, but a Cauchy horizon is only going to arise if it is possible to extend, but not extend in a way that's going to keep that surface a Cauchy surface. So the Cauchy surface, as I said, if there's a Cauchy surface, it's necessarily globally hyperbolic. So I think the way to think about these things first is to not term it a Cauchy surface, just call it a surface and say, you know, you put some data on this surface and then you evolve it. We're going to see if it's a Cauchy surface or not. We're going to evolve this data as far as we can. If we do so and we find ourselves in a situation where we've evolved it as far as we can, that's a globally hyperbolic spacetime, and this will be a Cauchy surface relative to that globally hyperbolic spacetime, but it still may be possible to extend further. And if it's possible to extend further, then your surface will no longer be Cauchy. So you get this behavior in space-times that violate the cosmic censorship hypothesis. One way to put the cosmic censorship hypothesis is that all physically reasonable space-times are globally hyperbolic. And what that means is that when you start off with the surface with the data, you evolve it all the way as far as you can. What you're going to end up with is a maximal globally hyperbolic space. If it's false, then what that means is you evolved it all the way as far as you can go, but that doesn't mean that it's necessarily a maximal space-time. You may be able to extend it further. It's just that any extension will ruin the global hyperbolicity property.

Suppose you evolve up until a point where you can't evolve further, and let's further suppose that global hyperbolicity isn't a necessary condition of our physical universe. That our physical universe may have solutions that are not globally hyperbolic. Now, I know that's anathema to some people who think that Malament-Hogarth space-times, for instance, are pathological or what have you. Okay, we don't know that a priori. Black holes were thought to be pathological, a priori, quote-unquote, by Einstein, I believe. Exactly. Exactly. So, okay, let's suppose that's the case. Then what do you think is happening after that point? Do you think the universe stops? Does it then choose one of many solutions? How does it choose it if there's no probability distribution? I don't know, man. It's a real problem. So here's an idea that traces its roots back to John Earman, who was a philosopher of physics. And one of my gurus, I guess. His idea was to think of a machine, like a time machine, or a Malament-Hogarth machine, as a type of thing where you evolve until you can't evolve anymore. Now you've got options for ways to extend that evolution. But like you said, it's not going to be unique. But what if we could show somehow that all of the extensions, every last one of them, satisfies some property? So either they have time travel in them, or they have the Malament-Hogarth property. Then you could say, well, I don't know which one is going to obtain, which one of those non-unique extensions will continue. But if all of them are Malament-Hogarth space-times, well, then you have now identified a Malament-Hogarth machine. And so that's something that I — you know, there was this time machine literature based on what John Earman did with respect to time machines. It's a fascinating literature because there's been several times theorems have been proven false by various people. But on the Malament-Hogarth side of things, yeah, I was wondering, if your space-time is Malament-Hogarth, it doesn't necessarily mean time travel is possible. The other direction does hold. If your space-time has time travel in it, then the Malament-Hogarth property obtains. And what that means is that time machines are going to be much harder to build than Malament-Hogarth machines. So I was thinking, well, there's all these problems with time machines. Can we build a Malament-Hogarth machine? Can I identify a space-time such that I'm looking at a globally hyperbolic space-time that cannot be extended anymore with that globally hyperbolic property? And so it's got all sorts of non-unique extensions around, but can I show that every one of them is Malament-Hogarth? And so that's one thing that I did. So these things do exist. They are physically reasonable in the sense that I've been able to find some that satisfy various energy conditions. They are causally well-behaved in the sense that they're stably causal. They're actually pretty causally well-behaved. So, you know, they might be unreasonable in some sense, but the idea is to give a proof of concept for this idea, and I think we do have a much better grip on Malament-Hogarth machines than we do on, say, something like time machines. Yeah.

Something you can do if a Malament-Hogarth space-time exists is run a machine, run a program which you don't know if it's going to halt or not. And in this universe, in a Minkowski universe purely, you would not be able to tell if your Turing machine is going to halt. But it's said that you could take a laptop, send it to some point in an MH space-time, and you'll be able to receive whether the laptop halts or not. But then there's issues of so-called blue-shifted photons. So, what is a blue-shifted photon? Is it actually a problem? Yeah, yeah. The blue shift happens because — well, so you're absolutely right in your characterization. What a Malament-Hogarth space-time is, is you identify a point, an event in the space-time, and you look at its past light cone, and in the past light cone you have an observer with an infinite future that's contained in your past light cone. So, for you, you're making an observation at a particular finite point in space and time, but part of what you're observing is an infinite future of some other observer. And so, that other observer could be a computer running some kind of program or something. What happens with the infinite blue shift situation is when the computer is sending signals back to that point, they can start to bunch up. And so, the signals can get very unreliable and they can also become very close together. So, the wavelength will go towards the blue side of things. And so, some people have said, well, you know, this really isn't physically reasonable because of blueshift situations like that. And I think I agree with that. What I've tried to do in some of my work has been to collect up a bunch of reasons that people want to exclude these things for. So, blueshift would be one of them. Another one would be, you know, whether the standard energy conditions are satisfied. Another would be, well, we don't want to have this computer having to accelerate really wildly and have infinite fuel to traverse whatever world line it's supposed to traverse. So, you can collect up a bunch of these things. And what I do is I say, look, here's a model. I go to work in my little workshop building a little model, and the model is a Malament-Hogarth space-time that has none of those unphysical properties. The irony is that this model is pretty unphysical. Even by your standards? Even by my standards, yeah. So, there's an old technique introduced by Penrose, and Geroch used it a lot in his work. Hawking used it with Ellis in his textbook. You might term it the cut-and-paste construction. It's where you take models of space-time, like relatively well-behaved models, like something like a Minkowski space-time, and what you're going to do is you're going to cut slits here, and you're going to glue things together. You're going to create wildly crazy topologies in this way. They're going to be standard models of GR. They're going to count as models of GR, but they're going to be wildly unphysical. Now, those folks, Penrose, Geroch, others, they emphasized early on that the point of these examples was not to exhibit a physically reasonable model of our universe. I mean, if you're studying something like logic or something like this, counterexamples are just so important. And so they serve the purpose of being counterexamples to certain lines of reasoning. And so in this case, I would say the example that I'm coming up with that's very unphysical is a counterexample to a certain line of reasoning that goes like this: Malament-Hogarth space-times can't possibly happen. Why? Because they have unphysical property X, Y, or Z. I'm going to say, well, I have a counterexample to that line of reasoning, because I've now exhibited a space-time which is Malament-Hogarth that doesn't have property X, Y, or Z. And so, what it does is it puts the ball in the other person's court again and says, if there's something else you're thinking of that makes it unphysical, then speak up. Come up with some other criticism here, but the criticisms you have so far aren't quite enough.

You know, hawk-eared — or hawk is the analogy for hawk-eyed, but for listeners, the eagle-eared, I'm not sure — listener may have noticed that you use the term "obtain." So, people who study philosophy are distinguished by their use of the word obtain. Physicists who haven't studied philosophy never use the word obtain. Really? Okay. So, "obtained" just means that something becomes the case. Yeah. I want to know, you mentioned earlier that there's a difference between not only the way that Malament does the philosophy of physics compared to other philosophers of physics, but you also mentioned that there's a difference between the way philosophy of physics is done and philosophy per se as such. So, what is that difference? Well, I mean, I think one thing to say is that philosophy of physics, even, is a very diverse thing. So, there's all sorts of styles of philosophy of physics. You've encountered some philosophers of physics, right? You had David Wallace on your program, Tim Maudlin, I think. So, you're familiar with — I mean, both of those folks have a very different style than I do. I'm in a style which is on the more mathematical, logical, rigorous side of things, I would say. The way I attack a problem is to try to formalize that problem and come up with a proof or a counterexample for a given philosophical claim. But if you look at the work of other philosophers, it's going to be very different. They're going to not necessarily be so rigorous. Here's an example. This is not my field at all, but something like quantum field theory, there's going to be some philosophers of physics who want to really make that into a really rigorous type thing. That's difficult. And so, what they might do is they might go to an algebraic approach to that. Maybe look at C*-algebras or something like that, whereas someone like David Wallace or others, they might want to dive into the practice of how folks are trying to make sense of quantum field theory within a physics department or something like that. So, these styles can clash and these clashes can pop up at a conference. There are more styles than that, too. I mean, there are styles where it's very disconnected from the actual practice of physics, where you're looking at something like time travel from a different starting point. You might bring in physics here or there, but you're not starting with, okay, let's assume we're working within the context of GR and let's ask this question. Rather, you're saying, I'm not going to start from GR. I'm going to start from my own system. And so there are a lot of different ways to do philosophy. And looking back on my own trajectory, when I was a youngster, I just wasn't aware of all of these ways. I mean, for me, it was just like, there's physics, there's philosophy, bringing them together, that's got to be like this one thing, but it's not one thing.

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There are a couple levels here. So, number one, when you tell people you go into math, if you tell laypeople who haven't studied any math, they may think that what a mathematician does is something like an accountant, where you're multiplying just larger and larger numbers together, and maybe something like the Pythagorean theorem. And they don't realize that it's much more than that. And then when you realize that it's much more than that, you don't think in terms of there are styles to that much more than that. That Grothendieck's style is different than Terence Tao's style. And something that didn't occur to me is that there are different styles of philosophy of physics until just speaking with you now and thinking about it. So is there a style of philosophy of physics that you envy? That you think, oh, I wish I could do that? Yeah, I mean, I certainly envy David — he sort of invented this style of doing philosophy of physics. Which David? David Malament. I would say, yeah, I'm very envious. I try in my own way to work in his style. I have my own style, though. My style is one where I don't put a sentence in a paper. I don't put a word in a paper unless it's necessary. So, my papers can be two, three pages. That's very unusual in philosophy. It's maybe more common in physics, although that's even pretty short for physics. But I like to let the results speak for themselves. I mean, I think a lot of philosophers, what they'll want to do is they might have a result. Most philosophers don't prove theorems, right? So, they're not actually making progress on the mathematical, physical side of things. What they're going to do is they're going to take existing statements and they're going to philosophize about them in some way. For me, my theorem, that is my philosophy. So, I let the results speak for themselves, and I try not to add on top of the results some flowery prose about whatever, or what the meaning of this is, or something. The result is what it is. That's the philosophy.

One of the reasons I'm excited to speak with you, and I have been excited to speak with you, is because you're so terse in your writing. And here you're not, and I like that because now you can expound and show me your thinking on a result. So, speaking of thinking, many people have a model of what GR is, many physicists, and you, I'm sure, have your own private model. So, how do you think about GR and what GR is and how that differs from many of the relativists that you encounter? I think I would approach GR as something that's not just one thing. So, I have a pluralistic notion of what this thing is. And so, if there's a question that depends on a background possibility space — so, for philosophers, the notion of possibility is very important for certain subjects. Something like determinism, for example. What counts as a possibility? Everything turns on that. And so, for me, a question like, what is a possible universe according to GR, that's a very deep question. I don't think we're going to get to an answer to that question. All we can do is a sort of relative style exploration where we say, well, let's assume that the background possibility space looks like this, then what follows from that? Let's assume that it's like that, what follows from that? And so we have a lot of relative results, I think. If someone wanted to show up to the scene and say, hey man, we really have just — it's just GR is this one thing, I'm going to kind of scratch my head and I'm going to go like, where did this come from? Like, what in the world?

One place you see this play out a lot is within the cosmic censorship hypothesis, right? Because this is something that Penrose came up with. One particularly nice way to phrase this is due to Robert Wald in his GR book. He says the cosmic censorship hypothesis, at least the strong version, says all physically reasonable space-times are globally hyperbolic. Well, you can see, given how that statement is formulated, that if your background possibility space is really small and you're kicking everything out, well then the cosmic censorship hypothesis is going to be true. If you're letting a bunch of misfits in, then it's going to be false. And so everything turns on how big or small is your collection going to be. And so all of these cosmic censorship hypothesis theorems or counterexamples, those are all trading on that game. So the people who are proving those results, what are they doing? Well, there's assumptions going into those theorems. What are those assumptions doing? They're paring down that collection, making it very small so that you don't have that riff-raff around. What are the counterexample folks doing? Well, they're considering things in a slightly more liberal way. They're letting in more guys, and those guys are potential counterexamples. So that's the game. And I don't think we're ever going to get to the answer of whether censorship is true or not, because that's how the game is set up.

I was speaking to John Norton of the hole argument. We'll speak about the hole argument later. And Norton had his dome. He said there's a similar question-begging that's happening in Newtonian mechanics when people say Newtonian mechanics is deterministic. He said, well, if you assume Lipschitz continuity, it is. But if you don't, then it's not necessarily. And then people would say, yeah, but physically realizable, maybe Lipschitz continuous. But then he said that's begging the question. We don't know. So you can assume — it amounts to assuming determinism in order to prove determinism. Yeah, yeah. I mean, one difference that I might point out is that I think Norton's Dome is operating in a non-standard type of Newtonian mechanics, right? So, Norton's Dome, for example, is not a smooth surface. That's why you can get it to work. This other thing that I've been talking about, it's all happening within the standard collection of models of GR. So one doesn't have to go outside of the standard context to show that determinism breaks down. Rather, one is looking at various subclasses of the standard collection.

Okay, so why don't you talk about again, what is GR specifically? I know you keep saying it's not one thing. Yeah, I don't know. What I mean is, when you're talking about the space of solutions, are you saying, okay, Curt, let's look at Einstein's field equations, let's write them down, what are all the potential metrics that could satisfy this equation? Is that what you mean by space of solutions? Do you mean something else? Yeah, yeah, okay, really good question, because I think it's important to lay the foundations of what are we even talking about here. You start with the idea of a four-dimensional manifold. A manifold is a certain kind of shape where locally it's going to look like Rⁿ. It's going to look flat. So a sphere, a cylinder, a torus, these are all examples of manifolds. Now, what you can do with a manifold is then you can define various types of smooth tensor fields or geometric objects on that manifold. And in GR, what you do is you add on a particular type of metric. It's a Lorentzian metric. So there are all sorts of metrics out there. There's a Riemannian metric that you could put on a manifold. In GR, what you do is you put a Lorentzian metric on there. And what that is, basically, it's like you put a little light cone at each point with some additional structure about how geodesics behave, but that's what it's telling you. It's giving you a little bit of extra structure on top of the manifold structure. And then what Einstein's equation — you know, people talk about Einstein's equations and solutions to Einstein's equations — that is an empty, vacuous concept until you tell me exactly what you mean by that. So if you give me a metric, any metric whatsoever, that's a Lorentzian metric, I can actually use Einstein's equation to kick out a matter field there that satisfies Einstein's equation. Because I'm defining my matter field to be one that is produced, in a sense, by the particular geometry that you give me with your metric. So, what do I mean by GR? Like, what is a solution? Well, you give me a manifold and a Lorentzian metric, and then I can use Einstein to add a matter field on top of that that satisfies Einstein's equation. Now, you can come on top of that and

Say, well, I don't just want any old matter field. I want a matter field that satisfies certain energy conditions. This is why the energy conditions are so important, because they give some kind of constraint on that matter field. And that gives content to Einstein's equation. Without any constraints, like the energy conditions, or restricting attention to vacuum solutions or something like that, without any constraints on the matter field, Einstein's equation doesn't have any content. It doesn't do anything.

So, the standard collection is just the collection of all manifolds that admit a Lorentzian metric. They all satisfy Einstein's equation in a minimal sense. And so then you go from there. How you go from there, or how you want to pin down some particular subclass of that larger class is up to you. You could use local conditions like requiring an energy condition or focusing on vacuum solutions or something like that. You could look at causal structure. That's another very common way to pare down that background class. You could say, well, maybe I'm not going to be so strict as to require global hyperbolicity, but maybe I'll require that there's no time travel, or maybe I'll require that my space-time is causal in a stable way. So there's a condition called stable causality. So there's local stuff, there's causal stuff.

Another common thing that you have to do — this is a little bit more on the philosophy side, because I think a lot of folks in GR maybe are not so familiar with it. You're familiar with this side of things, but what you want to do is come up with some way to rule out holes. So holes are kind of like singularities that are unphysical. In light of the singularity theorems, we think, well, there's a sense in which singularities are physically reasonable. You can't kick out all the singularities, so you've got to let some in, but you don't want to let them all in, because some singularities seem to be very unphysical.

Just a moment, the type of hole that you're talking about here is not the same type of hole as the hole argument, which we're going to get to. It's completely different. Yeah, yeah, I'm so sorry. It's completely different. The terminology is messed up. I don't think the hole argument should be called the hole argument because it's not really a hole. But that's beside the point. I can't change the name. But yes, absolutely right. This is a completely different idea.

Take Minkowski's space-time, the space-time of special relativity. Now remove one point from that space-time. Just a single point. Just remove it. The resulting structure, that's a model of GR. It's got a singularity in it. A lot of people are going to point to that and say, but that's unphysical. Like, what the heck is causing that thing? It's not as if there's no matter there that's causing some hole to form because of some curvature blow-up or something like that. No, it's just there. It just appeared out of nowhere. Bob Geroch in the 70s was one of the first people to look at solutions like that and say, what we need is a good condition to rule that kind of thing out. And so he came up with the idea of what's called hole-freeness. And some other people, Hawking, Ellis, other people, have come up with their own versions of a way to get rid of holes. So that's another way you can pare down this standard collection to something that's more physically reasonable.

Okay, let's forget about holes, although I do want to get to the hole argument. But forgetting about holes, let's speak about holy space-times, H-O-L-Y. Because you have a theorem about God's eye views. Okay, so what is your theorem? Do you say there can exist a God's eye view? There cannot exist a God's eye view? What is a God's eye view? Please.

When you mean God's eye view, are you meaning — sometimes I call it a God point? Is that what you mean? I think so, the point from which you can see the rest of the universe from that point. You can see everything. Yeah, yeah, yeah. So a God point is a point, it's an event in space-time such that the past light cone of the event is the entire universe. So you can see everything. And when I say everything, I mean everything, and that includes the future to this point. And so what that means is that there's necessarily going to be time travel going on, because if you can see your future, if you go back in your past far enough and you arrive at your future, there's some really weird causal stuff going on. And so, yeah, I would say space-time has a God point if there's a point like that. I use this idea as a very minimal condition. I say, let's restrict attention to space-times that don't have God points, and then I will go off and maybe prove a theorem or something. But the reason why the condition is introduced is because if you do have a God point, well, then a lot of this unknowability of the universe stuff doesn't go through because you can see everything. You have access to everything.

What is the ontology according to GR? What is the universe? Yeah, I don't know. I mean, I don't know what is meant by that. I mean, I'm sure you could make sense of that in a number of different ways. I'm very close — I like to tie myself very close to the physics and the mathematics that I'm working with. So, as I just mentioned, I'm working with a space-time manifold with a metric on top, via Einstein's equation. We have some matter field there. That's the ontology. If you want me to expound on that, well, what does that mean? Then this is where I get very uncomfortable and I say, I don't know. I don't want to go past that. The ontology of GR is you start with a manifold of events and you have a metric that provides some causal structure and some additional structure. You have a matter field that's related to the geometry via Einstein's equation, and that's what there is.

Interesting. Okay, so there are some people who are substantivalists and then there are some people who are relationalists. The substantivalists will say that space points exist and they exist independently of matter, and then some people who are on the relational side — and you can also correct me if I'm incorrect — will say that it's all spatiotemporal relations between events and that's what's real. Yeah. And my understanding is, historically, these can be seen as progenitors, broadly speaking, from Newton as the substance person, and then Leibniz as a relationalist. Now, whether or not that's historically accurate, that's at least the folklore. So, do you have a dog in that fight?

Yeah, yeah. See, this would be something that I maybe got from David Malament. I don't understand a lot of — there are just so many articles and books written about this, and it's occupied such an incredible part of the history of philosophy of physics. But I just — I'm not sure I understand the distinction there. I think when you try to articulate what exactly you mean by both of those positions, you very quickly come to something like, well, what's the argument about here? So, this would be an example where, yeah, I don't have a dog in the fight. I think folks who were interested in the hole argument, various folks, may have a dog in that fight. Part of the reason why I maybe avoided the hole argument for so long is because I just didn't want to get into this, what I viewed as a kind of a quagmire. I didn't see really any progress being made there. I didn't really see clear views being articulated and argued. What I saw were some hand-wavy, gestury type of stuff. That's just not my bag. My bag is to make things precise. And it took a really long time for me to really see something that I could say in a very precise way that has relevance to these issues. But even now, I don't think I'm on one side or another.

I'd like to start drilling into the unknowability of space-time. Okay. So, you have some theorems on this, and it says something — again, correct me if I'm incorrect — roughly like, from your local spot, you'll never be able to determine the global structure of space-time. Now said the way that I said it, that sounds trivially true about anything. From your local spot, you'll never be able to know some global structure. Well, you don't have access to the global from your local. So what have I said that's incorrect? What is your theorem and what are the implications?

Yeah, everything you've said is correct, but I would just go further. So it's not just that from this event I can't know what the global structure of space-time is like. It's not just that from that event I can't know. It's that it doesn't matter where in the universe one finds oneself, whether that's in the past, present, future, here, there, everywhere. It doesn't matter. Suppose that you give me all of that data. Suppose that you have an eyeball everywhere in the universe. Everywhere in the universe, there's an eyeball there. Okay. And then suppose that you're somehow able to relay the information that the eyeball is seeing at every event, everywhere. Not just where you are, but literally everywhere and every when, too. This is not just in the past and present. This is every possible event, even in the future. Now you give me all of that data, I'm going to go back in my workshop and I'm going to come up with a model that's going to reproduce all of that data, but it's going to be completely different from the model of the universe that we started with. Whatever structure that you may suppose that our universe has, I'm going to find one that's completely different from that, but that reproduces all of that data.

Moreover, that was conjectured back in the 70s that that was possible by David Malament. It was not proven until 2009. By yours truly. Yeah, yeah. So I was very fortunate. David helped a lot because he gave in his paper a sketch of a proof and I used that sketch. I built upon it. But one thing I want to emphasize that I don't think many people realize is that I went way beyond what David had conjectured. David said nothing about local structure. So he conjectured that if you give me a space-time, well, I'll be able to come up with a space — I'm going to come up with this nemesis space-time that reproduces all of the data, but that is completely different. Now that's one thing, but it's a completely extra level to say, well now, not only do you just give me the space-time, but you tell me that you want to preserve some kind of local structure. So, for example, if you give me a vacuum solution, maybe you want to preserve the fact that it's a vacuum solution. So I can't just go off and build whatever model I want as the nemesis. I'm now constrained. I have to build a model that is a vacuum solution. Or whatever other local property that you could possibly think of. And it turns out that the theorem does go through even under those additional constraints.

And so what that does is it closes off a common response, I think, to the work. Which is, I think folks think, well, of course, deductively, if you give me some data, it's not going to deductively prove that the universe has to be a certain way. Nobody ever thought that. Science doesn't work like that. Science works on induction, right? Science works in a way where you tell me what the local structure of the universe is like here, and even though I can't observe it over there, I'm just going to kind of extrapolate or do some kind of induction that says, well, you know, the local laws of physics aren't going to be completely different over there. They're going to match what's going on here. So I'm going to do a little induction. What this extra step, this extra level that I'm talking about with this unknowability theorem, is it shuts down any response that says, oh, well, you're not doing an inductive type thing. You're doing a deductive type thing. I want to say, no, I'm telling you to give me all of your data and I'm allowing you to do any kind of local induction that you want. You're still not going to be able to pin down what the universe is like. The only way to wiggle out of this is to do a global induction, is to start assuming certain global properties about the universe. But that's precisely what's at issue here. The whole point of the theorem is to say we don't have any idea what the global structure is like. So if you start whipping out global properties and saying, well, I can shut down your theorem by appealing to this global property, I want to say, where does that come from? It's not coming from any empirical data that we've gathered. It's not coming from any sort of local induction that we're doing. Where that's coming from is just an assumption about the way that you want the universe to be. So it puts the pressure back on the person who wants to say that we can somehow know the global structure.

Are there some inductions or some assumptions that are reasonable, like Occam's razor, to say that this one is more simple than something more bizarre that you may come up with? Yeah, that's an interesting question. Maybe you could somehow make that rigorous. I don't know, though. There's no natural sense in which there's a property of simplicity that has manifested here that's a useful thing to consider. I mean, I think rather what folks want to use are the properties that we've been talking about. So they might say, well, if you're dealing in globally hyperbolic space-times, then you're not going to be able to do this. Or if you are working with space-times without holes in some sense, you won't be able to do this. I have a response to that sort of thing. But the simplicity stuff, yeah, I mean, that's a type of non-empirical, almost metaphysical type of thing that you're going to have to insert into theory choice to get you to go this direction rather than this direction. I'm not aware of anything along those lines. I'm interested in this sort of thing. I think that probably the most promising approach isn't a metaphysical thing. It's more of assuming that space-time is radically asymmetric. And if it's very asymmetric, then it turns out that local structure is actually equivalent to global structure. And so... Right, right. Heraclitus, you're speaking of the space-times. Yeah, yeah. And so then you have an entry point because then if I am allowing you to do local inductions, what you can do is you can say, well, I want my local structure to be kind of like the local structure of this model over here, and that will uniquely determine the global structure. So that's one way through. It's a way that I've explored with co-author Thomas Barrett, where in a way it's kind of like arguing against myself in some way. And that's a trip to write a paper where you're kind of arguing against yourself. But I think it just shows the limits of the theorem. Every theorem has its limits and so this is going to be one of those. What I would say is that because local structure and global structure come together in this context of radically asymmetric spacetimes, you might want to consider weaker types of local structure inductions that you would be able to do that don't qualify. So what I mean by that is, you know, if you said, for example, okay, you can do local inductions like you can assume various types of standard energy conditions, you can talk about vacuum solutions, you can talk about that sort of thing. Well, then the theorem goes through just fine. So it's really that the fact that when I'm proving my theorem, I'm working with the most liberal notion of local property that I can possibly come up with. And when you're dealing with something like a Heraclitus space-time that's radically asymmetric, I've just let too many possibilities in. And the local structure actually turns out to be equivalent to global structure. It shuts the whole thing down.

What is a Heraclitus space-time? Yeah, good. So a Heraclitus space-time is a space-time where you give me any distinct events in the space-time and I claim that if it's Heraclitus, those two events are different in some invariant way. So they have different local structure. And so you know Heraclitus is famous for his constant theory of flux, this radical flux that's happening. Everything always changing. He famously said you can't step into the same river twice. And so that's the idea here. You can't step into the same river twice means that if you step into the river the first time and you try to do it later on, there's going to be some difference there. It might be the same with respect to some properties, but it's going to be different relative to some others. And so a Heraclitus space-time is something where, yeah, every event is different from every other.

Okay, so to the keen-eared listener, which is probably the better word than hawk-eared I said earlier, they may be thinking, well, what is it that you mean by local structure? Isn't the fact that we're on a manifold, doesn't that mean that everywhere already has the same local structure? If everywhere looks like Rⁿ, what is this local structure you keep saying can vary from point to point?

Oh, yeah, that's a great question. If you're just dealing with a manifold, then every point is just like every other, right? It's important that when we're working in GR, we're not working just with the manifold, we're working with additional geometric structures on top of it. In this case, a Lorentzian metric. So it's the Lorentzian metric that's going to be different here rather than here. The Lorentzian metric is telling you what's the geometry like at that point. And so in Heraclitus spacetime, it's really weird, but each point has its own geometry going on.

Okay. What if someone says, and I'm sure you've heard this, locally every point looks like Minkowski space? No, no, that's not true. I think that's a common misunderstanding. Minkowski spacetime is flat. There's no curvature there at all. What I mean by that is that if you look at various types of curvature tensors, like you may be familiar with the Ricci scalar field, on Minkowski spacetime, that's zero at every point. In an arbitrary spacetime, that's going to be varied over the spacetime. It's going to have curvature associated with it. And that curvature must be preserved by any mapping from one point to another. If you're going to say that these points are the same, then you're going to have some kind of isomorphism from one point to the other that's going to preserve that structure. And so what we do in Heraclitus spacetime is we look at properties just like this, the curvature properties, and say there's no two points with exactly the same curvature properties.

Some people may object and say, well, okay, if we're going to assume space-time is continuous, sure, we can construct such wildness. Maybe there's something about Zorn's Lemma, which we'll get to, that is necessary for these Heraclitus structures. But maybe — I mean, space-time, sorry — but maybe space-time is discontinuous, or we don't have a theory of quantum gravity. So we're speculating at something that's sub-Planckian, classically, but perhaps we shouldn't. How do you think about all that?

Yeah, yeah. Well, it's very common to have, for a variety of mathematical structures, one looks at the symmetries of those structures. One can do that for a set. One can then add some topological structure and look at symmetries of a topological space. One can do the same thing for a manifold structure. So there are all sorts of levels of structure that one has in mathematics. And at all of those levels of structure, what you want to do is look at the symmetries there. And what we're doing is basically drawing attention to the fact that, unlike a manifold, say, where each point is just like any other, in generic space-times, you just don't have symmetries. And you don't have symmetries in a global sense and you don't have symmetries in a local sense either. And so what that means is that these structures without symmetries are very common, not just within GR, but in basically any kind of mathematical object that you're going to want to look at. You start looking at the space of all of the objects, it's almost certainly going to be the case that generically a randomly picked arbitrary example is not going to have any symmetries. So that would be my response. This assumption is not some kind of special condition that happens only in very special cases. We think that probably almost every general relativistic space-time is Heraclitus. It's just that these are very hard to write down, and so you open up a textbook, you're not going to see these. You'll never see these in a textbook because in order for you to study these things, you're going to need symmetries to make them simple.

So let me ask you this, is it the case then that we can say that approximately Heraclitus becomes FLRW or something like that in order to infer the Big Bang? Because the Big Bang models have so many symmetries to them. So I'm not understanding, if the universe actually is Heraclitus, then does that put doubt into something as profound, important, seemingly fundamental as the Big Bang? How do you see that?

Yeah, yeah, yeah. It's a great question because I think what's going on is that there are different levels of scale involved. I mean, the Heraclitus point is that if you're really zooming in on each point and looking at its very particular curvature properties, that you're never going to find any two points with exactly those curvature properties. When you're dealing with something like FLRW models, you're zooming way out. You might talk about maybe that there are approximate symmetries in that sense. And there have been some philosophers and physicists who have tried to make that idea precise. So I think when you're working with an FLRW model, it's going to be an approximation of some other model that's Heraclitus that's in the background, that's maybe really what our universe might be best represented as. But when you're studying things on such a grand scale, maybe those small little differences in curvature aren't — they're just not important for whatever you're studying. So it's just a matter of what you're looking at there. I don't think there's any contradiction or any tension there. You just have to keep track of what are we studying, what are we trying to model. And when you're doing something, when you're trying to model something like FLRW, you're going to naturally reach for space-times with all sorts of symmetries. Because on a large enough scale, there is going to be all sorts of symmetries. But of course, it's not going to accurately model our universe. I mean, we have clumps of galaxies here and not there. And so our universe is not completely symmetric, right? It's not completely homogeneous or isotropic. Yes, yes, yes. So all those assumptions are great if you understand what you're doing with those assumptions. As long as you're not banging your fist on the table saying, no, no, but this FLRW model, this really is an accurate representation of our universe, then there's no problem. So if I had to bet money, I would bet money that if we're in a Heraclitus universe, that the curvature would average out to look like something like FLRW. But I know in effective field theories, there are precise theorems that show you that you can integrate out certain low-level details. I haven't encountered a Heraclitus space-time until your work, so I don't know if it's the case that something like chaos would occur where something small actually gives rise to some large change when you try to, quote, integrate out into an FLRW or into these other approximations. Do you know or do you suspect it to be the case that if our universe is indeed Heraclitus, that that wouldn't be the case? And it doesn't cause any tension with FLRW.

Yeah, I'm not aware. We're investigating some issues on the philosophical side of things. So we're investigating things like what philosophers have said for a long time about connections between, say, symmetry and structure and that sort of thing. So I haven't really looked at what the implications are for this sort of thing for something like quantum field theory. My guess is that you wouldn't run into problems. The examples that I know of that I've constructed in the past are really simple models. It's not like they're pathological in any way. It's just that they are varied everywhere. It's not like there are any curvature blow-ups. It's not like there are even singularities anywhere. You can have globally hyperbolic Heraclitus space-time. So I don't think it's messing anything up, really. It's just sort of like a fine-grained kind of model of the universe that could potentially have implications for certain philosophical things that maybe some people aren't interested in, but in philosophy of physics, for example, there is a long tradition of saying, well, the symmetries of an object tell you how much structure it has. If it's got tons of symmetries, it doesn't have very much structure. If it has few symmetries, it's got tons of structure. And so what we're doing with our Heraclitus stuff is kind of throwing a wrench into this and saying, well, at some point you get to the stage where your asymmetries kind of max out. You can't be any more asymmetric than a Heraclitus space-time, but your structure doesn't max out. You can still add on structure. You can still add on, you know, a little sign that says, here's the center of the universe. Or, you know, so you can pile on structure, but that's not going to decrease the asymmetries. That's something that may not be of interest to folks, but in philosophy this is decades and decades of folks writing about this, what we call dogma, this symmetry-structure dogma, and we're trying to blow that up.

Is there something about these maximally asymmetric space-times, otherwise known as Heraclitus space-times, that you've constructed — is there something about it that makes a certain direction privileged or something like that? Because it's said that in FLRW, there is no privileged center, there is no privileged direction, and there may be cosmological data on the experimental side that we're in an odd part of the universe. I've spoken to Subir Sarkar about that. I'll place the link on screen. But the point is that that's experimental data. Is there something about the theoretical Heraclitus space-time that says, oh, actually, because of this, there is a preferred something, a preferred direction, a preferred center?

Yeah, I would think it goes the other way. I would think that the fact that it's so asymmetric means that various tensor fields or geometric objects generally that you might consider on that space-time — there's a sense in which many, many, many geometric structures are, we would say, implicitly defined by the space-time. So, a space-time implicitly defines a whole set of structures when those structures are preserved by the symmetries. And what happens in a Heraclitus space-time, because there are no symmetries, that means that there are all sorts of geometric structures there that are kind of compatible with that structure. So, I would say it goes in the other direction. It's going to blow things up even more than, you know, might have in a very symmetric space-time like an FLRW space-time.

So, super interesting. Great, great response. Earman in a book called The World Is World Enough in Space — perfect, right — he has a chapter on various hierarchies of space. Yeah. And he said that as you add more structure, your symmetries become smaller but your absolute quantities increase. And questions of motion become more meaningful, something like that. Yeah, yeah. Okay, so give us an intuition. Why would it be the case that adding more structure — firstly, what does it mean to add more structure to your space-time? Why does that reduce symmetries? Why does that add meaningfulness to motion? How are these at all connected?

Yeah, yeah. Good question. I mean, in that chapter, he's looking at various types of classical space-times. So Newton, for example, he thought that there was an absolute space. Think of it like this. Imagine that you're even thinking that there is a center of the universe, right? So the center of the universe, that's a lot of structure. If you're pointing to a certain point in the universe and saying, this is the center of the universe, right? If you're not going to enter, that's an extra structure. And what that's going to do is it's going to limit the type of symmetries that are going to be allowed because symmetries are going to preserve all of the structure that you have there. So in a normal, you know, Newtonian space-time that we might think of now, you could shift the entire universe over four feet this direction. And what do you get back? You get back just what you started with. You don't change anything by shifting everything over if there's lots of symmetry. But if you've got, you know, the center of the universe is right here, now you no longer are able to shift over four feet because now things aren't going to line up anymore. So that additional structure shows that the symmetries have been decreased in some sense. And now, what is meant when Earman says that questions of motion become more meaningful?

Yeah, yeah. So you can actually start looking at models of Newtonian space-time where one of the symmetries is not just shifting over, you know, the whole universe in this way, but actually putting it in motion. Compare this universe that we have now with one that is moving five miles an hour that direction. Some space-time models will identify those as being the same structure. So, in Earman's book, these are the same structure. These are called Galilean space-times. Newton is going to want to say, no, these are different structures. So in a Galilean space-time, it makes sense to talk about acceleration, but it doesn't make sense to talk about, well, how fast am I moving? Because if you're moving, let's say I'm not moving in the original space-time, but now I've started considering my motion relative to this other model, this moving five miles an hour in this direction, well, now I am moving. And so if these models are identified, there's no matter of fact about whether I'm moving or not. Whereas in Newton's space-time, there is a matter of fact about whether you're moving or not. And so as you increase the structure, more and more things about motion make sense. So in this Galilean space-time, how fast am I going? That doesn't make sense. But you pile on some more structure, you have absolute space now. Now you're not identifying this space with the one that's moving five miles an hour in this direction. Those are not the same. That's not a symmetry anymore. Now, you can make sense of something like, you know, I am moving. Now, this is a bit technical, but remind me again of the difference between the Leibnizian space-time, the Newtonian space-time, and the Galilean space-time in Earman's book.

Yeah, it gets confusing. So I think, yeah, Leibnizian space is a space where each time slice is sort of its own thing. It's not really connected with the other time slices except for there's a temporal metric there which tells you how much time has passed between this time slice and this time slice. So you know, if there's an event happening here and one event happening at this one, maybe they're five years apart. But this space that looks like this is the same as this one in a Leibnizian thing. There's no connection. So in a Leibnizian space-time, there's no sense of acceleration because if I'm an accelerating body, well, if I'm not accelerating, what I can do is I can do this transformation and now I am accelerating. So just a moment, there's no sense of connection at all? Or is it just — you did a rotation with your hand? Yeah, I did a rotation. Could it have been like this and it could be discontinuous? I mean, you do have to stack these on top of each other, so there's a connection in the sense that there's a background manifold there that connects everything together. But there's no connection in the sense of a derivative operator that's going to allow you to make sense of something like acceleration. So, at this low level, there's tons of symmetries, so many symmetries. This is a symmetry, so is this, so is this, so is this, so is, you know. These are all symmetries. And what that says is that there's very little structure, and it means that very little makes sense in terms of motion. You can't even make sense of acceleration, let alone something like speed or my position or something like that. Then what you do from the Leibnizian thing is you go — you bump up a level, then you go to Galilean space-time. Now, this is a symmetry, so I can move things like this, but I can't rotate. Because rotation is acceleration. The whole point of a Galilean space-time is I want to preserve acceleration, but that's all I want to preserve. I want just enough structure so that Newton's laws make sense, F equals MA, so that that makes sense. But I don't want to add any more structure than I need to to make sense of F equals MA. So in a Galilean space-time, acceleration makes sense because, you know, think of a deck of cards, you know, it's all stacked up. If I bevel the deck, you know, that's essentially like putting everything in motion five miles an hour that way. Well, that's going to map straight lines to straight lines. Straight lines always get mapped to straight lines under those kinds of transformations. And so it's going to preserve acceleration. If you're not accelerating, under a symmetry, you're not going to be accelerating. Or if you are accelerating, under any symmetry, you will remain accelerating. And so the added structure means there's less symmetries than there were in the Leibnizian situation. There's more structure and that more structure allows us to talk about more motion, so we can talk about acceleration where we couldn't before. Then the next step is to bump up to what he calls a Newtonian space-time, and that's where we say, well, this is no longer a symmetry. This is a symmetry where you can shift things around like that, or if you want to rotate, you can rotate around like that, but no more can you — these things, you know, you're not allowed to move like that anymore. And so what that does is, now once you have these things connected in that way, you have this additional structure, you have less symmetries, and now you can talk about, am I moving? Am I moving? You know, what's my motion like relative to this absolute space thing? Because now this thing has kind of been glued together. Now this is Newton's idea of absolute space. You can keep going. You can do — I think he calls it an Aristotelian space. Interesting. Where you take Newtonian space and then you add the center of the universe to it. And now I can't even do translations like this. I can't, you know, I can't move everything over five feet because that will ruin this center of the universe structure. So that gives you a sense. It's very fascinating. And on some level it makes a whole lot of sense, this inverse relationship where you've got more structure, less symmetries. And for this context it makes a whole lot of sense to think about those things. But things start getting really crazy once you start looking at these Heraclitus-type models because then the asymmetry is so radical that it kind of gets maxed out. And then, you know, God knows you can pile on structure, but you're not decreasing the symmetries, and so then the whole idea kind of blows up.

Well, "God knows" is an interesting phrase there. Yeah. We can speak about God or we can speak about the nose part. In the Heraclitus space-time, you could know or you can infer what's going on at every other point because it's almost like — well, there's a photo which I'll show, which you will send me — that there's only one way that every puzzle piece is so jagged and so intricate that there's only one other puzzle piece that can fit on this side and one other puzzle piece that can fit on that side and so forth. Yeah. How would you put it?

Here's how I would put it. I would say, if you give me a Heraclitus space-time, we can play a game where you give me — where now you go and you cut up the space-time into tiny little pieces, little puzzle pieces for the universe. And you don't tell me which Heraclitus space-time you started with, but what you do is you hand me a box of puzzle pieces and you say, why don't you put the universe together? We're using these pieces. I mean, you can cut these things to be very small. Well, if I'm in a Heraclitus space-time, there is only one way to put the pieces together. So there's no room for any alternate way to put the puzzle together. If I'm putting — you know, if I find this piece, then I'm searching for another one that kind of matches up to it. There's only going to be one piece that matches up to it. So then I, you know, I put them together and then I keep going like this. There's only one solution. This contrasts from, you know, a situation like Minkowski space-time, where every point is just like every other. Every point can be mapped into any other point. And so if you were to cut up Minkowski space-time in a million pieces, hand me a box of pieces and say, put this puzzle together. Well, I could put it together like Minkowski space-time. I could also put it together in like a rolled up Minkowski space-time where there's time travel. I could also put it together where there's like a giant hole in the middle of Minkowski's space-time, some giant singularity. These are all possible ways of putting the puzzle pieces together. And so I think that's the sense in which local structure determines global structure. The local structure is given by those puzzle pieces. You're telling me what the local structure is like. And if there's symmetries around, like in Minkowski's space-time, the local structure just does not determine the global structure. There's all sorts of global structures consistent with the local structure, but in Heraclitus space-time there's just one way to put the pieces together.

Now, you've shown that Heraclitus structures implies knowability of the universe. Yeah. Okay. Have you shown that the only way for the universe to be knowable is via these Heraclitus space-times?

No. So, I mean, I think it might be possible to restrict attention to — look, if this is your background possibility space when you start, this is the standard collection, of course you can make that smaller and smaller and smaller to the point where you're going to get a uniqueness result. And you can do it by saying, well, let's just consider a collection where there's only one model. Got it. So, there's all sorts of ways in which you can get a uniqueness result, aside from this Heraclitus business. One route might be to do — you've got to do a tag team. It's not enough just to pick a causal condition or a local condition or something. You've got to kind of combine a bunch of them together. So I would say, if you were interested in proving a theorem like this, a positive theorem, look to globally hyperbolic space-times, which have very well-behaved local structure, like maybe they're vacuum solutions, which do not have holes. Maybe the combination of those three, in some sense, will allow you to get a unique model. But we know that even that's not going to work. We know that the general result may not be in the cards, but we do have examples of globally hyperbolic, really well-behaved models with no holes that have a cosmic underdetermination problem. De Sitter space-time is one example. So, in De Sitter space-time, your light cone doesn't do this. What it does is it kind of has these asymptotes. So there's an asymptote here and an asymptote here, so your light cone just does this. And so you don't get to see very much of the universe in a De Sitter space-time. And so De Sitter space-time, if you are in De Sitter space-time, you have no way of knowing if you're in De Sitter or some other unrolled De Sitter or some De Sitter with holes in it or some other De Sitter with, you know. So there's all sorts of possibilities around, even if one were to restrict attention just to really, really well-behaved space-times like that. I think talking about De Sitter is really important because that's what got this literature started when David Malament was writing his first article on this stuff. That's the example that he used to kind of illustrate the cosmic underdetermination problem. So, what you want to do is almost introduce the problem via something like De Sitter and say, look, you know, there's a good chance that on some level maybe we're living in a universe that is De Sitter-like. And so, in that sense, you have this cosmic underdetermination kind of lurking. What's the actual character of that? Well, now let's go investigate that and see how general things are. So, I think having the De Sitter in the background is a really good thing to do, and I'm glad we had a chance to talk about it.

So, to the people who care about science, who care about knowledge, knowability, and induction, the so what here is what? What is the consequence of your 2009 paper, I believe? Yeah. Yeah. So what? So what?

Well, I mean, I think it's natural. I love science. And that's what I do every day of my life. I'm doing physics. But science has its limits. And so what I'm trying to articulate is that one can use science to show that science has limits. And that's what I'm trying to do here. And so, the so what is, even in principle, there are things that science just isn't going to be able to help us better understand. And in this case, one of those things is the big structure of the universe that we're living in. Interesting. You can use science to show the limits of science. Yeah. Please explain more.

Well, I mean, in philosophy, underdetermination has been a big idea for a very long time. So you can go back to something like Descartes thinking about, well, how do I know this is what the universe is like and I'm not being deceived by some evil demon? We have this idea. The Matrix is a movie that I often bring up in class because most students are familiar with this idea. Underdetermination is this fascinating idea that pops up all over in philosophy. And there's something called the Duhem thesis which, you know, basically says that for any belief that I have, you give me whatever data that you want, I'm going to be able to keep that belief. As long as I'm, you know, tweaking auxiliary hypotheses in just the right ways. I may have to do some drastic stuff. I may have to maybe plead hallucination. I may have to, you know, assume that, you know, my eyeballs aren't working properly. Maybe I'm in the matrix, whatever it is. But it's certainly logically a possibility always to keep whatever one wants to think about the universe in spite of whatever empirical data is coming in. And so that's one type of underdetermination that's around. And what I'm drawing attention to now, back to your question, is I'm saying there's another type of underdetermination that ought to keep us up at night. That hallucination stuff, the matrix stuff, I mean, it's fun to think about, it's fun to talk about, but at the end of the day, I don't lose sleep about that stuff because it's not coming from our best scientific theories. What I'm doing here is I'm saying, hey, let's focus attention on our best science. Let's see what our best science says about what we can know. Let's not just start philosophizing about, you know, what can I know? You know, maybe I'm being deceived. That's a recipe for a lot of confusion, I think. Science is wonderful and let's use what we've got to try to figure out what we can know and what we can't. And what I'm saying is just that within certain contexts, and GR is one of them, you dive into the details of the actual science and try to answer the question, can we know what the universe is like via our best tools that we have? And the answer is no. Ah, okay. So, would you say that the former, what you talked about at the beginning of that answer, is not science showing the limits of science, it was more like philosophy showing the limits of science. Yeah. But what you're doing is taking science seriously, whatever that means, taking the theories of science seriously. Exactly. And then following their consequences and showing them. Exactly. That's a great way to put it. Yeah. That other kind is like using logic or philosophy or fun little, very wild kinds of, almost conspiracy theory kind of situations to prove the limits of science. I'm not doing that.

So, this is so late in the conversation for me to ask you this. And I have the luxury of going through your lectures and your papers, which I'll put in the description for people to go through because you write wonderfully. And that's not a quality that many modern writers have. So, thank you. Yeah. It's great to read your papers. And like I mentioned, this is a bit late for me to ask you this question, but because I want to move on from this topic, people have heard the term Heraclitus over and over and over. And I have the context of the Heraclitus space-times and all.

of that from your lectures. So, apologies to the audience member who still has question marks over their head. But before we move on, what simply is a Heraclitus space-time? In simple language for someone who doesn't know what GR is.

Yeah. Yeah. It's a simple idea that you don't even need GR to say it. It just means a Heraclitus universe or a Heraclitus world is one where any two events that you pick are different in some way. That's it.

Perfect. Okay. Did you construct the Heraclitus space, or Heraclitus universe, whatever we want to call it, because you wanted to rescue knowability? No. Because you were so disturbed by the unknowability? No, no. That wasn't the angle. The angle was some of these questions about what is the connection between symmetry and structure. Philosophers talk about this a lot. Jill North is a philosopher who wrote a book recently about this. And me and a buddy Thomas Barrett, we just started talking about the relationship between symmetry and structure. I think at one point he said, like, well, it would be impossible really to find a space-time like this, but what would really be cool is if you found some kind of universe where every two events are locally just completely different. And I don't know, it just sparked something in me. I was like, I've got to figure this out.

And so, you know, I went to the drawing board and tried — well, you know, actually what I did is I wrote some code back when I was in graduate school for mathematics, where I could enter in some space-time metrics and it would pop out some curvature tensors. And so what I would do is I would just play around and play around with various kinds of metrics to kind of get these curvature properties to kind of play off of each other. So what we did was we found — we limited attention to a really simple model. I love making things simple, as simple as they can possibly be. And so for us, we're like, okay, well, let's just focus on two-dimensional universes, one space dimension, one time dimension, and let's make this conformally flat. So it has the same causal structure as Minkowski's space-time. But let's just tweak things a little bit so that we identify two curvature functions where, you know, we'll call them P and Q, say. Points can share the same P-values and they can share the same Q-values, but we set things up in just this right way where they can't share the same P- and Q-values at the same time.

Does it bother you that, at least as far as I can tell, the axiom of choice is required for Heraclitus space-times?

Okay, good. Yeah. I'm glad you asked about that. We have a separate paper that talks about maximal Heraclitus space-times, which are space-times which are maximal with respect to this Heraclitus property. And there we do need Zorn's Lemma to prove their existence. It's not uncommon to use Zorn's Lemma to show maximality properties within space-time. But just the Heraclitus property, we don't need Zorn at all to do that. So the one that we construct, I literally can just tell you what the metric is. I don't need to rely on any existence result that's mysterious like Zorn there.

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Much of the audience may be coming from the consciousness side or the philosophy of mind side. If space's global structure is unknowable from within it, does that have any bearing at all on how first-person knowledge is irreducibly from within?

I don't see a connection there. I mean, I see a connection only insofar as there's limits to what one can know. And I think that there is a similar type of underdetermination problem with respect to the self. It's a completely different context. But I think some of the rough contours of the situations are the same. But I don't see any direct relationship there.

Tell me about the underdetermination of the self.

Yeah, yeah. I mean, this is something that — I have an intro philosophy of science class going right now. We're just getting to the part where we talk about this. You know, essentially, we have set things up in this class so that we start out with kind of like philosophy of science 101. This is where we learn about basic logic and the basic underdetermination problem. Then what we do is we do a case study where we look at the underdetermination, how it pops up in physics, in particular with respect to this cosmic underdetermination, this large-scale underdetermination. But now we're at the end of the quarter and we're looking at a second case study, and the second case study is itself. And so, what we're doing there is we're saying, you know, if you look at the writings of folks like Hume, other empiricists, what they're doing is they're saying, when I look, when I try to observe what I am, when I ask the question, who am I, what am I, and I try to observe myself, I find that I don't have any observation of myself, right? So, you know, Hume is someone who, as an empiricist, he's saying — he's looking at billiard balls and he's trying to look carefully to see the cause of this billiard ball going over here when it's hit by this other one. And he says, try as I might, I see all sorts of empirical data going on here. I see correlations with movements and stuff, but I never see the cause. And that's because this idea of causation is a very slippery, metaphysical thing. And I think Hume, in this context, is saying the same kind of thing. He's saying, when I look carefully for myself, I see all sorts of empirical data. I have all sorts of perceptions. I think there's a famous quote where he talks about — he says, I see love in me. I see hate in me. I see pleasure. I see pain. I have all these perceptions, but I never see me. And some folks have thought — I mean, this is very, very similar to what Buddhists say about the self. Of course, Buddhism is a very vast, diverse thing. But what draws them to the self is a very vast, diverse thing. But what draws them together are just a few very important principles like impermanence. But the idea of non-self is one of those. And so, you know, you look at what an empiricist like David Hume is saying and you compare it to what a Buddhist might say about the self and you see that they're both kind of picking up on a type of underdetermination. A type of underdetermination that starts with empirical data about the self and then not being able to pin down what that self is like.

So, as far as I know, Hume also wrote that these investigations left him frightened and confounded. I think he said something about retreating to backgammon to escape the distress. And also that Hume was arguing for an epistemological claim that there's limits to what we can observe through introspection. As far as my limited readings are, I didn't see an ontological claim. But the Buddhist claim to me seems ontological about the no-self. So, how do you make that bridge between Hume and Buddhism?

Yeah, I mean, there's different ways in which you could make sense of the Buddhist idea of non-self. And I think it's probably going to be — you could make sense of it in different ways depending on which type of Buddhism you're looking at. The tradition that I am a part of, the Zen tradition, the idea of non-self is not that — first of all, anytime you're working with any kind of words in Zen, you're almost always for sure going to mess up whatever you're trying to say, right? I mean, a part of Zen is a sort of skepticism towards being able to articulate the deep truths with words. So, something like non-self, I think, for a Zen Buddhist, it's — as far as I would think of it, a Zen Buddhist would be not so much grasping onto an idea of, hey, non-self. It's a letting go of the idea of self. In that I don't want to be trying to grasp foolishly at stuff that goes beyond my direct experience or the empirical data that I may find myself with. Right. So, it's a way to be in light of a type of underdetermination. It's not — it's saying no to the idea of grasping onto something, even if that something is a negation like non-self.

I'm sorry. My question was more about how do you get from Hume? Because I heard you say, I think it was in one of your lectures, that Hume can be read as somehow validating Buddhist ideas. But I don't see how one goes from that we can't observe the self to that we don't have the self. It also seems more like one cannot find the self is also consistent with just introspection has systemic blind spots. And it doesn't warrant a Buddhist conclusion necessarily. So, I just wanted to know how that bridge was made.

Yeah, yeah. And I guess my response is that the Buddhists should not be viewed as someone who's grasping onto the idea of non-self.

Ah, okay. So, do you see that as a Western misinterpretation then? The idea of the no-self almost as a grasping of the no-self?

Yeah, it may be. I mean, certainly within the Zen tradition, you have this happen over and over again where you have some student trying to articulate something that he feels like he's, you know, come to or something. And then the master, in some usually funny way, says, you've missed the whole point. And so you can easily imagine, you know, someone saying like, oh, I figured it out. Buddhism is about non-self. And the Zen master saying, you've missed it. You've missed the whole point. What do you mean? Because the master might be saying, you're grasping too hard on this idea of non-self. Whenever you're grasping on anything, you're not doing the Zen thing.

What's the difference between Zen Buddhism and then — why don't you outline some of the other major tenets of Buddhism, two or three, if you like, just so that people can get an understanding?

Yeah, so, you know, I'm not an expert on any of this stuff. I'm just somebody who found a little group back in the day and I would like to go meditate because it allowed for a space where — yeah, those words and those theories, those isms were — I think depending on your tradition of Buddhism, those words and theories and, you know, practices, certain beliefs, whether it be about reincarnation or whatever it is, may be different. I'm drawn to the Zen because it seems to be the most simple. In my opinion, it's the one that allows me to almost be a spiritual person without being a spiritual person. I think one characteristic of Zen is that there is this skepticism towards expressing truths through words and theories. It's saying that's missing the whole point. So, it's like the finger pointing at the moon or whatever. Like the words, that's just the finger. What it's supposed to be doing is getting you to have a direct experience about something. And that's something that's very hard to talk about. But that's the point of it. I see that as the real thing that draws me to Buddhism.

To Zen Buddhism in particular?

To Zen Buddhism in particular, yeah.

So, what was it that initially drew you? Tell me if I'm putting words in your mouth. I don't mean to. I understand that that may have sustained you, that you liked that, but what drew you to it initially?

I think, yeah, I mean, I think I mentioned I grew up Mormon, so I had this spiritual side. And then when I got to grad school, this went away. I didn't want to be a part of that group anymore, but I still had this side. I had, fortunately, as an undergrad, a philosophy professor who practiced meditation as a practicing Mormon as well. But he would talk about Buddhist ideas and benefits of meditation. So I kind of had that rolling around in my head for a number of years. It wasn't really until — you know, I was working really, really hard as one does as a grad student and early professor. You know, I had, you know, I've got to get tenure. I've got to get tenure. Like that was such a big thing for me for a few years there. It just was taking up so much of my energy in my head. And then when I got it, it was like, okay, I have a sort of a space now and I want to kind of get some clarity. And so that was initially when I really was drawn to Buddhism, right after getting tenure, and recognizing that I had this opportunity now to kind of calm down a bit. And so, there just happened to be some folks who sat zazen in the neighborhood. It was just — I just walked down the street and there was a group there. And so, I was able to join them and learn about some of the basics and things. And I went on some retreats with those guys. Yeah, it's just been a part of my life ever since. I mean, every night before bed, we're in my daughter's room. The whole family gathers. My wife, my daughter, the dog. We're all on the bed together. And that meditation moment, you know, it's a big part of life around here. And it's how we may frame certain things that might come up in life as well. So, I think of it very much as a kind of a religion for me. I mean, it provides a kind of framework and a structure. And that's what I wanted and needed, I guess.

So the uniqueness of every moment — going back to the limits of science, science requires repeatability. Does a Heraclitus space, or the impermanence of everything, does that mean science is not possible in this framework, technically speaking? Is there a way around it by using the word approximately similar? But then how do you make approximately similar rigorous without having some topology on some space of experiments?

Yeah, yeah. How do you view that? Yeah, yeah, that's a good question. I mean, one thing to keep in mind is that it's not as if at these two different events that everything is different about them. So, there's going to be all sorts of structures that are going to be exactly the same in some sense, right? So, in both cases, if you're looking at what the manifold structure looks like at those points, those are going to be exactly the same. If you're looking for a light cone structure, both of those points are going to have a light cone structure. So there's going to be a causal structure there that's very similar. So there's so much that's going to be preserved. There might even be a bunch of curvature properties that are going to be preserved as well. It's just the only point about the Heraclitus space-time, the only requirement is just that it can't be the case that everything is the same at these two points. So there's going to be all sorts of opportunities to kind of do inductions about all sorts of other properties. You could, in principle, even do something like put some data on a surface. The data is going to be kind of like a Heraclitus data, so every point is going to be slightly different. And then evolve that data in just the same way that these theorems of Choquet-Bruhat and Geroch show. Presumably, that all works out, even if you're working within the context of Heraclitus spacetimes. So, it doesn't limit science at all. All the theorems that everyone knows and loves are presumably all intact. It's just that Heraclitus-based theorems are so hard to study just because they don't have those symmetries.

What are you excited about right now?

Yeah, what am I excited about? You know, I'm working on a new paper with my buddy Thomas. We write papers via text message. So, every night we — you know, I'm going to sleep and I'm thinking about some question concerning symmetry and structure these days. In the morning I'm texting him my idea. He's texting me back. You know, that's what's getting me really excited.

You called him your symmetry buddy.

Yeah, he's my symmetry buddy. And it's a wonderful thing. First part of my career — and Jim, I think. Well, Thomas, Jim Weatherall, and then Hans Halvorson. Those three, I've co-authored a number of papers with them about symmetry and structure. And yeah, I mean, the first part of my career, I wrote all these papers by myself. That was just me, just, you know, and this is so much better just to have symmetry buddies around, be able to, you know, bounce ideas off of each other. And I think we're all in this style where we're all coming from the Malament style of things. And so, we all have a shared space where we're working and it's allowing for some wonderful things. As far as like, that's on the horizon. Just recently, you know, I've been excited. I just got a paper accepted with some Hungarians. It was a problem that I had been thinking about for like 10 years and I hadn't been able to figure it out. And I started working with the Logic Group in Budapest. We figured out this problem. So it's easy to state. The problem was posed by Bob Geroch more than 50 years ago. And he asked, if you have an extendable space, so a space that's not as big as it can be, it's extendable, and let's say that there are no closed timelike curves in this space, so time travel is not possible in this space, can you find a model like that such that every time you extend it, every extension has time travel in it? And man, I thought about this question for so many years, so long. I had a sketch of an idea, which was to take Minkowski's space-time and roll it up so that there are closed timelike curves there, but then kind of remove a fractal, a Cantor set, like a delicate little thing where I'm poking little delicate little holes and it's this little fractal. And the fractal is going to block those closed timelike curves from forming. But then every time you want to extend it, you're going to have to extend a bit of that delicate structure, and that's going to allow a little path for the closed timelike curve to form. And anyway, that was my idea and I just randomly said that to the Budapest folks and I was like, if you guys can see a way of doing this, you know, here's an idea. And they did it. They went to work and we just, you know, a couple weeks ago it was published. So that's the latest thing that I've been excited about in terms of research.

So, speaking of looping back, a question I asked you near the beginning was, what is it that you envy others of in terms of style? And then you gave an example, I believe, of Malament's style. But now, I want to ask you, it's always easier to look at one's own deficiencies than it is to look at the positive aspects of oneself. But what is it that, if you were being honest, other people look at you and say, why is that so easy for him?

I don't know, man. Because everything seems hard for me. I don't — I don't know.

I think — How about this? Let me ask it to you in the form of an interview question. Okay. What is your greatest — okay, so what is your greatest weakness, anyone? Okay, whatever. What is your greatest strength?

Yeah. One thing that I can do that I think is a strength would be — I'm able to — yeah — tenaciously come back to the same idea again and again and again. I think where some people might give up on an idea, I just don't give up. I just am relentless and it may take years for me to figure something out. And I'm very slow. I really — I'm not just saying that. I really am very slow. I've learned this stuff over the course of 20 years now. So I'm building up very slowly. Things don't come very quickly to me. I'm not comfortable in a let's shoot the shit kind of situation about physics or philosophy or really anything. But this is a blessing and it's a strength in the sense that it allows me to — well, let's just go back to the basics. What is asymmetry? What are we doing here? And it lends itself well to philosophical questions because I think philosophers — that's kind of what a philosopher does, is go back again and again to some basic, basic stuff. And so I think that's probably it.

So you said you don't enjoy or aren't skilled at real-time sparring in physics or philosophy.

No.

Okay, great. So then, what does it look like for you? What does the slow thinking look like for you? Is it you alone writing on your whiteboard, writing with a pen and paper? And I mean, to actually tell me specifically, is it you in the shower? So for instance, for myself, I like to take showers and I have a note, a way of taking notes in the shower. It's actually a device that we got sponsored by called Plaud. Oh, that's cool. Super great for it. They're not paying me for this. But it's fantastic. So I like to — I think on long walks, I think by speaking aloud. I don't enjoy pen and paper. But I do think pen and paper is better for me. I just don't like it, so I don't do it. But if I was to be my own coach, I would say you need to actually think more with a paper and pen. But anyhow, so how does — what is this slow thinking? What does slow thinking look like for you?

Yeah, yeah. It's a great question. I mean, I do have a clipboard. It's one of my cherished possessions. I've always got paper on that clipboard and a mechanical pencil. A lot of my work lends itself well to pictures. I think in pictures, that's how I've been able to prove a lot of this stuff, is because I'm imagining it. So I'm drawing a lot. I'll usually have a question. I'm trying to figure it out. And I'm just, you know, going back over it. Over and over and over, trying to see if there's something there. And if it's not there that day, then, you know, I'm coming back the next day. Like I said, fortunately now I have buddies where I can bounce ideas off of them and they can, you know, we can do like a little conversation or a little dialogue which is really lovely. Mainly, yeah, it's just like you said, it's in the shower. It's when I'm walking to and from school. It's always having that idea or that problem in your life. And you're trying to figure it out. For me it's incessant. I can't help it. Almost destructively so. Yeah, it's hard for me to sleep sometimes. It's destructive in the sense that, yeah, it'll pull me away from maybe a conversation I should be paying more attention to or something. I'm thinking about this other thing. So it can be destructive sometimes. But yeah, I mean, in the shower, sure. Yeah, I've come up with ideas in the shower.

What's a life lesson you wish you could impart to your younger self?

Yeah, good question. I would just say, like, do not give up. I mean, these are things — I have an 11-year-old daughter, so these are things that I feel like I'm trying to teach her. Just don't give up. Keep going. Keep trying. That makes all the difference in the world in terms of like, just other people may give up at this stage or this place. And it can make a big difference if you just don't give up. Even when things are not looking good, you just keep going. So that would be the one thing.

Professor, thank you for spending so much time with me.

Thank you, Curt. I really appreciate it. Yeah, it's been lovely. Yeah. Thank you so much. This is a blast. It's been an honor.

And how can people find out more about you?

I think the best place would be just to go to my website. In particular, I like to draw attention to a couple of non-technical pieces. One on the unknowability of the universe and then another on Heraclitus space-times.

Hi there, Curt here. If you like more content from Theories of Everything and the very best listening experience, then be sure to check out my Substack at Curt Jaimungal. Some of the top perks are that every week you get brand new episodes ahead of time. You also get bonus written content exclusively for our members. That's C-U-R-T-J-A-I-M-U-N-G-A-L.org. You can also just search my name and the word Substack on Google. Since I started that Substack, it somehow already became number two in the science category. Now, Substack, for those who are unfamiliar, is like a newsletter. One that's beautifully formatted. There's zero spam. This is the best place to find out I'm going to follow the content of this channel that isn't anywhere else. It's not on YouTube. It's not on Patreon. It's exclusive to the Substack. It's free. There are ways for you to support me on Substack if you want, and you'll get special bonuses if you do. Several people ask me, like, hey, Curt, you've spoken to so many people in the fields of theoretical physics, of philosophy, of consciousness. What are your thoughts, man? Well, while I remain impartial in interviews, this Substack is a way to peer into my present deliberations on these topics. And it's the perfect way to support me directly. Curt Jaimungal, or search Curt Jaimungal Substack on Google. Oh, and I received several messages, emails, and comments from professors and researchers saying that they recommend Theories of Everything to their students. That's fantastic. If you're a professor or a lecturer or what have you, and there's a particular standout episode that students can benefit from, or your friends, please do share them. Share. And of course, a huge thank you to our advertising sponsor, The Economist. Visit economist.com/TOE to get a massive discount on their annual subscription. I subscribe to The Economist and you'll love it as well. TOE is actually the only podcast that they currently partner with. So it's a huge honor for me. And for you, you're getting an exclusive discount. That's economist.com/TOE. T-O-E. And finally, you should know this podcast is on iTunes, it's on Spotify, it's on all the audio platforms. All you have to do is type in Theories of Everything and you'll find it. I know my last name is complicated, so maybe you don't want to type in Jaimungal, but you can type in Theories of Everything and you'll find it. Personally, I gain from rewatching lectures and podcasts. I also read in the comments that TOE listeners also gain from replaying. So how about instead you re-listen on one of those platforms? If you're using platforms like iTunes, Spotify, Google Podcasts, whatever podcast catcher you use, I'm there with you. Thank you for listening.