Transcription
I've always wondered why space is three-dimensional. We don't know, but it is my very strong suspicion that that is a consequence of our nature as observers. Let's dive into some of the details, first of all about space and time. I really like your idea that the concept of space-time together might have been the biggest wrong turn in physics in the 20th century, whereas in your model, space and time are so simple and separate. Space is the hypergraph, and time is involved in the evolution of the hypergraph, and maybe we can get into some details of timelines later. But sticking with space for the moment, I've always wondered why space is three-dimensional. In your model, is that an emergent property of the model, or we don't know yet? It's a position in the rad that like we interpret space as three-dimensional.
So, so here's what I suspect. So, so first thing to say is that, you know, if we think about what is space for much of human history, that hasn't been a question that anybody thinks to ask. You know, Euclid just sort of said space is this background, and we put points at different places in space. You know, the very first common notion of Euclid is a point is that which has no part; there are indivisible points, that which can be placed sort of anywhere in space. So, in a sense, the first thing you have to get over is the idea that space, there's something space is made of something. You know, in a sense, I've now kind of understood a little bit of the arc of history that leads to this. You know, back in antiquity, everybody was arguing, is everything discrete, made of atoms, or something continuous, kind of flowing? And for millennia, nobody knew. It wasn't clear whether matter was made of discrete things or was continuous. Eventually, by the end of the 19th century, yes, you know, molecules, atoms, and so on exist; matter is discrete. Then, you know, people became clear that you could think of photons as existing; light is discrete as well. What about space? A thing that I've only learned in the last couple of years, really, is how convinced most physicists were at the beginning of the 20th century that space would turn out to be discrete. But it just didn't work. There wasn't the relativity; you couldn't sort of fit relativity in with this idea of discreteness. And so by about 1930, people kind of gave up, and people, and sort of then it was, well, let's just do it the way Euclid set it up and have space be continuous and this sort of background kind of thing.
But so one of the key ideas is, well, let's actually have space have a structure. Let's have it be made of things. That's something that's kind of natural if you think computationally, because we know sort of this robust notion of computation on discrete kinds of things. We don't really know how to set up computation on purely continuous kinds of things; it doesn't seem to be as robust a notion there. So one's kind of led by the idea of computation to think, well, could space be discrete? And the sort of what people had tried to do in the early part of the 20th century was to make sort of a lattice, like a crystal in space, and that just didn't work. And that was a thing I realized in the early '90s, that, you know, if you think about space as this kind of network of discrete points, these kind of atoms of space, where the only thing you can say about them is how they're related to other atoms of space, that's a more promising foundation for thinking about space.
So the sort of this idea, space is made of something, it's made of these discrete atoms of space related to other atoms of space, making up this graph or this hypergraph. The only real difference there is a sort of technical one, which was important for me, although in retrospect, if I had known what I know now, I could have just steamed forward with the things I was doing with graphs in the early 1990s. It was, it was one of these things where you just have to understand more and then go back and see that you didn't actually need the extra pieces. But the idea of hypergraphs is, instead of just having atoms of space where in a graph you would have one node is connected by an edge to another node, you can have multiple nodes related by a hyperedge in a hypergraph. But so the notion is that's sort of the structure of space is this hypergraph.
And then the question is, well, what's time? Well, time, as far as I'm concerned, is the progress of computation. It is what happens to the hypergraph. It is what happens in general in thinking about time and in all kinds of systems; it's this sort of progress of computation, the application of computational rules. And the thing that's sort of important about that is you might have thought, well, once you know the rules, once you have the setup of the system, then you have determined everything about the system. And in some sense, you have, but in the sense that you can know what will happen, but you can't know it by any process in many cases that is more efficient than just following each of those individual steps. So it's an important idea, this notion of computational irreducibility, that I first came up with back in 1984. Now, so it's been more than 40 years that, this idea that there is sort of an irreducible computational process that has to be gone through to work out what systems will do. And it's that irreducible computational process that kind of is the thing that happens when time passes. And so the passage of time is this process of progressively applying computation to a system, in this case, to this hypergraph. So the extent of space is going from one atom of space that's related to another that's related to another, that's kind of the extent of space. And then time is this progressive rewriting, this computational process.
Now, the thing that gets tricky and a bit more technical is, so we know that there are mathematical relations that seem to work well in relativity between space and time. How does that emerge from a system where space and time are sort of in some ontological sense very different, where one is this extent of a hypergraph, the other is this progression of computation? Well, the thing that gets tricky, and I knew this back in the '90s actually, is what if you imagine yourself as an observer in the system? There's a question of what do you actually know about what's going on. And what you realize is the only thing you can know is kind of the causal relationships between rewriting events. Kind of the analogy I used back in the 1990s is imagine that only one place in the universe ever got updated at a given moment, but that place is kind of zipping around all over the place. And so there's a moment at which I'm getting updated, there's a moment at which you're getting updated. And looked at from the outside, these are different moments because the active site that's updating things has been zipping around. But the fact is, to me, I can't tell whether you've been updated or not until I've been updated. The only thing that I can really be sensitive to is this causal graph of the relationship between your updates and my updates. And that's that's kind of the, so what ends up happening is that what matters is the causal graph of updating events. And the result of that is you have this causal graph. This causal graph defines the relation between these events. And you're slicing that causal graph to determine what counts as space, what counts as time. Roughly, the idea is that when you have different events, those events are time-like connected if one event produces output that feeds into another event. Then those events are successive in time in some sense. Space is kind of what's orthogonal to that; it's kind of what is not sort of related by this leads to that leads to that. Are things that can happen simultaneously, and those define the kind of space-like slices. And there are many different ways you can make those space-like slices. Those correspond to kind of the reference frames of relativity theory and so on. They're different interpretations of what counts as space and what counts as time. And that's what leads to kind of the mathematical structure of relativity and so on in these models.
So it's something where, you know, I think what happened historically, I don't think Einstein ever really thought that space and time were the same kind of thing. No, he always thought that, you know, he had these arguments about the relation between space and time. But then Minkowski, who'd done a bunch of work in number theory, this I think 1909 or so, came along and said, look, I can see this quadratic form, it's x squared minus t squared, t squared minus x squared, whatever. This just reminds me of something that is has this perfect symmetry between space and time. Let's talk about this thing that is spacetime. And then everybody kind of had to stand on their head, in a sense, to say, well, time is really the same as space. I mean, it is a matter of common experience that time operates in a very different way from space. And in our models, time and space are fundamentally different. But nevertheless, there is a relation between them that comes through these causal graphs and so on, that is what gives you the kind of mathematical physics results of relativity theory and such like.
Yeah, so you say you don't know yet how space condenses to be three-dimensional, if it condenses to be three-dimensional, or if we're just perceiving it to be three-dimensional space because that's the kind of observers we are. Do you have any clues? Do you have anything that's given you a hint as to which way that might go?
Yes, so, so here's my guess. The we're talking about kind of the dimension of space. It is a, it is first of all, I think the way things are going to come out in the end is, in effect, space is going to start at the beginning of the universe infinite-dimensional. Everything's sort of connected easily to everything, and then gradually it's kind of cooling down. Now, observers like us don't have a happy time in the very, very early moments of the universe. It's tough to have an observer like us at that moment. So exactly, you know, we're looking at it kind of from the outside at that point as a theoretical matter. Now, the question of why we perceive space to be more or less three-dimensional. I mean, in our models, dimension is a dynamical variable. So we expect there to be, and this is important in sort of potential experimental implication, we expect there to be fluctuations in dimension, places in the universe, early universe, perhaps current universe, where space isn't exactly three-dimensional. It might be 3.01 dimensional or something like this. But the question is, why is it even roughly three-dimensional? And we don't know. But it is my very strong suspicion that that is a consequence of our nature as observers. In other words, it's perfectly possible that I could describe the world in terms of the way that I was, where it's just like, instead of saying I gulp in this three-dimensional lump of space as the thing that I'm perceiving, I could be kind of like this little creature that's kind of exploring it on a one-dimensional line and eventually discovering, oh yes, things fit together in this way and that way, without having said, you know, it is something where I can immediately perceive these three dimensions. And it's tricky because by the time you kind of explored your region of space, things have changed. It's not so obvious that you can just sort of gulp it in. The fact that we can gulp it in is a consequence, I think, of pretty detailed features of us. So, you know, we are the size we are, meter tall, give or take. We look around us, you know, we can see 10 meters, 100 meters away, whatever. In our normal kind of, in our normal life, and the speed of light is such that from the distances we can readily see away from us, it takes only a microsecond for light to come from there to us. Our brains take milliseconds to process that information. The result is that for us, we perceive this sort of gulp of a region of space to be just, it's that state of space. And then, by the time we've thought about it, it's we've kind of ingested that instantaneous state of space, and then later on, it may be different. If our brains ran a million times faster, as they would if they were digital electronics, for example, this idea that we sort of gulp in the structure of space as we see around us in one gulp before we've thought about the next thing, it wouldn't work that way. You know, we would instead be seeing the individual photons arriving and could be doing that and sort of building things up that way. My guess is that in the end, with respect to why space seems three-dimensional to us, we'll kind of kick ourselves because it'll be kind of obvious that it's going to be some feature of our nature as observers that is completely obvious to us. We take it completely for granted, like this idea that we can kind of gulp in the state of space. That's not something you usually think, oh, is that really how it works? If you were an astronomer, for example, used to dealing with huge distances, then yes, you have to pay attention to the fact that there are time delays and all this kind of thing. But in our sort of normal life, it's just something we take for granted.
There are other things we take for granted. We take for granted things like that we can distinguish objects, that there are independent objects in the world, that it isn't the case that everything we deal with is intimately connected to every other thing we deal with, that we have the idea that we can sort of independently do an experiment on this one thing without entraining everything else in the world, so to speak. So there are a bunch of these implicit assumptions, and I've been trying to sort of tease out what those are. My guess is that one or a combination of those will be like, oh, okay, that's why we think space is three-dimensional, even though, for example, our computers, for instance, I don't know, they think space is three-dimensional. They're connected to the internet, which has all of these, you know, different, you know, they might be looking at the ping times to different machines, and you know, can you really conclude that the surface of the Earth is two-dimensional from ping times? I don't think quite so, because of the way that, you know, different fiber cables work and all this kind of thing. So the perception of space is a bit different, probably, if we think about it in terms of computers with networks and so on.
But I think, I think that one of the things that is sort of confusing, in a sense, is to us, sort of reality feels so solid that it, it, the idea that you could sort of deconstruct that reality seems impossible. I mean, for example, in working on the physics project, some of the things, you know, I've had to kind of get over some hang-ups that I've had, like, for example, one hang-up is how much of a profligate waster of computational resources can the universe really be? You know, in other words, I, for a while, and this is something when I was working on *A New Kind of Science* in the 1990s, I was not, I certainly knew of the idea of multi-way systems and thinking about quantum mechanics in terms of branching histories and so on, but I was very much, you know, it can't be serious that the universe is just treeing out all of these histories and sort of wasting that much computation. It didn't feel right to me that that could be the case. But I've now sort of got over that hang-up, and I, and I, you know, in the end, having gotten over that, I realize that I shouldn't have had it in the first place for all kinds of reasons. But I think, you know, this notion that one can have a solid reality built from things that seem so fragile is one that is intuitively difficult. I mean, I think we all have one piece of intuition that's obvious from that, which is, you know, our computer screens are made from discrete pixels and so on, yet we're pretty sure that we're experiencing the actual video conference or something with an actual sort of person at the other end, type thing, even though, even though it's, in the end, it's all represented as these sort of fragile discrete pixels and so on. And I think that's, but in terms of this question about the three-dimensionality of the world, it is my strong belief that that is an observer-related thing, and that if we were different from the way we are, we would not perceive the dominant feature of the universe to be its three-dimensionality.
Now, having said that, it's sort of a complicated thing to say, okay, when we make the right observations in the universe to see fluctuations from three dimensions, that's a thing we kind of expect in our models. It's a, it's one of those very unexpected predictions of a model like this, where space has this kind of underlying structure. It's something where, that's sometime to say, you know, the idea that space has an underlying structure, it's kind of like saying, well, matter has an underlying structure, it's made of molecules. Can we see those molecules? People were pretty lucky at the end of the 19th century that the experimental apparatus that existed was such that you could actually see molecules given the size that molecules are. And it might not have been the case. We may or may not be lucky in terms of finding that really clever experiment that lets us kind of see the microscope into the structure of space, to see the discreteness of space in that kind of more or less direct way. But the thing that's really nice about things like dimension fluctuations is they're an indirect sign that there's something other than just this Euclidean-style kind of manifold structure of space. They're an indirect way of seeing that. And that phenomenon doesn't depend on how big the elementary length is. It doesn't require that we're lucky that the elementary length is not, let's say, 10 to the minus 100 meters, but 10 to the minus 92 meters, which happens to be what we can access. I don't know the numbers, but, you know, which happens to be the thing that, with exactly the right kind of black hole observation, we might be able to get to, whereas we wouldn't if it was another 10 orders of magnitude smaller, so to speak.
Yeah, that's fascinating. I know what you mean about getting your head around the idea that the universe can be profuse with computation and compute the entire multi-graph. It just seems crazy when you first come across that idea, as I have reading your stuff. And then when you get your head around it a little bit more, it's like, well, why not? Why shouldn't a theory of physics work on that basis? It is fascinating.