The following is a conversation with Stephen Wolfram, his second time on the podcast. He's a computer scientist, mathematician, theoretical physicist, and the founder and CEO of Wolfram Research, a company behind Mathematica, Wolfram Alpha, Wolfram Language, and the new Wolfram Physics Project. He's the author of several books, including A New Kind of Science, and the new book, A Project to Find the Fundamental Theory of Physics. This second round of our conversation is primarily focused on this latter endeavor of searching for the physics of our universe in simple rules that do their work on hypergraphs, and eventually generate the infrastructure from which space, time, and all of modern physics can emerge. Quick summary of the sponsors. SimplySafe, SunBasket, and Masterclass. Please check out these sponsors in the description to get a discount and to support this podcast. As a side note, let me say that to me, the idea that seemingly infinite complexity can arise from very simple rules and initial conditions is one of the most beautiful and important mathematical and philosophical mysteries in science. I find that both cellular automata and the hypergraph data structure that Stephen and team are currently working on to be the kind of simple, clear mathematical playground within which fundamental ideas about intelligence, consciousness, and the fundamental laws of physics can be further developed in totally new ways. In fact, I think I'll try to make a video or two about the most beautiful aspects of these models in the coming weeks. Especially, I think, trying to describe how fellow curious minds like myself can jump in and explore them, either just for fun, or potentially for publication of new, innovative research in math, computer science, and physics. But honestly, I think the emerging complexity in these hypergraphs can capture the imagination of everyone, even if you're someone who never really connected with mathematics. That's my hope, at least, to have these conversations that inspire everyone to look up to the skies and into our own minds in awe of our amazing universe. Let me also mention that this is the first time I ever recorded a podcast outdoors, as a kind of experiment to see if this is an option in times of COVID. I'm sorry if the audio is not great. I did my best, and promise to keep improving and learning, as always. If you enjoy this thing, subscribe on YouTube, review it with Fat Stars on Apple Podcasts, follow on Spotify, support on Patreon, or connect with me on Twitter @LexFriedman. As usual, I'll do a few minutes of ads now, and no ads in the middle. I tried to make these interesting, but I do give you timestamps, so you're welcome to skip. 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In fact, when I was younger, the idea of being a hit man or targeting evil in a skilled way, which is how I thought about it, really appealed to me. The skill of it, the planning, the craftsmanship. In another life, perhaps, if I didn't love engineering and science so much, I could see myself being something like a Navy SEAL. And in general, I love the idea of serving my country, of serving society, by contributing my skill in some small way. Anyway, go to simplysafe.com/lex to get a free HD camera and to support this podcast. They're a new sponsor, and this is a trial run, so you know what to do. This show is also sponsored by Sun Basket, a meal delivery service. Visit sunbasket.com/lex and use code LEX to get $30 off your order and to support this podcast. This is the last read of the trial they're doing, so this is the time to get them if you're considering it. And if you do, it'll help ensure that they decide to support this podcast long-term. 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So it really does mean a lot, the love and support I get as I try to get better at this thing, at this talking thing. Anyway, go to masterclass.com/lex to get a discount and to support this podcast. And now, finally, here's my conversation with Stephen Wolfram. You said that there are moments in history of physics, and maybe mathematical physics or even mathematics, where breakthroughs happen and then a flurry of progress follows. So if you look back through the history of physics, what moments stand out to you as important such breakthroughs where a flurry of progress follows? - So the big famous one was 1920s, the invention of quantum mechanics, where in about five or 10 years, lots of stuff got figured out. That's now quantum mechanics. - Can you mention the people involved? - Yeah, it was kind of the Schrodinger, Heisenberg, Einstein had been a key figure, originally Planck, then Dirac was a little bit later. That was something that happened at that time, that's sort of before my time, right? In my time was in the 1970s, there was this sort of realization that quantum field theory was actually going to be useful in physics and QCD, quantum carbon dynamics theory of quarks and gluons and so on was really getting started. And there was again, sort of big flurry of things happened then. I happened to be a teenager at that time and happened to be really involved in physics. And so I got to be part of that, which was really cool. - Who were the key figures aside from your young selves at that time? - You know, who won the Nobel prize for QCD, okay? People, David Gross, Frank Wilczek, David Politzer, the people who are the sort of the slightly older generation, Dick Feynman, Murray Gell-Mann, people like that, who were Steve Weinberg, Gerhard Hoft, he's younger, he's in the younger group actually, but these are all characters who were involved. I mean, it's funny because those are all people who are kind of in my time and I know them and they don't seem like sort of these historical, iconic figures. They seem more like everyday characters, so to speak. So it's always, when you look at history from long afterwards, it always seems like everything happened instantly. And that's usually not the case. There was usually a long buildup, but usually there's some methodological thing happens and then there's a whole bunch of low-hanging fruit to be picked and that usually lasts five or 10 years. We see it today with machine learning and deep learning neural nets and so on, methodological advance, things actually started working in 2011, 2012 and so on. And there's been this sort of rapid picking of low-hanging fruit, which is probably some significant fraction of the way done, so to speak. - Do you think there's a key moment? Like if I had to really introspect, like what was the key moment for the deep learning quote unquote revolution? I mean-- - It's probably the AlexNet business. - AlexNet with ImageNet. So is there something like that with physics where, so deep learning neural networks have been around for a long time. - Absolutely, since the 1940s, yeah. - There's a bunch of little pieces that came together and then all of a sudden everybody's eyes lit up. Like, wow, there's something here. Like even just looking at your own work, just your thinking about the universe, that there's simple rules can create complexity. At which point was there a thing where your eyes light up? It's like, wait a minute, there's something here. Is it the very first idea or is it some moment along the line of implementations and experiments and so on? - Yeah, there's a couple of different stages to this. One is the think about the world computationally. Can we use programs instead of equations to make models of the world? That's something that I got interested in in the beginning of the 1980s. I did a bunch of computer experiments. When I first did them, I didn't really, I could see some significance to them, but it took me a few years to really say, wow, there's a big important phenomenon here that lets sort of complex things arise from very simple programs. That kind of happened back in 1984 or so. Then a bunch of other years go by, then I start actually doing a lot of much more systematic computer experiments and things and find out that this phenomenon that I could only have said occurs in one particular case is actually something incredibly general. And then that led me to this thing called principle of computational equivalence. And that was a long story. And then as part of that process, I was like, okay, you can make simple programs, can make models of complicated things. What about the whole universe? That's our sort of ultimate example of a complicated thing. And so I got to thinking, could we use these ideas to study fundamental physics? I happen to know a lot about traditional fundamental physics. My first, I had a bunch of ideas about how to do this in the early 1990s. I made a bunch of technical progress. I figured out a bunch of things I thought were pretty interesting. I wrote about them back in 2002. - With the new kind of science in the cellular automata world. - Right. - There's echoes in the cellular automata world with your new Wolfram physics project. We'll get to all that. Allow me to sort of romanticize a little more on the philosophy of science. So Thomas Kuhn, a philosopher of science, describes that the progress in science is made with these paradigm shifts. And so to linger on the sort of original line of discussion, do you agree with this view that there is revolutions in science that just kind of flip the table? - What happens is it's a different way of thinking about things. It's a different methodology for studying things. And that opens stuff up. - There's this idea of, he's a famous biographer, but I think it's called the Innovators. The biographer of Steve Jobs, of Albert Einstein. He also wrote a book, I think it's called The Innovators, where he discusses how a lot of the innovations in the history of computing has been done by groups. There's a complicated group dynamic going on. But there's also a romanticized notion that the individual is at the core of the revolution. Like where does your sense fall? Is ultimately like one person responsible for these revolutions that creates the spark? Or one or two, whatever. Or is it just the big mush and mess and chaos of people interacting, of personalities interacting? - I think it ends up being like many things, there's leadership and there ends up being, it's a lot easier for one person to have a crisp new idea than it is for a big committee to have a crisp new idea. And I think, but I think it can happen that you have a great idea, but the world isn't ready for it. And you can, I mean, this has happened to me plenty, right? You have an idea, it's actually a pretty good idea, but things aren't ready. Either you're not really ready for it, or the ambient world isn't ready for it. And it's hard to get the thing to get traction. - It's kind of interesting. I mean, when I look at a new kind of science, you're now living inside the history, so you can't tell the story of these decades. But it seems like the new kind of science has not had the revolutionary impact I would think it might. Like, it feels like at some point, of course it might be, but it feels at some point people will return to that book and say there was something special here. This was incredible. - Well, look, what happened-- - Or do you think that's already happened? - Oh yeah, it's happened, except that people aren't, you know, the sort of the heroism of it may not be there. But what's happened is for 300 years, people basically said, if you want to make a model of things in the world, mathematical equations are the best place to go. Last 15 years, doesn't happen. You know, new models that get made of things most often are made with programs, not with equations. Now, you know, was that sort of going to happen anyway? Was that a consequence of, you know, my particular work and my particular book? It's hard to know for sure. I mean, I am always amazed at the amounts of feedback that I get from people where they say, oh, by the way, you know, I started doing this whole line of research because I read your book, blah, blah, blah, blah, blah. It's like, well, can you tell that from the academic literature? You know, was there a chain of, you know, academic references? Probably not. - One of the interesting side effects of publishing in the way you did this tome is it serves as an education tool and an inspiration to hundreds of thousands, millions of people, but because it's not a single, it's not a chain of papers with 50 titles, it doesn't create a splash of citations. - It's had plenty of citations, but it's, you know, I think that the, people think of it as probably more, you know, conceptual inspiration than kind of a, you know, this is a line from here to here to here in our particular field. I think that the, you know, the thing which I am disappointed by and which will eventually happen is this kind of study of the, this sort of pure computationalism, this kind of study of the abstract behavior of the computational universe. That should be a big thing that lots of people do. - You mean in mathematics purely, almost like- - It's like pure mathematics, but it isn't mathematics. - But it isn't. - It's a new kind of mathematics. Is it a title of a book? - Yeah, right. - Good title. - That's why the book is called that, right? That's not coincidental. - Yeah. It's interesting that I haven't seen really rigorous investigation by thousands of people of this idea. I mean, you look at your competition around rule 30. I mean, that's fascinating. If you can say something, is there some aspect of this thing that could be predicted? That's the fundamental question of science. That's the core. - Well, that has been a question of science. I think that is some people's view of what science is about, and it's not clear that's the right view. In fact, as we live through this pandemic full of predictions and so on, it's an interesting moment to be pondering what science's actual role in those kinds of things is. - Oh, you think it's possible that in science, clean, beautiful, simple prediction may not even be possible in real systems. That's the open question. - Right, I don't think it's open. I think that question is answered, and the answer is no. - Well, no, no. The answer could be just humans are not smart enough yet. We don't have the tools yet. - No, no, that's the whole point. I mean, that's sort of the big discovery of this principle of computational equivalence of mind. And this is something which is kind of a follow-on to Godel's theorem, to Turing's work on the halting problem, all these kinds of things, that there is this fundamental limitation built into science, this idea of computational irreducibility, that says that even though you may know the rules by which something operates, that does not mean that you can readily sort of be smarter than it and jump ahead and figure out what it's going to do. - Yes, but do you think there's a hope for pockets of computational reducibility? - Yes, yes. - So, and then a set of tools and mathematics that help you discover such pockets. - That's where we live, is in the pockets of reducibility. That's why, and this is one of the things that sort of come out of this physics project and actually something that, again, I should have realized many years ago, but didn't, it could very well be that everything about the world is computationally irreducible and completely unpredictable. But in our experience of the world, there is at least some amount of prediction we can make. And that's because we have sort of chosen a slice of, probably talk about this in much more detail, but I mean, we've kind of chosen a slice of how to think about the universe in which we can kind of sample a certain amount of computational reducibility. And that's sort of where we exist. And it may not be the whole story of how the universe is, but it is the part of the universe that we care about and we sort of operate in. And that's, in science, that's been sort of a very special case of that. That is science has chosen to talk a lot about places where there is this computational reducibility that it can find, the motion of the planets can be more or less predicted. Something about the weather is much harder to predict. Something about other kinds of things are much harder to predict. And it's, these are, but science has tended to concentrate itself on places where its methods have allowed successful prediction. - So you think rule 30, if we could linger on it, because it's just such a beautiful, simple formulation of the essential concept underlying all the things we're talking about. Do you think there's pockets of reducibility inside rule 30? - Yes. That is the question of how big are they? What will they allow you to say? And so on. And that's, and figuring out where those pockets are, I mean, in a sense, that's the, that's sort of a, you know, that is an essential thing that one would like to do in science. But it's also, the important thing to realize that has not been, you know, is that science, if you just pick an arbitrary thing, you say, what's the answer to this question? That question may not be one that has a computationally reducible answer. That question, if you choose, you know, if you walk along the series of questions and you've got one that's reducible and you get to another one that's nearby and it's reducible too, if you stick to that kind of, stick to the land, so to speak, then you can go down this chain of sort of reducible, answerable things. But if you just say, I'm just pick a question at random, I'm going to have my computer pick a question at random. - Yeah, most likely it's going to be reducible. - Most likely it will be irreducible. And what we're throwing in the world, so to speak, we, you know, when we engineer things, we tend to engineer things to sort of keep in the zone of reducibility. When we're throwing things by the natural world, for example, not at all certain that we will be kept in this kind of zone of reducibility. - Can we talk about this pandemic then? - Sure. (laughing) - For a second. Is this, so how do we, there's obviously huge amount of economic pain that people are feeling. There's a huge incentive and medical pain, health, just all kinds of psychological. There's a huge incentive to figure this out, to walk along the trajectory of reducibility. There's a lot of disparate data. You know, people understand generally how virus is spread, but it's very complicated because there's a lot of uncertainty. There could be a lot of variability, like so many, obviously, a nearly infinite number of variables that represent human interaction. And so you have to figure out, from the perspective of reducibility, figure out which variables are really important in this kind of, from an epidemiological perspective. So why aren't we, you kind of said that we're clearly failing. - Well, I think it's a complicated thing. So, I mean, you know, when this pandemic started up, you know, I happened to be in the middle of being about to release this whole physics project thing. But I thought, you know- - The timing is just cosmically absurd. - A little bit bizarre. But, you know, but I thought, you know, I should do the public service thing of, you know, trying to understand what I could about the pandemic. And, you know, we've been curating data about it and all that kind of thing. But, you know, so I started looking at the data and started looking at modeling, and I decided it's just really hard. You need to know a lot of stuff that we don't know about human interactions. It's actually clear now that there's a lot of stuff we didn't know about viruses and about the way immunity works and so on. And it's, you know, I think what will come out in the end is there's a certain amount of what happens that we just kind of have to trace each step and see what happens. There's a certain amount of stuff where there's gonna be a big narrative about this happened because, you know, of T cell immunity. This happened because there's this whole giant sort of field of asymptomatic viral stuff out there. You know, there will be a narrative, and that narrative, whenever there's a narrative, that's kind of a sign of reducibility. But when you just say, "Let's from first principles figure out what's going on," then you can potentially be stuck in this kind of mess of irreducibility where you just have to simulate each step, and you can't do that unless you know details about, you know, human interaction networks and so on and so on and so on. The thing that has been very sort of frustrating to see is the mismatch between people's expectations about what science can deliver and what science can actually deliver, so to speak, because people have this idea that, you know, it's science, so there must be a definite answer, and we must be able to know that answer. And, you know, this is, it is both, you know, when you've, after you've played around with sort of little programs in the computational universe, you don't have that intuition anymore. You know, it's, I always, I'm always fond of saying, you know, the computational animals are always smarter than you are. That is, you know, you look at one of these things, and it's like, it can't possibly do such and such a thing. Then you run it, and it's like, wait a minute, it's doing that thing. How does that work? Okay, now I can go back and understand it. - But that's the brave thing about science, is that in the chaos of the irreducible universe, we nevertheless persist to find those pockets. That's kind of the whole point. That's, like you say that the limits of science, but that, you know, yes, it's highly limited, but there's a hope there. And like, there's so many questions I wanna ask here. So one, you said narrative, which is really interesting. So obviously from a, at every level of society, you look at Twitter, everybody's constructing narratives about the pandemic, about not just the pandemic, but all the cultural tension that we're going through. So there's narratives, but they're not necessarily connected to the underlying reality of these systems. So our human narratives, I don't even know if they're, I don't like those pockets of reducibility, 'cause we're, it's like constructing things that are not actually representative of reality, and thereby not giving us like good solutions to how to predict the system. - Look, it gets complicated because, you know, people want to say, explain the pandemic to me. Explain what's gonna happen. - In the future, like predict. - Yes, but also, can you explain it? Is there a story to tell? - What already happened in the past. - Yeah, or what's going to happen, but I mean, it's similar to sort of explaining things in AI or in any computational system. It's like, you know, explain what happened. Well, it could just be this happened because of this detail and this detail and this detail and a million details, and there isn't a big story to tell. There's no kind of big arc of the story that's going to happen. There's no kind of big arc of the story that says, oh, it's because, you know, there's a viral field that has these properties and people start showing symptoms. You know, when the seasons change, people will show symptoms and people don't even understand, you know, seasonal variation of flu, for example. It's something where, you know, there could be a big story or it could be just a zillion little details that mount up. - See, but, okay, let's pretend that this pandemic, like the coronavirus, resembles something like the 1D Rule 30 cellular automata, okay? So, I mean, that's how epidemiologists model virus spread. - Indeed, yes. They sometimes use cellular automata, yes. - And, okay, so you could say it's simplistic, but, okay, let's say it's representative of actually what happens. You know, the dynamic of, you have a graph. It probably is closer to the hypergraph model. - It is, yes. It's actually, that's another funny thing. As we were getting ready to release this physics project, we realized that a bunch of things we'd worked out about foliations of causal graphs and things were directly relevant to thinking about contact tracing and interactions with cell phones and so on, which is really weird. - But, like, it just feels like we should be able to get some beautiful core insight about the spread of this particular virus on the hypergraph of human civilization, right? - I tried. I didn't manage to figure it out. - But you're one person. - Yeah, but, I mean, I think, actually, it's a funny thing 'cause it turns out the main model, you know, this SIR model, I only realized recently, was invented by the grandfather of a good friend of mine from high school, so that was just a, you know, it's a weird thing, right? The question is, you know, okay, so you know, you know, on this graph of how humans are connected, you know something about what happens if this happens and that happens. That graph is made in complicated ways that depends on all sorts of issues that, where we don't have the data about how human society works well enough to be able to make that graph. There's actually, one of my kids did a study of sort of what happens on different kinds of graphs and how robust are the results. Okay, his basic answer is there are few general results that you can get that are quite robust, like, you know, a small number of big gatherings is worse than a large number of small gatherings, okay? That's quite robust, but when you ask more detailed questions, it seemed like it just depends. It depends on details, in other words, it's kind of telling you, in that case, you know, the irreducibility matters, so to speak. It's not, there's not gonna be this kind of one sort of master theorem that says, and therefore this is how things are gonna work. - Yeah, but there's a certain kind of, from a graph perspective, the certain kind of dynamic to human interaction, so like large groups and small groups, I think it matters who the groups are. For example, you could imagine, depends how you define large, but you can imagine groups of 30 people, as long as they are cliques or whatever, as long as the outgoing degree of that graph is small or something like that, like you can imagine some beautiful underlying rule of human dynamic interaction where I can still be happy, where I can have a conversation with you and a bunch of other people that mean a lot to me in my life and then stay away from the bigger, I don't know, not going to a Miley Cyrus concert or something like that, and figuring out mathematically some nice rule of behavior. - See, this is an interesting thing, so I mean, this is the question of what you're describing is kind of the problem of the many situations where you would like to get away from computational irreducibility. A classic one in physics is thermodynamics. The second law of thermodynamics, the law that says entropy tends to increase, things that start orderly tend to get more disordered, or which is also the thing that says, given that you have a bunch of heat, it's hard, heat is the microscopic motion of molecules, it's hard to turn that heat into systematic mechanical work. It's hard to just take something being hot and turn that into, oh, all the atoms are gonna line up in the bar of metal and the piece of metal is gonna shoot in some direction. That's essentially the same problem as how do you go from this computationally irreducible mess of things happening and get something you want out of it? It's kind of mining, you're kind of, now, actually, I've understood in recent years that the story of thermodynamics is actually precisely a story of computational irreducibility, but it is already an analogy. - It's interesting. You can kind of see that, is can you take the, what you're asking to do there is you're asking to go from the kind of, the mess of all these complicated human interactions and all this kind of computational processes going on, and you say, I want to achieve this particular thing out of it. I want to kind of extract from the heat of what's happening, I want to kind of extract this useful piece of sort of mechanical work that I find helpful. I mean-- - Do you have a hope for the pandemic? So we'll talk about physics, but for the pandemic, can that be extracted? Do you think? - Well, I think-- - What's your intuition? - The good news is the curves basically, for reasons we don't understand, the curves, the clearly measurable mortality curves and so on for the Northern Hemisphere have gone down. - Yeah, but the bad news is that it could be a lot worse for future viruses, and what this pandemic revealed is we're highly unprepared for the discovery of the pockets of reducibility within a pandemic that's much more dangerous. - Well, my guess is the specific risk of viral pandemics, that the pure virology and immunology of the thing, this will cause that to advance to the point where this particular risk is probably considerably mitigated. But it's, does, is the structure of modern society robust to all kinds of risks? Well, the answer is clearly no. And it's surprising to me the extent to which people, as I say, it's kind of scary actually how much people believe in science. That is people say, oh, because the science says this, that and the other, we'll do this and this and this, even though from a sort of common sense point of view, it's a little bit crazy. And people are not prepared, and it doesn't really work in society as it is for people to say, well, actually, we don't really know how the science works. People say, well, tell us what to do. - Yeah, because then, yeah, what's the alternative? For the masses, it's difficult to sit, it's difficult to meditate on computational reducibility. It's difficult to sit, it's difficult to enjoy a good dinner meal while knowing that you know nothing about the world. - Well, I think this is a place where, this is what politicians and political leaders do for a living, so to speak, is you gotta make some decision about what to do. And it's-- - Tell some narrative. - Right. - While amidst the mystery and knowing not much about the past or the future, still telling a narrative that somehow gives people hope that we know what the heck we're doing. - Yeah, and get society through the issue. Even though the idea that we're just gonna be able to get the definitive answer from science, and it's gonna tell us exactly what to do, unfortunately, it's interesting, because let me point out that if that was possible, if science could always tell us what to do, then in a sense, that would be a big downer for our lives. If science could always tell us what the answer was gonna be, it's like, well, it's kind of fun to live one's life and just sort of see what happens. If one could always just say, let me check my science, oh, I know the result of everything is gonna be 42, I don't need to live my life and do what I do, it's just we already know the answer. It's actually good news in a sense that there is this phenomenon of computational irreducibility that doesn't allow you to just sort of jump through time and say this is the answer, so to speak. And so that's a good thing. The bad thing is it doesn't allow you to jump through time and know what the answer is. - It's scary. Do you think we're gonna be okay as a human civilization? You said we don't know. - Absolutely. Absolutely. - Do you think we'll prosper or destroy ourselves? - In general? - In general. - I'm an optimist. No, I think that it'll be interesting to see, for example, with this pandemic. To me, when you look at organizations, for example, having some kind of perturbation, some kick to the system, usually the end result of that is actually quite good unless it kills the system. It's actually quite good usually. And I think in this case, people, I mean, my impression, it's a little weird for me because I've been a remote tech CEO for 30 years. This is bizarrely, and the fact that this coming to see you here is-- - One of the rare moments that human interaction-- - Is the first time in six months that I've been in a building other than my house. - Yeah, that's amazing. - So I'm a kind of ridiculous outlier in these kinds of things. - But overall, your sense is when you shake up the system and throw in chaos, that you challenge the system, we humans emerge better. - Seems to be that way. Who's to know? But I think that people, my sort of vague impression is that people are sort of, oh, what's actually important? What is worth caring about and so on? And that seems to be something that perhaps is more emergent in this kind of situation. - It's so fascinating that on the individual level, we have our own complex cognition. We have consciousness, we have intelligence. We're trying to figure out little puzzles. And then that somehow creates this graph of collective intelligence, where we figure out, and then you throw in these viruses of which there's millions different, there's entire taxonomy, and the viruses are thrown into the system of collective human intelligence. And the little humans figure out what to do about it. We get like, we tweet stuff about information. There's doctors as conspiracy theorists. And then we play with different information. I mean, the whole of it is fascinating. I am, like you, also very optimistic, but there's a, just you said the computational reducibility. There's always a fear of the darkness of the uncertainty before us. - Yeah, I know. - And it's scary. - I mean, the thing is, if you knew everything, it will be boring. (Dave laughs) And it would be, and then, and worse than boring, so to speak. It would be, it would reveal the pointlessness, so to speak. And in a sense, the fact that there is this computational reducibility, it's like as we live our lives, so to speak, something is being achieved. We're computing what our lives, what happens in our lives. - That's funny. So the computational reducibility is kind of like, it gives the meaning to life. It is the meaning of life. Computational reducibility is the meaning of life. There you go. - It gives it meaning, yes. I mean, it's what causes it to not be something where you can just say, you know, you went through all those steps to live your life, but we already knew what the answer was. - Right. - Hold on one second. I'm going to use my handy Wolfram Alpha sunburn computation thing, so long as I can get network here. There we go. Oh, actually, you know what? It says sunburn unlikely. This is a QA moment. (laughing) - This is a good moment. - Okay, well, let me just check what it thinks. I can see why it thinks that. It doesn't seem like my intuition. This is one of these cases where we can, the question is, do we trust the science or do we use common sense? - The UV thing is cool. - Yeah, yeah, well, we'll see. This is a QA moment, as I say. (laughing) Do we trust the product? Yes, we trust the product. - And then there'll be a data point either way. - If I'm desperately sunburned, I will send in an angry feedback. - Because we mentioned the concept so much, and a lot of people know it, but can you say what computation reducibility is? - Yeah, right. So the question is, if you think about things that happen as being computations, you think about some process in physics, something that you compute in mathematics, whatever else, it's a computation in the sense it has definite rules. You follow those rules, you follow them many steps, and you get some result. So then the issue is, if you look at all these different kinds of computations that can happen, whether they're computations that are happening in the natural world, whether they're happening in our brains, whether they're happening in our mathematics, whatever else, the big question is, how do these computations compare? Is, are there dumb computations and smart computations? Or are they somehow all equivalent? And the thing that I kind of was sort of surprised to realize from a bunch of experiments that I did in the early '90s, and now we have tons more evidence for it, this thing I call the principle of computational equivalence, which basically says, when one of these computations, one of these processes that follows rules, doesn't seem like it's doing something obviously simple, then it has reached the sort of equivalent level of computational sophistication of everything. So what does that mean? That means that, you know, you might say, gosh, I'm studying this little tiny, you know, tiny program on my computer, I'm studying this little thing in nature, but I have my brain, and my brain is surely much smarter than that thing. I'm gonna be able to systematically outrun the computation that it does, because I have a more sophisticated computation that I can do. But what the principle of computational equivalence says is that doesn't work. Our brains are doing computations that are exactly equivalent to the kinds of computations that are being done in all these other sorts of systems. And so what consequences does that have? Well, it means that we can't systematically outrun these systems. These systems are computationally irreducible, in the sense that there's no sort of shortcut that we can make that jumps to the answer. - In a general case. - Right, right. But the, so what has happened, you know, what science has become used to doing is using the little sort of pockets of computational reducibility, which by the way are an inevitable consequence of computational irreducibility, that there have to be these pockets scattered around of computational reducibility to be able to find those particular cases where you can jump ahead. I mean, one thing, sort of a little bit of a parable type thing that I think is fun to tell, you know, if you look at ancient Babylon, they were trying to predict three kinds of things. They tried to predict, you know, where the planets would be, what the weather would be like, and who would win or lose a certain battle. And they had no idea which of these things would be more predictable than the other. - That's funny. - And, you know, it turns out, you know, where the planets are is a piece of computational reducibility that, you know, 300 years ago or so, we pretty much cracked. I mean, it's been technically difficult to get all the details right, but it's basically, we got that. You know, who's gonna win or lose the battle? No, we didn't crack that one. That one, that one. - I can't. - Right. - Game theorists are trying. - Yes. - And then the weather. - It's kind of halfway on that one. - Halfway? - Yeah, I think we're doing okay on that one. You know, long-term climate, different story. But the weather, you know, we're much closer on that. - But do you think eventually we'll figure out the weather? So do you think eventually most think we'll figure out the local pockets in everything, essentially, the local pockets of reducibility? - No, I think that it's an interesting question, but I think that the, you know, there is an infinite collection of these local pockets. We'll never run out of local pockets. And by the way, those local pockets are where we build engineering, for example. That's how we, you know, when we, if we want to have a predictable life, so to speak, then, you know, we have to build in these sort of pockets of reducibility. Otherwise, you know, if we were, if we were sort of existing in this kind of irreducible world, we'd never be able to, you know, have definite things to know what's gonna happen. You know, I have to say, I think one of the features, you know, when we look at sort of today from the future, so to speak, I suspect one of the things where people will say, "I can't believe they didn't see that," is stuff to do with the following kind of thing. So, you know, if we describe, oh, I don't know, something like heat, for instance, we say, "Oh, you know, the air in here, "it's, you know, it's this temperature, this pressure." That's as much as we can say. Otherwise, just a bunch of random molecules bouncing around. People will say, "I just can't believe "they didn't realize that there was all this detail "and how all these molecules were bouncing around "and they could make use of that." And actually, I realized, there's a thing I realized last week, actually, was a thing that people say, you know, one of the scenarios for the very long-term history of our universe is a so-called heat death of the universe, where basically everything just becomes thermodynamically boring. Everything's just this big kind of gas in thermal equilibrium. People say, "That's a really bad outcome." But actually, it's not a really bad outcome. It's an outcome where there's all this computation going on and all those individual gas molecules are all bouncing around in very complicated ways, doing this very elaborate computation. It just happens to be a computation that right now, we haven't found ways to understand. We haven't found ways, you know, our brains haven't, you know, and our mathematics and our science and so on, haven't found ways to tell an interesting story about that. It just looks boring to us. - You're saying there's a hopeful view of the heat death, quote-unquote, of the universe, where there's actual beautiful complexity going on, similar to the kind of complexity we think of that creates rich experience in human life and life on Earth? - Yes. - So those little molecules interact in complex ways. There could be intelligence in that. There could be-- - Absolutely. I mean, this is what you learn from this principle. - Wow, that's a hopeful message. - Right, I mean, this is what you kind of learn from this principle of computational equivalence. You learn it's both a message of sort of hope and a message of kind of, you know, you're not as special as you think you are, so to speak. I mean, because, you know, we imagine that with sort of all the things we do with human intelligence and all that kind of thing, and all of the stuff we've constructed in science, it's like, we're very special, but actually it turns out, well, no, we're not. We're just doing computations like things in nature do computations, like those gas molecules do computations, like the weather does computations. The only thing about the computations that we do that's really special is that we understand what they are, so to speak. In other words, we have a, you know, to us, they're special because kind of, they're connected to our purposes, our ways of thinking about things and so on. And that's some, but so-- - That's very human-centric. That's, we're just attached to this kind of thing. So let's talk a little bit of physics. Maybe let's ask the biggest question. What is a theory of everything in general? What does that mean? - Yeah, so I mean, the question is, can we kind of reduce what has been physics as a something where we have to sort of pick away and say, do we roughly know how the world works to something where we have a complete formal theory where we say, if we were to run this program for long enough, we would reproduce everything, you know, down to the fact that we're having this conversation at this moment, et cetera, et cetera, et cetera. - Any physical phenomena, any phenomena in this world? - Phenomena in the universe. But the, you know, because of computational irreducibility, it's not, you know, that's not something where you say, okay, you've got the fundamental theory of everything. Then, you know, tell me whether, you know, lions are going to eat tigers or something. You know, that's a, no, you have to run this thing for, you know, 10 to the 500 steps or something to know something like that, okay? So at some moment, potentially, you say, this is a rule and run this rule enough times and you will get the whole universe, right? That's what it means to kind of have a fundamental theory of physics, as far as I'm concerned, is you've got this rule, it's potentially quite simple. We don't know for sure it's simple, but we have various reasons to believe it might be simple. And then you say, okay, I'm showing you this rule. You just run it only 10 to the 500 times and you'll get everything. In other words, you've kind of reduced the problem of physics to a problem of mathematics, so to speak. It's like, it's as if, you know, you'd like you'd generate the digits of pi. There's a definite procedure, you just generate them. And it'd be the same thing if you have a fundamental theory of physics of the kind that I'm imagining, you know, you get this rule and you just run it out and you get everything that happens in the universe. - So a theory of everything is a mathematical framework within which you can explain everything that happens in the universe kind of in a unified way. It's not there's a bunch of disparate modules of... Does it feel like if you create a rule, and we'll talk about the Wolfram physics model, which is fascinating, but if you have a simple set of rules with a data structure like a hypergraph, does that feel like a satisfying theory of everything? Because then you really run up against the irreducibility, computational reducibility. - Right, so that's a really interesting question. So I, you know, what I thought was gonna happen is I thought we, you know, I thought we had a pretty good, I had a pretty good idea for what the structure of this sort of theory that's sort of underneath space and time and so on might be like. And I thought, gosh, you know, in my lifetime, so to speak, we might be able to figure out what happens in the first 10 to the minus 100 seconds of the universe. And that would be cool, but it's pretty far away from anything that we can see today. And it will be hard to test whether that's right and so on and so on and so on. To my huge surprise, although it should have been obvious, and it's embarrassing that it wasn't obvious to me, but to my huge surprise, we managed to get unbelievably much further than that. And basically what happened is that it turns out that even though there's this kind of bed of computational irreducibility that sort of all these simple rules run into, there are certain pieces of computational reducibility that quite generically occur for large classes of these rules. And, and this is the really exciting thing as far as I'm concerned, the big pieces of computational reducibility are basically the pillars of 20th century physics. That's the amazing thing, that general relativity and quantum field theory, the sort of the pillars of 20th century physics turn out to be precisely the stuff you can say. There's a lot you can't say. There's a lot that's kind of at this irreducible level where you kind of don't know what's going to happen. You have to run it. You know, you can't run it within our universe, et cetera, et cetera, et cetera, et cetera, et cetera. But the thing is there are things you can say and the things you can say turn out to be very beautifully, exactly the structure that was found in 20th century physics, namely general relativity and quantum mechanics. - And general relativity and quantum mechanics are these pockets of reducibility that we think of as, that 20th century physics is essentially pockets of reducibility. And then it is incredibly surprising that any kind of model that's generative from simple rules would have such pockets. - Yeah, well, I think what's surprising is we didn't know where those things came from. It's like general relativity, it's a very nice mathematically elegant theory. Why is it true? You know, quantum mechanics, why is it true? What we realized is that from this, that these theories are generic to a huge class of systems that have these particular, very unstructured underlying rules. And that's the thing that is sort of remarkable. And that's the thing to me that's just, it's really beautiful. I mean, it's, and the thing that's even more beautiful is that it turns out that, you know, people have been struggling for a long time. You know, how does general relativity, theory of gravity relate to quantum mechanics? They seem to have all kinds of incompatibilities. It turns out what we realized is at some level, they are the same theory. And that's just, it's just great as far as I'm concerned. - So maybe like taking a little step back from your perspective, not from the low, not from the beautiful hypergraph, well, from physics model perspective, but from the perspective of 20th century physics, what is general relativity? What is quantum mechanics? How do you think about these two theories from the context of the theory of everything? Like, just even definitions. - Yeah, yeah, yeah, right. So I mean, you know, a little bit of history of physics. Right, so I mean the, you know, okay, very, very quick history of physics, right? So, I mean, you know, physics, you know, in ancient Greek times, people basically said, we can just figure out how the world works. As, you know, we're philosophers, we're gonna figure out how the world works. You know, some philosophers thought there were atoms. Some philosophers thought there were, you know, continuous flows of things. People had different ideas about how the world works. And they tried to just say, we're gonna construct this idea of how the world works. They didn't really have sort of notions of doing experiments and so on quite the same way as developed later. So that was sort of an early tradition for thinking about sort of models of the world. Then by the time of 1600s, time of Galileo and then Newton, sort of the big idea there was, you know, title of Newton's book, you know, Principia Mathematica, Mathematical Principles of Natural Philosophy. We can use mathematics to understand natural philosophy, to understand things about the way the world works. And so that then led to this kind of idea that, you know, we can write down a mathematical equation and have that represent how the world works. So Newton's, one of his most famous ones is his universal law of gravity, inverse square law of gravity that allowed him to compute all sorts of features of the planets and so on. Although some of them he got wrong and it took another hundred years for people to actually be able to do the math to the level that was needed. But so that had been, this sort of tradition was we write down these mathematical equations. We don't really know where these equations come from. We write them down, then we figure out, we work out the consequences and we say, yes, that agrees with what we actually observe in astronomy or something like this. So that tradition continued. And then the first of these two sort of great 20th century innovations was, well, the history is actually a little bit more complicated, but let's say that there were two, quantum mechanics and general relativity. Quantum mechanics, the kind of 1900 was kind of the very early stuff done by Planck that led to the idea of photons, particles of light. But let's take general relativity first. One feature of the story is that special relativity, thing Einstein invented in 1905, was something which surprisingly was a kind of logically invented theory. It was not a theory where, it was something where given these ideas that were sort of axiomatically thought to be true about the world, it followed that such and such a thing would be the case. It was a little bit different from the kind of methodological structure of some existing theories in the more recent times, where it had just been, we write down an equation and we find out that it works. So what happened there-- - So there's some reasoning about the light. The basic idea was the speed of light appears to be constant. Even if you're traveling very fast, you shine a flashlight, the light will come out. Even if you're going at half the speed of light, the light doesn't come out of your flashlight at one and a half times the speed of light. It's still just the speed of light. And to make that work, you have to change your view of how space and time work to be able to account for the fact that when you're going faster, it appears that length is foreshortened and time is dilated and things like this. - And that's special relativity. - That's special relativity. So then Einstein went on with sort of vaguely similar kinds of thinking. In 1915, invented general relativity, which is a theory of gravity. And the basic point of general relativity is it's a theory that says when there is mass in space, space is curved. And what does that mean? You usually you think of what's the shortest distance between two points, like ordinarily on a plane in space, it's a straight line. Photons, light goes in straight lines. Well, then the question is, if you have a curved surface, a straight line is no longer straight. On the surface of the earth, the shortest distance between two points is a great circle. It's a circle. So Einstein's observation was maybe the physical structure of space is such that space is curved. So the shortest distance between two points, the path, the straight line in quotes, won't be straight anymore. And in particular, if a photon is traveling near the sun or something or if a particle is going, something is traveling near the sun, maybe the shortest path will be one that is something which looks curved to us because it seems curved to us because space has been deformed by the presence of mass associated with that massive object. So the kind of the idea there is think of the structure of space as being a dynamical changing kind of thing. But then what Einstein did was he wrote down these differential equations that basically represented the curvature of space and its response to the presence of mass and energy. - And that ultimately is connected to the force of gravity, which is one of the forces that seems to, based on its strength, operate on a different scale than some of the other forces. So it operates at a scale that's very large. - What happens there is just this curvature of space which causes the paths of objects to be deflected. That's what gravity does. It causes the paths of objects to be deflected. And this is an explanation for gravity, so to speak. And the surprise is that from 1915 until today, everything that we've measured about gravity precisely agrees with general relativity. And that it wasn't clear black holes were sort of a, well, actually the expansion of the universe was an early potential prediction, although Einstein tried to sort of patch up his equations to make it not cause the universe to expand 'cause it was kind of so obvious the universe wasn't expanding. And it turns out it was expanding and he should have just trusted the equations. And that's a lesson for those of us interested in making fundamental theories of physics is you should trust your theory and not try and patch it because of something that you think might be the case, that might turn out not to be the case. - Even if the theory says something crazy is happening. - Yeah, right. - Like the universe is expanding. - Like the universe is expanding, right. But then it took until the 1940s, probably even really until the 1960s, until people understood that black holes were a consequence of general relativity and so on. But that's some, the big surprise has been that so far, this theory of gravity has perfectly agreed with these collisions of black holes seen by the gravitational waves, it all just works. So that's been kind of one pillar of the story of physics. It's mathematically complicated to work out the consequences of general relativity, but it's not, there's no, I mean, and some things are kind of squiggly and complicated, like people believe energy is conserved. Okay, well, energy conservation doesn't really work in general relativity in the same way as it ordinarily does. And it's all a big mathematical story of how you actually nail down something that is definitive that you can talk about and not specific to the reference frames you're operating in and so on and so on and so on. But fundamentally, general relativity is a straight shot in the sense that you have this theory, you work out its consequences. - And that theory is useful in terms of basic science and trying to understand the way black holes work, the way the creation of galaxies work, sort of all of these kind of cosmological thing, understanding what happened, like you said, at the Big Bang. - Yeah. - Like all those kinds of, well, no, not at the Big Bang, actually, right? - Well, features of the expansion of the universe, yes. I mean, and there are lots of details where we don't quite know how it's working. You know, where's the dark matter? Is there dark energy? You know, et cetera, et cetera, et cetera. But fundamentally, the testable features of general relativity, it all works very beautifully. And it's, in a sense, it is mathematically sophisticated, but it is not conceptually hard to understand in some sense. - Okay, so that's general relativity. And what's its friendly neighbor? Like you said, there's two theories. Quantum mechanics. - Quantum mechanics, right. So quantum mechanics, the sort of, the way that that originated was, one question was, is the world continuous or is it discrete? You know, in ancient Greek times, people have been debating this. People debated it throughout history. Is light made of waves? Is it continuous? Is it discrete? Is it made of particles, corpuscles, whatever? What had become clear in the 1800s is that atoms, that materials are made of discrete atoms. You know, when you take some water, the water is not a continuous fluid, even though it seems like a continuous fluid to us at our scale. But if you say, let's look at it, smaller and smaller and smaller and smaller scale, eventually you get down to these molecules and then atoms. It's made of discrete things. So the question is sort of how important is this discreteness? Just what's discrete, what's not discrete? Is energy discrete? Is, you know, what's discrete, what's not? And so-- - Does it have mass? Those kinds of questions. - Yeah, yeah, right. Well, there's a question, I mean, for example, is mass discrete is an interesting question, which is now something we can address. But, you know, what happened in the, coming up to the 1920s, there was this kind of mathematical theory developed that could explain certain kinds of discreteness in particularly in features of atoms and so on. And, you know, what developed was this mathematical theory that was the theory of quantum mechanics, theory of wave functions, Schrodinger's equation, things like this. That's a mathematical theory that allows you to calculate lots of features of the microscopic world, lots of things about how atoms work, et cetera, et cetera, et cetera. Now, the calculations all work just great. The question of what does it really mean is a complicated question. Now, I mean, to just explain a little bit historically, the, you know, the early calculations of things like atoms worked great in 1920s, 1930s, and so on. There was always a problem there were in quantum field theory, which is a theory of, in quantum mechanics, you're dealing with a certain number of atoms, a certain number of electrons, and you fix the number of electrons. You say, I'm dealing with a two electron thing. In quantum field theory, you allow for particles being created and destroyed. So you can emit a photon that didn't exist before. You can absorb a photon, things like that. That's a more complicated, mathematically complicated theory. And it had all kinds of mathematical issues and all kinds of infinities that cropped up. And it was finally figured out more or less how to get rid of those. But there were only certain ways of doing the calculations. And those didn't work for atomic nuclei, among other things. And that led to a lot of development up until the 1960s of alternative ideas for how one could understand what was happening in atomic nuclei, et cetera, et cetera, et cetera, end result. In the end, the kind of most quotes, obvious mathematical structure of quantum field theory seems to work, although it's mathematically difficult to deal with, but you can calculate all kinds of things. You can calculate to, you know, a dozen decimal places, certain things, you can measure them. It all works. It's all beautiful. Now you say- - By the way, the underlying fabric is the model of that particular theory is fields. Like you keep saying fields. - Those are quantum fields. Those are different from classical fields. A field is something like you say, there's like you say, the temperature field in this room. It's like there is a value of temperature at every point around the room. That's, or you can say the wind field would be the vector direction of the wind at every point. - It's continuous. - Yes, and that's a classical field. The quantum field is a much more mathematically elaborate kind of thing. And I should explain that one of the pictures of quantum mechanics that's really important is, you know, in classical physics, one believes that sort of definite things happen in the world. You pick up a ball, you throw it. The ball goes in a definite trajectory that has certain equations of motion. It goes in a parabola, whatever else. In quantum mechanics, the picture is definite things don't happen. Instead, sort of what happens is this whole sort of structure of all, you know, many different paths being followed. And we can calculate certain aspects of what happens, certain probabilities of different outcomes and so on. And you say, well, what really happened? What's really going on? What's the sort of, what's the underlying, you know, what's the underlying story? What, how do we, how do we turn this mathematical theory that we can calculate things with into something that we can really understand and have a narrative about? And that's been really, really hard for quantum mechanics. My friend Dick Feynman always used to say, nobody understands quantum mechanics, even though he'd made his whole career out of calculating things about quantum mechanics. And, you know, so it's a little-- - But nevertheless, it's what the quantum field theory is very, very accurate at predicting a lot of the physical phenomena. So it works. - Yeah, but there are things about it, you know, it has certain, when we apply it, the standard model of particle physics, for example, we, you know, which we apply to calculate all kinds of things, it works really well. And you say, well, it has certain parameters. It has a whole bunch of parameters, actually. You say, why is the, you know, why does the muon particle exist? Why is it 206 times the mass of the electron? We don't know, no idea. - But so the standard model of physics is one of the models that's very accurate for describing three of the fundamental forces of physics. And it's looking at the world of the very small. - Right. - And then there's back to the neighbor of gravity, of general relativity. So, and in the context of a theory of everything, what's traditionally the task of the unification of these theories? And why is it hard? - Well, the issue is, you try to use the methods of quantum field theory to talk about gravity, and it doesn't work. Just like there are photons of light. So there are gravitons, which are sort of the particles of gravity. And when you try and compute sort of the properties of the particles of gravity, the kind of mathematical tricks that get used in working things out in quantum field theory don't work. And that's, so that's been a sort of fundamental issue. And when you think about black holes, which are a place where sort of the structure of space is, you know, has sort of rapid variation, and you get kind of quantum effects mixed in with effects from general relativity, things get very complicated, and there are apparent paradoxes and things like that. And people have, you know, there have been a bunch of mathematical developments in physics over the last, I don't know, 30 years or so, which have kind of picked away at those kinds of issues and got hints about how things might work. And, but it hasn't been, you know, and the other thing to realize is, as far as physics is concerned, it's just like, here's general relativity, here's quantum field theory, you know, be happy. - Yeah, so do you think there's a quantization of gravity, so quantum gravity, what do you think of efforts that people have tried to, yeah, what do you think in general of the efforts of the physics community to try to unify these laws? - So I think what's, it's interesting. I mean, I would have said something very different before what's happened with our physics project. I mean, you know, the remarkable thing is, what we've been able to do is to make from this very simple, structurally simple, underlying set of ideas, we've been able to build this, this, you know, very elaborate structure that's both very abstract and very sort of mathematically rich. And the big surprise, as far as I'm concerned, is that it touches many of the ideas that people have had. So in other words, things like string theory and so on, twister theory, it's like, you know, we might've thought, I had thought we're out on a prong, we're building something that's computational, it's completely different from what other people have done. But actually, it seems like what we've done is to provide essentially the machine code that, you know, these things are various features of domain-specific languages, so to speak, that talk about various aspects of this machine code. And I think there's a, this is something that to me is very exciting because it allows one both for us to provide sort of a new foundation for what's been thought about there and for all the work that's been done in those areas to, you know, to give us, you know, more momentum to be able to figure out what's going on. Now, you know, people have sort of hoped, oh, we're just gonna be able to get, you know, string theory to just answer everything. That hasn't worked out. And I think we now kind of can see a little bit about just sort of how far away certain kinds of things are from being able to explain things. Some things, one of the big surprises to me, actually, I literally just got a message about one aspect of this is the, you know, it's turning out to be easier. I mean, this project has been so much easier than I could ever imagine it would be. That is, I thought we would be, you know, just about able to understand the first 10 to the minus 100 seconds of the universe. And, you know, it would be 100 years before we get much further than that. It's just turned out it actually wasn't that hard. I mean, we're not finished, but, you know-- - So you're seeing echoes of all the disparate theories of physics in this framework? - Yes, yes. I mean, it's a very interesting, you know, sort of history of science-like phenomenon. I mean, the best analogy that I can see is what happened with the early days of computability and computation theory. You know, Turing machines were invented in 1936. People sort of understand computation in terms of Turing machines, but actually there had been preexisting theories of computation, combinators, general recursive functions, lambda calculus, things like this. But people hadn't, those hadn't been concrete enough that people could really wrap their arms around them and understand what was going on. And I think what we're gonna see in this case is that a bunch of these mathematical theories, including some very, I mean, one of the things that's really interesting is one of the most abstract things that's come out of sort of mathematics, higher category theory, things about infinity groupoids, things like this, which to me always just seemed like they were floating off into the stratosphere, ionosphere of mathematics, turn out to be things which our sort of theory anchors down to something fairly definite and says are super relevant to the way that we can understand how physics works. - Give me a sec. By the way, I just threw a hat on. You've said that, this metaphor analogy that theory of everything is a big mountain. And you have a sense that however far we are up the mountain, that the Wolfram physics model, a view of the universe is at least the right mountain. - We're the right mountain, yes, without question. - So which aspect of it is the right mountain? So for example, I mean, so there's so many aspects to just the way of the Wolfram physics project, the way it approaches the world, that's clean, crisp, and unique and powerful. So there's a discrete nature to it. There's a hypergraph. There's a computational nature. There's a generative aspect. You start from nothing, you generate everything. Do you think the actual model is actually a really good one? Or do you think this general principle from simplicity generating complexity is the right? Like what aspect of the mountain is the correct? - I think that the kind of the meta idea about using simple computational systems to do things, that's the ultimate big paradigm that is sort of super important. The details of the particular model are very nice and clean and allow one to actually understand what's going on. They are not unique. And in fact, we know that. We know that there's a large number of different ways to describe essentially the same thing. I mean, I can describe things in terms of hypergraphs. I can describe them in terms of higher category theory. I can describe them in a bunch of different ways. They are in some sense all the same thing, but our sort of story about what's going on and the kind of cultural mathematical resonances are a bit different. I mean, I think it's perhaps worth sort of saying a little bit about kind of the, the foundational ideas of these models and things. - Great. So can you maybe, can we like rewind? We've talked about it a little bit, but can you say like what the central idea is of the Wolfram Physics Project? - Right. So the question is we're interested in finding sort of simple computational rule that describes our whole universe. - Can we just pause on that? It's just so beautiful. That's such a beautiful, that's such a beautiful idea that we can generate our universe from a data structure, simple structure, simple set of rules, and we can generate our entire universe. - Yes. - That's awe-inspiring. - Right. But so, so, you know, the question is how do you actualize that? What might this rule be like? And so one thing you quickly realize is if you're going to pack everything about a universe into this tiny rule, not much that we are familiar with in our universe will be obvious in that rule. So you don't get to fit all these parameters of the universe, all these features of, you know, this is how space works, this is how time works, et cetera, et cetera, et cetera. You don't get to fit that all in. It all has to be sort of packed in to this, this thing, something much smaller, much more basic, much lower level machine code, so to speak, than that. And all the stuff that we're familiar with has to kind of emerge from the operation of- - So the rule in itself, because of the computational reducibility, is not going to tell you the story. It's not going to give you the answer to, it's not going to let you predict what you're going to have for lunch tomorrow. - Right. - And it's not going to let you predict basically anything about your life, about the universe. - Right, but, and you're not going to be able to see in that rule, oh, there's the three for the number of dimensions of space and so on. - Right. - That's not going to be just- - The problem is not going to be obviously- - Right, so the question is then, what is the universe made of? That's a basic question. And we've had some assumptions about what the universe is made of for the last few thousand years, that I think in some cases just turn out not to be right. And the most important assumption is that space is a continuous thing. That is, that you can, if you say, let's pick a point in space. We're going to do geometry. We're going to pick a point. We can pick a point absolutely anywhere in space. Precise numbers, we can specify of where that point is. In fact, Euclid, who kind of wrote down the original kind of axiomatization of geometry back in 300 BC or so, his very first definition, he says, "A point is that which has no part." A point is this indivisible, infinitesimal thing. Okay, so we might've said that about material objects. We might've said that about water, for example. We might've said water is a continuous thing that we can just pick any point we want in some water. But actually we know it isn't true. We know that water is made of molecules that are discrete. And so the question, one fundamental question is what is space made of? And so one of the things that's sort of a starting point for what I've done is to think of space as a discrete thing, to think of there being sort of atoms of space just as there are atoms of material things, although very different kinds of atoms. And by the way, I mean, this idea, there were ancient Greek philosophers who had this idea. There were, Einstein actually thought this is probably how things would work out. I mean, he said repeatedly, he thought that's the way it would work out. We don't have the mathematical tools in our time, which was 1940s, 1950s, and so on, to explore this. - Like the way he thought, you mean that there is something very, very small and discrete that's underlying space? - Yes, and that means that, so the mathematical theory, mathematical theories in physics assume that space can be described just as a continuous thing. You can just pick coordinates and the coordinates can have any values, and that's how you define space. Space is this just sort of background sort of theater on which the universe operates. - But can we draw a distinction between space as a thing that could be described by three values, coordinates, and how you're, are you using the word space more generally when you say? - No, I'm just talking about space as in what we experience in the universe. - So you think this 3D aspect of it is fundamental? - No, I don't think that 3D is fundamental at all, actually. I think that the thing that has been assumed is that space is this continuous thing where you can just describe it by, let's say, three numbers, for instance. But the most important thing about that is that you can describe it by precise numbers, because you can pick any point in space, and you can talk about motions, any infinitesimal motion in space. - And that's what continuous means? - That's what continuous means. That's what, you know, Newton invented calculus to describe these kind of continuous, small variations and so on. That's kind of a fundamental idea. From Euclid on, that's been a fundamental idea about space. And so-- - Is that right or wrong? - It's not right. It's not right. It's right at the level of our experience most of the time. It's not right at the level of the machine code, so to speak. And so-- - Machine code. Yeah, of the simulation, that's right, that's right. The very lowest level of the fabric of the universe, at least under the Wolfram physics model, is your sense is discrete. - Right, so now, what does that mean? So it means, what is space then? So in models, the basic idea is you say, there are these sort of atoms of space, there are these points that represent, you know, represent places in space, but they're just discrete points. And the only thing we know about them is how they're connected to each other. We don't know where they are. They don't have coordinates. We don't get to say, this is a position such and such. It's just, here's a big bag of points. Like in our universe, there might be 10 to the 100 of these points. And all we know is this point is connected to this other point. So it's like, you know, all we have is the friend network, so to speak. We don't have, you know, people's physical addresses. All we have is the friend network of these points. - Yeah, the underlying nature of reality is kind of like a Facebook. We don't know their location, but we have the friends. - Yeah, yeah, right. We know which point is connected to which other points. And that's all we know. And so you might say, well, how on earth can you get something which is like our experience of, you know, what seems like continuous space? Well, the answer is, by the time you have 10 to the 100 of these things, those connections can work in such a way that on a large scale, it will seem to be like continuous space in, let's say, three dimensions or some other number of dimensions or 2.6 dimensions or whatever else. - Because they're much, much, much, much larger. So like the number of relationships here we're talking about is just a humongous amount. So the kind of thing you're talking about is very, very, very small relative to our experience of daily life. - Right, so it's, I mean, you know, we don't know exactly the size, but maybe 10 to the minus, maybe around 10 to the minus 100 meters. So, you know, the size of, to give a comparison, you know, the size of a proton is 10 to the minus 15 meters. And so this is something incredibly tiny compared to that. - And the idea that from that would emerge the experience of continuous space is mind-blowing. What's your intuition why that's possible? Like, first of all, I mean, we'll get into it, but I don't know if we will through the medium of conversation, but the construct of hypergraphs is just beautiful. - Right. - Cellular automata are beautiful. We'll talk about it, but okay. - Right, but this thing about, you know, continuity arising from discrete systems is in today's world is actually not so surprising. I mean, you know, your average computer screen, right? Every computer screen is made of discrete pixels, yet we have the, you know, we have the idea that we're seeing these continuous pictures. I mean, it's, you know, the fact that on a large scale, continuity can arise from lots of discrete elements. This is at some level unsurprising now. - But, wait, wait, wait, wait, wait, but the pixels have a very definitive structure of neighbors on a computer screen. - Right. - There's no concept of spatial, of space, inherent in the underlying fabric of reality. - Right, right, right. So the point is, but there are cases where there are. So for example, let's just imagine you have a square grid, okay? And at every point on the grid, you have one of these atoms of space, and it's connected to four other atoms of space on the, you know, northeast, southwest corners, right? There you have something where if you zoom out from that, it's like a computer screen. - Yeah, so the relationship creates the spatial, like the relationship creates a constraint, which then in an emergent sense creates a, like, yeah, like a, basically a spatial coordinate for that thing. - Yes, yeah, right. - Even though the individual point doesn't have a spatial. - Even though the individual point doesn't know anything, it just knows what its, you know, what its neighbors are. The, on a large scale, it can be described by saying, oh, it looks like it's a, you know, this grid, this zoomed out grid. You can say, well, you can describe these different points by saying they have certain positions, coordinates, et cetera. Now, in the sort of real setup, it's more complicated than that. It isn't just a square grid or something. It's something much more dynamic and complicated, which we'll talk about. But so, you know, the first idea, the first key idea is, you know, what's the universe made of? It's made of atoms of space, basically, with these connections between them. What kind of connections do they have? Well, so the simplest kind of thing you might say is we've got something like a graph where every atom of space, where we have these edges that go between, out of these connections that go between atoms of space. We're not saying how long these edges are. We're just saying there is a connection from this place to the, from this atom to this atom. - Just a quick pause, 'cause there's a lot of very people that listen to this. Just to clarify, 'cause I did a poll, actually. What do you think a graph is a long time ago? And it's kind of funny how few people know the term graph outside of computer science. - Let's call it a network. I think that's-- - Let's call it a network is better. So, but every time, I like the word graph, though. So let's define, let's just say that a graph will use terms nodes and edges, maybe. And it's just nodes represent some abstract entity, and then the edges represent relationships between those entities. - Right, exactly. - So that's what a graph, sorry, so. So there you go. So that's the basic structure. - That is the simplest case of a basic structure. Actually, it tends to be better to think about hypergraphs. So a hypergraph is just, instead of saying there are connections between pairs of things, we say there are connections between any number of things. So there might be ternary edges. So instead of just having two points are connected by an edge, you say three points are all associated with a hyperedge, are all connected by a hyperedge. That's just, at some level, that's a detail. It's a detail that happens to make the, for me, you know, sort of in the history of this project, the realization that you could do things that way broke out of certain kinds of arbitrariness that I felt that there was in the model before I had seen how this worked. - I mean, a hypergraph can be mapped to a graph. It's just a convenient representation, mathematically speaking. - Right, that's correct. That's correct. But so then, so, okay, so the first question, the first idea of these models of ours is space is made of these, you know, connected sort of atoms of space. The next idea is space is all there is. There's nothing except for this space. So in traditional ideas in physics, people have said there's space, it's kind of a background, and then there's matter, all these particles, electrons, all these other things, which exist in space, right? But in this model, one of the key ideas is there's nothing except space. So in other words, everything that exists in the universe is a feature of this hypergraph. So how can that possibly be? Well, the way that works is that there are certain structures in this hypergraph where you say that little twisty, knotted thing, we don't know exactly how this works yet, but we have sort of idea about how it works mathematically. This sort of twisted, knotted thing, that's the core of an electron. This thing over there that has this different form, that's something else. - So the different peculiarities of the structure of this graph are the very things that we think of as the particles inside the space, but in fact, it's just the property of the space. Mind-blowing, first of all. It's mind-blowing, and we'll probably talk, in its simplicity and beauty. - Yes, I think it's very beautiful. - But that's space, and then there's another concept we didn't really kind of mention, but you think of computation as a transformation. - Let's talk about time in a second. Let's just, I mean, on the subject of space, there's this question of kind of what, there's this idea, there is this hypergraph, it represents space, and it represents everything that's in space. The features of that hypergraph, you can say, certain features in this part we do know, certain features of the hypergraph represent the presence of energy, for example, or the presence of mass or momentum, and we know what the features of the hypergraph that represent those things are, but it's all just the same hypergraph. So one thing you might ask is, if you just look at this hypergraph and you say, and we're gonna talk about sort of what the hypergraph does, but if you say, how much of what's going on in this hypergraph is things we know and care about, like particles and atoms and electrons and all this kind of thing, and how much is just the background of space? So it turns out, so far as in one rough estimate of this, or everything that we care about in the universe is only one part in 10 to the 120 of what's actually going on. The vast majority of what's happening is purely things that maintain the structure of space. That, in other words, the things that are the features of space that are the things that we consider notable, like the presence of particles and so on, that's a tiny little piece of froth on the top of all this activity that mostly is just intended to, mostly, I can't say intended, there's no intention here, that just maintains the structure of space. - Let me load that in. It just makes me feel so good as a human being. To be the froth on the one in the 10 to the-- - 120 or something of, well-- - And also just humbling how, in this mathematical framework, how much work needs to be done on the infrastructure of our universe. - Right, yes. Right, to maintain the infrastructure of our universe is a lot of work. We are merely writing little tiny things on top of that infrastructure. But you were just starting to talk a little bit about, we talked about space, that represents all the stuff that's in the universe. The question is, what does that stuff do? And for that, we have to start talking about time and what is time and so on. And one of the basic idea of this model is time is the progression of computation. So in other words, we have a structure of space and there is a rule that says how that structure of space will change, and it's the application, the repeated application of that rule that defines the progress of time. - And what does the rule look like in the space of hypergraphs? - Right, so what the rule says is something like, if you have a little tiny piece of hypergraph that looks like this, then it will be transformed into a piece of hypergraph that looks like this. So that's all it says. It says you pick up these elements of space and you can think of these edges, these hyper edges as being relations between elements in space. You might pick up these two relations between elements in space. And we're not saying where those elements are or what they are, but every time there's a certain arrangement of elements in space, then arrangement in the sense of the way they're connected, then we transform it into some other arrangement. - So there's a little tiny pattern and you transform it into another little pattern. - That's right. - And then because of this, I mean, again, it's kind of similar to cellular automata in that like, on paper, the rule looks like super simple. It's like, yeah, okay. Yeah, like, yeah, right, from this, the universe can be born. But like, once you start applying it, beautiful structure starts being, potentially can be created. And what you're doing is you're applying that rule to different parts, like anytime you match it within the hypergraph. - Exactly. - And then one of the like incredibly beautiful and interesting things to think about is the order in which you apply that rule. Because that pattern appears all over the place. - Right, so this is a big, complicated thing, very hard to wrap one's brain around. Okay, so you say the rule is, every time you see this little pattern, transform it in this way. But yet, as you look around the space that represents the universe, there may be zillions of places where that little pattern occurs. So what it says is, just do this, apply this rule wherever you feel like. And what is extremely non-trivial is, well, okay, so this is happening sort of in computer science terms sort of asynchronously. You're just doing it wherever you feel like doing it. And the only constraint is that if you're going to apply the rule somewhere, the things to which you apply the rule, the little elements to which you apply the rule, if they have to be, okay, you can think of each application of the rule as being kind of an event that happens in the universe. And the input to an event has to be ready for the event to occur. That is, if one event occurred, if one transformation occurred, and it produced a particular atom of space, then that atom of space has to already exist before another transformation that's going to apply to that atom of space can occur. - Yeah, so that's like the prerequisite for the event. - That's right, that's right. So that defines a kind of this sort of set of causal relationships between events. It says this event has to have happened before this event. But that is-- - But that's not a very limited constraint. - No, it's not. And what's interesting-- - You still get the zillion, that's the technical term, options. - That's correct. But, okay, so this is where things get a little bit more elaborate, but-- - But they're mind-blowing, so. - Right, but so what happens is, so the first thing you might say is, well, okay, so this question about the freedom of which event you do when, well, let me sort of state an answer and then explain it. Okay? The validity of special relativity is a consequence of the fact that in some sense it doesn't matter in what order you do these underlying things so long as they respect this kind of set of causal relationships. - So that's, the part that's in a certain sense is a really important one, but the fact that it sometimes doesn't matter, that's a, I don't know, that's another beautiful thing. - So there's this idea of what I call causal invariance. - Causal invariance, exactly. So that's a really, really powerful, powerful idea. - Powerful idea which has actually arisen in different forms many times in the history of mathematics, mathematical logic, even computer science, has many different names. I mean, our particular version of it is a little bit tighter than other versions, but it's basically the same idea. Here's how to think about that idea. So imagine that, well, let's talk about it in terms of math for a second. Let's say you're doing algebra and you're told, multiply out this series of polynomials that are multiplied together, okay? You say, well, which order should I do that in? Say, well, do I multiply the third one by the fourth one and then do it by the first one? Or do I do the fifth one by the sixth one and then do that? Well, it turns out it doesn't matter. You can multiply them out in any order, you'll always get the same answer. That's a property, if you think about kind of making a kind of network that represents in what order you do things, you'll get different orders for different ways of multiplying things out, but you'll always get the same answer. Same thing if you, let's say you're sorting, you've got a bunch of A's and B's, they're in some random order, you know, BAA, BBBAA, whatever, and you have a little rule that says every time you see BA, flip it around to AB, okay? Eventually you apply that rule enough times, you'll have sorted the string so that it's all the A's first and then all the B's. Again, there are many different orders in which you can do that, many different sort of places where you can apply that update. In the end, you'll always get the string sorted the same way. - I know with sorting a string, it sounds obvious. That's to me, surprising. That there is, in complicated systems, obviously with a string, but in a hypergraph, that the application of the rule, a synchronous rule can lead to the same results sometimes. - Yes, yes, that is not obvious, and it was something that, you know, I sort of discovered that idea for these kinds of systems and back in the 1990s, and for various reasons, I was not satisfied by how sort of fragile finding that particular property was. And let me just make another point, which is that it turns out that even if the underlying rule does not have this property of causal invariance, it can turn out that every observation made by observers of the rule, they can impose what amounts to causal invariance on the rule. We can explain that, it's a little bit more complicated. I mean, technically that has to do with this idea of completions, which is something that comes up in term rewriting systems, automated theorem proving systems and so on, but let's ignore that for a second. We can come to that later. - Is it useful to talk about observation? - Not yet, not yet. - Okay, so great. So there's some concept of causal invariance as you apply these rules in an asynchronous way, you can think of those transformations as events. So there's this hypergraph that represents space and all of these events happening in the space, and the graph grows in interesting, complicated ways, and eventually the froth arises of what we experience as human existence. - That's some version of the picture, but let's explain a little bit more. - Exactly, a little more detail. - Right, so one thing that is sort of surprising in this theory is one of the sort of achievements of 20th century physics was kind of bringing space and time together. That was special relativity, people talk about space-time, this sort of unified thing where space and time kind of are mixed, and there's a nice mathematical formalism in which space and time sort of appear as part of the space-time continuum, the space-time four vectors and things like this. We talk about time as the fourth dimension and all these kinds of things. It's, you know, and it seems like the theory of relativity sort of says space and time are fundamentally the same kind of thing. So one of the things that took a while to understand in this approach of mine is that in my kind of approach, space and time are really not fundamentally the same kind of thing. Space is the extension of this hypergraph, time is the kind of progress of this inexorable computation of these rules getting applied to the hypergraph. So they seem like very different kinds of things. And so that, at first, seems like how can that possibly be right? How can that possibly be Lorentz invariant? That's the term for things being, you know, following the rules of special relativity. Well, it turns out that when you have causal invariance, that, and let's see, we can, it's worth explaining a little bit how this works. It's a little bit elaborate, but the basic point is that even though space and time sort of come from very different places, it turns out that the rules of sort of space time that special relativity talks about come out of this model when you're looking at large enough systems. So a way to think about this, you know, in terms of when you're looking at large enough systems, the part of that story is when you look at some fluid like water, for example, there are equations that govern the flow of water. Those equations are things that apply on a large scale. If you look at the individual molecules, they don't know anything about those equations. It's just the sort of the large scale effect of those molecules turns out to follow those equations. And it's the same kind of thing happening in our models. - I know this might be a small point, but it might be a very big one. We've been talking about space and time at the lowest level of the model, which is space. The hypergraph time is the evolution of this hypergraph. But there's also space time that we think about in general relativity for your special relativity. Like what, how do you go from the lowest source code of space and time we're talking about to the more traditional terminology of space and time? - Yeah, right. So the key thing is this thing we call the causal graph. So the causal graph is the graph of causal relationships between events. So every one of these little updating events, every one of these little transformations of the hypergraph happens somewhere in the hypergraph, happens at some stage in the computation. That's an event. That event has a causal relationship to other events in the sense that if another event needs as its input, the output from the first event, there will be a causal relationship of the future event will depend on the past event. So you can say it has a causal connection. So you can make this graph of causal relationships between events. That graph of causal relationships, causal invariance implies that that graph is unique. It doesn't matter, even though you think, oh, I'm, you know, let's say we were sorting a string. For example, I did that particular transposition of characters at this time. And then I did that one. Then I did this one. Turns out if you look at the network of connections between those updating events, that network is the same. It's the, if you were to-- - I see. - The structure, so in other words, if you were to draw that, if you were to put that network on a picture of where you're doing all the updating, the places where you put the nodes of the network will be different, but the way the nodes are connected will always be the same. - But the causal graph is, I don't, it's kind of an observate, it's not enforced. It's just emergent from a set of events. - It's a feature of, okay, so what it is-- - It's characteristic, I guess, of the way events happen. - Right, it's an event can't happen until its input is ready. And so that creates this network of causal relationships. And that's the causal graph. And the thing, the next thing to realize is, okay, we, when you're going to observe what happens in the universe, you have to sort of make sense of this causal graph. So, and you are an observer who yourself is part of this causal graph. And so that means, so let me give you an example of how that works. So imagine we have a really weird theory of physics of the world where it says this updating process, there's only going to be one update at every moment in time. And there's just going to be like a Turing machine that has a little head that runs around and just is always just updating one thing at a time. So you say, I have a theory of physics and the theory of physics says, there's just this one little place where things get updated. You say, that's completely crazy because it's plainly obvious that things are being updated sort of at the same time. - Asynchronously, yeah, or at the same time, yeah. - But the fact is that the thing is that if I'm talking to you and you seem to be being updated as I'm being updated, but if there's just this one little head that's running around updating things, I will not know whether you've been updated or not until I'm updated. So in other words, when you draw this causal graph of the causal relationship between the updatings in you and the update is in me, it'll still be the same causal graph, whether even though the underlying sort of story of what happens is, oh, there's just this one little thing and it goes and updates in different places in the universe. - Is that clear or is that a hypothesis? Is that clear that there's a unique causal graph? - If there's causal invariance, there's a unique causal graph. - So it's okay to think of what we're talking about as a hypergraph and the operations on it as a kind of Turing machine with a single head, like a single guy running around updating stuff. Is that safe to intuitively think of it this way? - Let me think about that for a second. Yes, I think so. I think there's nothing, it doesn't matter. I mean, you can say, okay, there is one, the reason I'm pausing for a second is that I'm wondering, well, when you say running around, depends how far it jumps every time it runs around. - Yeah, yeah, that's right. - But I mean like one operation at a time. - Yeah, you can think of it as one operation at a time. - It's easier for the human brain to think of it that way as opposed to simultaneous. - Well, maybe it's not, okay, but the thing is that's not how we experience the world. What we experience is we look around, everything seems to be happening at successive moments in time everywhere in space. That is the, and that's partly a feature of our particular construction. I mean, that is the speed of light is really fast compared to, we look around, I can see maybe a hundred feet away right now. My brain does not process very much in the time it takes light to go a hundred feet. - The brain operates at a scale of hundreds of milliseconds or something like that, I don't know. - Right. - And speed of light is much faster. - Right, light goes in a billionth of a second, light has gone a foot. So it goes a billion feet every second. - There's certain moments through this conversation where I imagine the absurdity of the fact that there's two descendants of apes modeled by a hypergraph that are communicating with each other and experiencing this whole thing as a real-time simultaneous update with, I'm taking in photons from you right now, but there's something much, much deeper going on here. - Right, it does have a-- - It's paralyzing sometimes just to remember that. - Right, no, I mean, you know, as a small little tangent, I just remembered that we're talking about, I mean, this, about the fabric of reality. - Right, so we've got this causal graph that represents the sort of causal relationships between all these events in the universe. That causal graph kind of is a representation of space-time, but our experience of it requires that we pick reference frames. This is kind of a key idea, Einstein had this idea that what that means is we have to say what are we going to pick as being the, sort of what we define as simultaneous moments in time. So for example, we can say, you know, how do we set our clocks? You know, if we've got a spacecraft landing on Mars, you know, do we say that it, you know, what time is it landing at? Was it, you know, even though there's a 20-minute speed of light delay or something, you know, what time do we say it landed at? How do we set up sort of time coordinates for the world? And that turns out to be, there's kind of this arbitrariness to how we set these reference frames that define sort of what counts as simultaneous. And what is the essence of special relativity is to think about reference frames going at different speeds and to think about sort of how they assign what counts as space, what counts as time, and so on. That's all a bit technical, but the basic bottom line is that this causal invariance property, that means that it's always the same causal graph, independent of how you slice it with these reference frames, you'll always sort of see the same physical processes go on. And that's basically why special relativity works. - So there's something like special relativity, like everything around space and time that fits this idea of the causal graph. - Right, well, you know, one way to think about it is given that you have a basic structure that just involves updating things in these connected updates and looking at the causal relationships between connected updates, that's enough. When you unravel the consequences of that, that together with the fact that there are lots of these things and that you can take a continuum limit and so on, implies special relativity. And so that, it's kind of a, not a big deal, because it's kind of a, you know, it was completely unobvious when you started off with saying we've got this graph, it's being updated in time, et cetera, et cetera, et cetera, that just looks like nothing to do with special relativity. And yet you get that. And what, I mean, then the thing, I mean, this was stuff that I figured out back in the 1990s. The next big thing you get is general relativity. And so in this hypergraph, the sort of limiting structure, when you have a very big hypergraph, you can think of as being just like, you know, water seems continuous on a large scale. So this hypergraph seems continuous on a large scale. One question is, you know, how many dimensions of space does it correspond to? So one question you can ask is, if you've just got a bunch of points and they're connected together, how do you deduce what effective dimension of space that bundle of points corresponds to? And that's pretty easy to explain. So basically, if you say, you've got a point and you look at how many neighbors does that point have? Okay, imagine it's on a square grid. Then it'll have four neighbors. Go another level out. How many neighbors do you get then? What you realize is, as you go more and more levels out, as you go more and more distance on the graph out, you're capturing something which is essentially a circle in two dimensions so that, you know, the number of the area of a circle is pi r squared. So it's the number of points that you get to goes up like the distance you've gone squared. And in general, in d-dimensional space, it's r to the power d. It's the number of points you get to if you go r steps on the graph grows like the number of steps you go to the power of the dimension. And that's a way that you can estimate the effective dimension of one of these graphs. - So what does that grow to? So how does the dimension grow? 'Cause, I mean, obviously the visual aspect of these hypergraphs, they're often visualized in three dimensions. - Right. - And then there's a certain kind of structure. Like you said, there's, I mean, a circle, a sphere. There's a planar aspect to it, to this graph, to where it kinda, it almost starts creating a surface, like a complicated surface, but a surface. So how does that connect to effective dimension? - Okay, so I mean, if you can lay out the graph in such a way that the points in the graph, points that are neighbors on the graph are neighbors as you lay them out, and you can do that in two dimensions, then it's gonna approximate a two-dimensional thing. If you can't do that in two dimensions, if everything would have to fold over a lot in two dimensions, then it's not approximating a two-dimensional thing. Maybe you can lay it out in three dimensions. Maybe you have to lay it out in five dimensions to have it be the case that it sort of smoothly lays out like that. - Well, but, okay, so, and I apologize for the different tangent questions, but there's an infinity number of possible rules. So we have to look for rules that create the kind of structures that are reminiscent for, that have echoes of the different physics theories in them. So what kind of rules, is there something simple to be said about the kind of rules that you have found beautiful, that you have found powerful? - Right, so, I mean, what, you know, one of the features of computational irreducibility is it's very, you can't say in advance what's gonna happen with any particular, you can't say, I'm gonna pick these rules from this part of rule space, so to speak, 'cause they're gonna be the ones that are gonna work. That's, you can make some statements along those lines, but you can't generally say that. Now, you know, the state of what we've been able to do is, you know, different properties of the universe, like dimensionality, you know, integer dimensionality, features of other features of quantum mechanics, things like that. At this point, what we've got is we've got rules that any one of those features, we can get a rule that has that feature. - Yeah, so the-- - But we don't have the sort of, the final, here's a rule which has all of these features. We do not have that yet. - So if I were to try to summarize the Wolfram Physics Project, which is, you know, something that's been in your brain for a long time, but really has just exploded in activity, you know, only just months ago. - Yes. - So it's an evolving thing, and next week, I'll try to publish this conversation as quickly as possible because by the time it's published, already new things will probably have come out. So if I were to summarize it, we've talked about the basics of, there's a hypergraph that represents space, there is transformations in that hypergraph that represents time. - Progress of time. - That progress of time, there's a causal graph that's a characteristic of this, and the basic process of science, of, yeah, of science within the Wolfram Physics model is to try different rules and see which properties of physics that we know of, known physical theories, are, appear within the graphs that emerge from that rule. - That's what I thought it was going to be. - Uh-oh, okay. So what, so what is it? - It turns out we can do a lot better than that. It turns out that using kind of mathematical ideas, we can say, and computational ideas, we can make general statements, and those general statements turn out to correspond to things that we know from 20th century physics. In other words, the idea of you just try a bunch of rules and see what they do, that's what I thought we were going to have to do. But in fact, we can say, given causal invariance and computational irreducibility, we can derive, and this is where it gets really pretty interesting, we can derive special relativity, we can derive general relativity, we can derive quantum mechanics. And that's where things really start to get exciting, is it wasn't at all obvious to me that even if we were completely correct, and even if we had, this is the rule, even if we found the rule, to be able to say, yes, it corresponds to things we already know, I did not expect that to be the case. And-- - So for somebody who is a simple mind and definitely not a physicist, not even close, what does derivation mean in this case? - Okay, so let me, this is an interesting question. Okay, so one thing-- - In the context of computational irreducibility. - Yeah, yeah, right, right. So what you have to do, let me go back to, again, the mundane example of fluids and water and things like that, right? So you have a bunch of molecules bouncing around. You can say, just as a piece of mathematics, I happened to do this from cellular automata back in the mid-1980s, you can say, just as a matter of mathematics, you can say the continuum limit of these little molecules bouncing around is the Navier-Stokes equations. That's just a piece of mathematics. It's not, it doesn't rely on, you have to make certain assumptions that you have to say there's enough randomness in the way the molecules bounce around that certain statistical averages work, et cetera, et cetera, et cetera. Okay, it is a very similar derivation to derive, for example, the Einstein equations. Okay, so the way that works, roughly, the Einstein equations are about curvature of space. Curvature of space, I talked about sort of how you can figure out dimension of space. There's a similar kind of way of figuring out if you just sort of say, you know, you're making a larger and larger ball or larger and larger, if you draw a circle on the surface of the Earth, for example, you might think the area of a circle is pi r squared, but on the surface of the Earth, because it's a sphere, it's not flat, the area of a circle isn't precisely pi r squared, as the circle gets bigger, the area is slightly smaller than you would expect from the formula pi r squared as a little correction term that depends on the ratio of the size of the circle to the radius of the Earth. Okay, so it's the same basic thing, allows you to measure from one of these hypergraphs, what is its effective curvature? And that was- - So, the little piece of mathematics that explains special general relativity can map nicely to describe fundamental property of the hypergraphs, of the curvature of the hypergraphs. - Okay, so, special relativity is about the relationship of time to space. General relativity is about curvature and the space represented by this hypergraph. - So, what is the curvature of a hypergraph? - Okay, so, first I have to explain, what I'm explaining is, first thing you have to have is a notion of dimension. You don't get to talk about curvature of things. If you say, "Oh, it's a curved line, "but I don't know what a line is yet." So- - Yeah, what is the dimension of a hypergraph then? Where, from where, we've talked about effective dimension, but- - Right, that's what this is about. What this is about is, you have your hypergraph, it's got a trillion nodes in it. What is it roughly like? Is it roughly like a grid, a two-dimensional grid? Is it roughly like all those nodes are arranged on a line? What's it roughly like? And there's a pretty simple mathematical way to estimate that by just looking at the, this thing I was describing, this sort of the size of a ball that you construct in the hypergraph. That's a, you just measure that, you can just compute it on a computer for a given hypergraph, and you can say, "Oh, this thing is wiggling around, "but it's roughly corresponds to two," or something like that, or roughly corresponds to 2.6, or whatever. So, that's how you have a notion of dimension in these hypergraphs. Curvature is something a little bit beyond that. It's, if you look at the, how the size of this ball increases as you increase its radius, curvature is a correction to the size increase associated with dimension. It's a sort of a second order term in determining size, just like the area of a circle is roughly pi r squared, so it goes up like r squared, the two is because it's in two dimensions. But when that circle is drawn on a big sphere, the actual formula is pi r squared times one minus r squared over a squared, and some coefficient. So, in other words, there's a correction to, and that correction term, that gives you curvature. And that correction term is what makes this hypergraph correspond, have the potential to correspond to curved space. Now, the next question is, is that curvature, is the way that curvature works, the way that Einstein's equations for general relativity, is it the way they say it should work? And the answer is yes. - Really? - And so, how does that work? - The calculation of the curvature of this hypergraph for some set of rules? - No, it doesn't matter what the rules are. It doesn't, so long as they have causal invariance and computational irreducibility, and they lead to finite dimensional space. Non-infinite dimensional space. - Oh, oh, dimensional. - Non-infinite dimensional. It can grow infinitely, but it can't be infinite dimensional. So-- - What does a infinitely dimensional hypergraph look like? So that means-- - For example, so in a tree, you start from one root of the tree, it doubles, doubles again, doubles again, doubles again, and that means if you ask the question, starting from a given point, how many points do you get to? Remember, like a circle, you get to r squared, the two there. On a tree, you get to, for example, two to the r. It's exponential dimensional, so to speak, or infinite dimensional. - Do you have a sense of, in the space of all possible rules, how many lead to infinitely dimensional hypergraphs? Is that-- - No. - Okay. Is that an important thing to know? - Yes, it's an important thing to know. I would love to know the answer to that. But it gets a little bit more complicated, because, for example, it's very possibly the case that in our physical universe, that the universe started infinite dimensional, and it only, as the big bang, it was very likely infinite dimensional. And as the universe sort of expanded and cooled, its dimension gradually went down. And so one of the bizarre possibilities, which actually there are experiments you can do to try and look at this, the universe can have dimension fluctuations. So in other words, we think we live in a three-dimensional universe, but actually there may be places where it's actually 3.01 dimensional, or where it's 2.99 dimensional. And it may be that in the very early universe, it was actually infinite dimensional, and it's only a late-stage phenomenon that we end up getting three-dimensional space. - But from your perspective of the hypergraph, one of the underlying assumptions you kind of implied, but you have a sense, a hope, set of assumptions that the rules that underlie our universe or the rule that underlies our universe is static. Is that one of the assumptions you're currently operating under? - Yes, but there's a footnote to that, which we should get to, 'cause it requires a few more steps. - Well, actually then, but let's backtrack to the curvature, because we're talking about as long as it's finite dimensional. - Finite dimensional computational irreducibility and causal invariance, then it follows that the large-scale structure will follow Einstein's equations. And now let me, again, qualify that a little bit more. There's a little bit more complexity to it. Okay, so Einstein's equations in their simplest form apply to the vacuum, no matter just the vacuum. And they say, in particular, what they say is if you have, so there's this term geodesic. That's a term that means shortest path, comes from measuring the shortest paths on the earth. So you look at a bunch of, a bundle of geodesics, a bunch of shortest paths. It's like the paths that photons would take between two points. Then the statement of Einstein's equations is basically a statement about a certain, that as you look at a bundle of geodesics, the structure of space has to be such that although the cross-sectional area of this bundle may, although the actual shape of the cross-section may change, the cross-sectional area does not. That's a version, that's the most simple-minded version of R mu nu minus a half R, g mu nu equals zero, which is the more mathematical version of Einstein's equations. It's a statement of the thing called the Ritchie tensor is equal to zero. That's Einstein's equations for the vacuum. Okay, so we get that as a result of this model, but footnote, big footnote, because all the matter in the universe is the stuff we actually care about, the vacuum is not stuff we care about. So the question is, how does matter come into this? And for that, you have to understand what energy is in these models. And one of the things that we realized, you know, last, late last year, was that there's a very simple interpretation of energy in these models, okay? And energy is basically, well, intuitively it's the amount of activity in these hypergraphs and the way that that remains over time. So a little bit more formally, you can think about this causal graph as having these edges that represent causal relationships. You can think about, oh boy, there's one more concept that we didn't get to. There's the notion of space-like hyper surfaces. So this is not as scary as it sounds. It's a common notion in general relativity. The notion is you're defining what is a possibly, what is, what, where in space-time might be a particular moment in time. So in other words, what is a consistent set of places where you can say this is happening now, so to speak, and you make this series of sort of slices through the space-time, through this causal graph, to represent sort of what we consider to be successive moments in time. It's somewhat arbitrary because you can deform that if you're going at a different speed in special activity, you tip those things. There are different kinds of deformations, but only certain deformations are allowed by the structure of the causal graph. Anyway, be as it may, the basic point is there is a way of figuring out, you say what is the energy associated with what's going on in this hypergraph? And the answer is there is a precise definition of that. And it is, the formal way to say it is it's the flux of causal edges through space-like hyper surfaces. The slightly less formal way to say it, it's basically the amount of activity. See, the reason it gets tricky is you might say it's the amount of activity per unit volume in this hypergraph, but you haven't defined what volume is. So it's a little bit, you have to be a little bit more-- - But this hyper surface gives some more formalism to that. - Yeah, yeah, it gives a way to connect that. - But intuitively we should think about as the just-- - The amount of activity. Right, so the amount of activity that kind of remains in one place in the hypergraph corresponds to energy. The amount of activity that is kind of where an activity here affects an activity somewhere else corresponds to momentum. And so one of the things that's kind of cool is that I'm trying to think about how to say this intuitively. The mathematics is easy, but the intuitive version, I'm not sure, but basically the way that things sort of stay in the same place and have activity is associated with rest mass. And so one of the things that you get to derive is E equals MC squared. That is a consequence of this interpretation of energy in terms of the way the causal graph works, which is the whole thing is sort of a consequence of this whole story about updates and hypergraphs and so on. - So can you linger on that a little bit? How do we get E equals MC squared? So where does the mass come from? - Okay, okay. - I mean, is there an intuitive, it's okay. First of all, you're pretty deep in the mathematical explorations of this thing right now. We're in a very, we're in a flux currently. So maybe you haven't even had time to think about intuitive explanations. But-- - Yeah, I mean, this one is, look, roughly what's happening, that derivation is actually rather easy. And everybody, and I've been saying we should pay more attention to this derivation because it's such, you know, 'cause people care about this one. But everybody says, "It's just easy." - It's easy. So there's some concept of energy that can be intuitively thought of as the activity, the flux, the level of changes that are occurring based on the transformations within a certain volume, however the heck do you find the volume. - Okay, so, and then mass-- - Well, mass is-- - What? - Mass is associated with kind of the energy that does not cause you to, that does not somehow propagate through time. Yeah, I mean, one of the things that was not obvious in the usual formulation of special relativity is that space and time are connected in a certain way, energy and momentum are also connected in a certain way. The fact that the connection of energy to momentum is analogous to the connection to space between space and time is not self-evident in ordinary relativity. It is a consequence of the way this model works. It's an intrinsic consequence of the way this model works. And it's all to do with that, with unraveling that connection that ends up giving you this relationship between energy and, well, it's energy, momentum, mass, they're all connected. - And so, like, hence the general relativity, you have a sense that it appears to be baked in to the fundamental properties of the way these hypergraphs are evolved. - Well, I didn't yet get to, so I got as far as special relativity and E equals MC squared. The one last step is, in general relativity, the final connection is energy and mass cause curvature in space. And that's something that, when you understand this interpretation of energy and you kind of understand the correspondence to curvature and hypergraphs, then you can finally sort of, the big final answer is you derive the full version of Einstein's equations for space, time, and matter. And that's-- - Is that, have you, that last piece with curvature, have, is that, have you arrived there yet? - Oh yeah, we're there, yes. And here's the way that we, here's how we're really, really going to know we've arrived, okay? So, you know, we have the mathematical derivation, it's all fine, but, you know, mathematical derivations, okay, so one thing that's sort of a, you know, we're taking this limit of what happens when, the limit, you have to look at things which are large compared to the size of an elementary length, small compared to the whole size of the universe, large compared to certain kinds of fluctuations, blah, blah, blah. There's a tower of many, many of these mathematical limits that have to be taken. So if you're a pure mathematician saying, where's the precise proof? It's like, well, there are all these limits, we can, you know, we can try each one of them computationally and we could say, yeah, it really works, but the formal mathematics is really hard to do. I mean, for example, in the case of deriving the equations of fluid dynamics from molecular dynamics, that derivation has never been done. There is no rigorous version of that derivation. So, so-- - Because you can't do the limits? - Yeah, 'cause you can't do the limits. - But so the limits allow you to try to describe something general about the system and very, very particular kinds of limits that you need to take with these very-- - Right, and the limits will definitely work the way we think they work, and we can do all kinds of computer experiments-- - It's just a hard derivation. - Yeah, it's just the mathematical structure kind of ends up running right into computational irreducibility, and you end up with a bunch of difficulty there. But here's the way that we're getting really confident that we know completely what we're talking about, which is when people study things like black hole mergers using Einstein's equations, what do they actually do? Well, they actually use Mathematica a whole bunch to analyze the equations and so on, but in the end, they do numerical relativity, which means they take these nice mathematical equations and they break them down so that they can run them on a computer, and they break them down into something which is actually a discrete approximation to these equations. Then they run them on a computer, they get results, then you look at the gravitational waves and you see if they match, okay? Turns out that our model gives you a direct way to do numerical relativity. So in other words, instead of saying you start from these continuum equations from Einstein, you break them down into these discrete things, you run them on a computer, you say, "We're doing it the other way around. "We're starting from these discrete things "that come from our model, "and we're just running big versions on a computer, "and what we're saying is, "and this is how things will work." So the way I'm calling this is proof by compilation, so to speak. - Proof by comp-- - That is, in other words, you're taking something where we've got this description of a black hole system, and what we're doing is we're showing that what we get by just running our model agrees with what you would get by doing the computation from the Einstein equations. - As a small tangent, or actually a very big tangent, but proof by compilation is a beautiful concept. In a sense, the way of doing physics with this model is by running it or compiling it. - At some level, yes. - Have you thought about, and these things can be very large, is there totally new possibilities of computing hardware and computing software which allows you to perform this kind of compilation? - Well-- - Algorithms, software, hardware? - So first comment is, these models seem to give one a lot of intuition about distributed computing, a lot of different intuition about how to think about parallel computation, and that particularly comes from the quantum mechanics side of things, which we didn't talk about much yet, but the question of what, given our current computer hardware, how can we most efficiently simulate things? That's actually partly a story of the model itself, because the model itself has deep parallelism in it. The ways that we are simulating it, we're just starting to be able to use that deep parallelism to be able to be more efficient in the way that we simulate things. But in fact, the structure of the model itself allows us to think about parallel computation in different ways, and one of my realizations is that, so it's very hard to get in your brain how you deal with parallel computation, and you're always worrying about, if multiple things can happen on different computers at different times, oh, what happens if this thing happens before that thing, and we have these race conditions where something can race to get to the answer before another thing, and you get all tangled up because you don't know which thing is gonna come in first. And usually when you do parallel computing, there's a big obsession to lock things down to the point where you've had locks and mutexes and God knows what else, where you've arranged it so that there can only be one sequence of things that can happen, so you don't have to think about all the different kinds of things that can happen. Well, in these models, physics is throwing us into, forcing us to think about all these possible things that can happen, but these models, together with what we know from physics, is giving us new ways to think about all possible things happening, about all these different things happening in parallel. And so I'm guessing-- - They have built-in protection for some of the parallelism. - Well, causal invariance is the built-in protection. Causal invariance is what means that even though things happen in different orders, it doesn't matter in the end. - As a person who struggled with concurrent programming in like Java, with all the basic concepts of concurrent programming, if there could be built up a strong mathematical framework for causal invariance, that's so liberating. And that could be not just liberating, but really powerful for massively distributed computation. - Absolutely. No, I mean, what's eventual consistency in distributed databases is essentially the causal invariance idea. - Yeah. - Okay, so that's-- - But have you thought about really large simulations? - Yeah, I mean, I'm also thinking about, look, the fact is, I've spent much of my life as a language designer, right? So I can't possibly not think about, what does this mean for designing languages for parallel computation? In fact, another thing that's one of these, I'm always embarrassed at how long it's taken me to figure stuff out. But back in the 1980s, I worked on trying to make up languages for parallel computation. I thought about doing graph rewriting. I thought about doing these kinds of things, but I couldn't see how to actually make the connections to actually do something useful. I think now physics is kind of showing us how to make those things useful. And so my guess is that in time, we'll be talking about, we do parallel programming, we'll be talking about programming in a certain reference frame, just as we think about thinking about physics in a certain reference frame. It's a certain co-ordinatization of what's going on. We say, we're gonna program in this reference frame. Oh, let's change the reference frame to this reference frame. And then our program will seem different and we'll have a different way to think about it, but it's still the same program underneath. - So let me ask on this topic, 'cause I put out that I'm talking to you. I got way more questions than I can deal with, but what pops to mind is a question somebody asked on Reddit, I think, is, please ask Dr. Wolfram, what are the specs of the computer running the universe? So we're talking about specs of hardware and software for simulations of a large scale thing. What about a scale that is comparative to something that eventually leads to the two of us talking about? - Right, right, right. So actually I did try to estimate that and we actually have to go a couple more stages before we can really get to that answer because we're talking about this thing. This is what happens when you build these abstract systems and you're trying to explain the universe. They're quite a number of levels deep, so to speak. But the-- - You mean conceptually or like literally, 'cause you're talking about small objects and there's 10 to the 120 something. - Yeah, right. It is conceptually deep. And one of the things that's happening sort of structurally in this project is, you know, there were ideas, there's another layer of ideas, there's another layer of ideas to get to the different things that correspond to physics. They're just different layers of ideas. And they are, you know, it's actually probably, if anything, getting harder to explain this project 'cause I'm realizing that the fraction of way through that I am so far in explaining this to you is less than, you know, it might be because, 'cause we know more now, you know, and every week basically we know a little bit more. And like-- - Those are just layers on the initial fundamental-- - Yes, but the layers are, you know, you might be asking me, you know, how do we get, you know, the difference between fermions and bosons, the difference between particles that can be all in the same state and particles that exclude each other, okay? Last three days we've kind of figured that out, okay? But, and it's very interesting, it's very cool. And it's very-- - And those are some kind of properties at a certain level, layer of abstraction on the hydrograph. - Yes, yes, and there's, but the layers of abstraction are kind of, they're compounding-- - Stacking up, so it's difficult, but-- - But okay, but this thing-- - But the specs nevertheless remain the same. - The specs underneath, so I have an estimate. So the question is, what are the units? So we've got these different fundamental constants about the world. So one of them is the speed of light, which is the, so the thing that's always the same in all these different ways of thinking about the universe is the notion of time, because time is computation. And so there's an elementary time, which is sort of the amount of time that we ascribe to elapsing in a single computational step, okay? So that's the elementary time. So then there's an elementary-- - That's a parameter or whatever, that's a constant. - It's whatever we define it to be, because I mean, we don't, you know-- - I mean, it's all relative, right? It doesn't matter. - Yes, it doesn't matter what it is, because we could be, it could be slow, it's just a number which we use to convert that to seconds, so to speak, because we are experiencing things, and we say this amount of time has elapsed, so to speak. - But we're within this thing, so it doesn't-- - It doesn't matter, right? But what does matter is the ratio, what we can, the ratio of the spatial distance in this hypergraph to this moment of time, again, that's an arbitrary thing, but we measure that in meters per second, for example, and that ratio is the speed of light. So the ratio of the elementary distance to the elementary time is the speed of light, okay? - Perfect. - And so there's another, there are two other levels of this, okay? So there is a thing which we can talk about, which is the maximum entanglement speed, which is a thing that happens at another level in this whole sort of story of how these things get constructed, that's a sort of maximum speed in quantum, in the space of quantum states, just as the speed of light is a maximum speed in physical space, this is a maximum speed in the space of quantum states, there's another level which is associated with what we call ruleal space, which is another one of these maximum speeds, we'll get to this. - So these are limitations on the system that are able to capture the kind of physical universe which we live in, the quantum mechanical-- - They are inevitable features of having a rule that has only a finite amount of information in the rule. So long as you have a rule that only involves a bounded amount, a limited amount of, only involving a limited number of elements, limited number of relations, it is inevitable that there are these speed constraints. We knew about the one for speed of light, we didn't know about the one for maximum entanglement speed, which is actually something that is possibly measurable, particularly in black hole systems and things like this. But anyway, this is long story short, you're asking what the processing specs of the universe, of the sort of computation of the universe, there's a question of even what are the units of some of these measurements, okay? So the units I'm using are Wolfram language instructions per second, okay? Because you gotta have some, you know, what computation are you doing? - There gotta be some kind of frame of reference. - Right, right, so because it turns out in the end, there will be, there's sort of an arbitrariness in the language that you use to describe the universe. So in those terms, I think it's like 10 to the 500 Wolfram language operations per second, I think, is the, I think it's of that order. You know, basically-- - So that's the scale of the computation. What about memory, if there's an interesting thing to say about storage and memory? - Well, there's a question of how many sort of atoms of space might there be? You know, maybe 10 to the 400. We don't know exactly how to estimate these numbers. I mean, this is based on some, I would say, somewhat rickety way of estimating things. When there start to be able to be experiments done, if we're lucky, there will be experiments that can actually nail down some of these numbers. - And because of computation reducibility, there's not much hope for very efficient compression, like very efficient representation of this atom space? - Good question, good question. I mean, there's probably certain things, you know. The fact that we can deduce, okay, the question is how deep does the reducibility go? - Right. - Okay? And I keep on being surprised that it's a lot deeper than I thought, okay? And so one of the things is that there's a question of sort of how much of the whole of physics do we have to be able to get in order to explain certain kinds of phenomena? Like, for example, if we want to study quantum interference, do we have to know what an electron is? Turns out I thought we did, turns out we don't. I thought to know what energy is, we would have to know what electrons were. We don't. - So you can get a lot of really powerful shortcuts. - Right, there's a bunch of sort of bulk information about the world. The thing that I'm excited about last few days, okay, is the idea of fermions versus bosons, fundamental idea that, I mean, it's the reason we have matter that doesn't just self-destruct is because of the exclusion principle that means that two electrons can never be in the same quantum state. - Is it useful for us to maybe first talk about how quantum mechanics fits into the Wolfram physics model? - Yes. - Let's go there. So we talked about general relativity. Now, what have you found from quantum mechanics? - What's the story of quantum mechanics? Right. - Within and outside of the Wolfram physics. - Right, so I mean, the key idea of quantum mechanics, that sort of the typical interpretation is classical physics says a definite thing happens. Quantum physics says there's this whole set of paths of things that might happen, and we are just observing some overall probability of how those paths work. Okay, so when you think about our hypergraphs and all these little updates that are going on, there's a very remarkable thing to realize, which is if you say, well, which particular sequence of updates should you do? Say, well, it's not really defined. You can do any of a whole collection of possible sequences of updates. Okay, that set of possible sequences of updates defines yet another kind of graph that we call a multi-way graph. And a multi-way graph just is a graph where at every node, there is a choice of several different possible things that could happen. So for example, you go this way, you go that way, those are two different edges in the multi-way graph, and you're building up the set of possibilities. So actually, like for example, I just made the one, the multi-way graph for tic-tac-toe, okay? So tic-tac-toe, you start off with some board that, you know, everything is blank, and then somebody can put down an X somewhere, an O somewhere, and then there are different possibilities. At each stage, there are different possibilities. And so you build up this multi-way graph of all those possibilities. Now, notice that even in tic-tac-toe, you have the feature that there can be something where you have two different things that happen, and then those branches merge because you end up with the same shape, you know, the same configuration of the board, even though you got there in two different ways. So the thing that's sort of an inevitable feature of our models is that just like quantum mechanics suggests, definite things don't happen. Instead, you get this whole multi-way graph of all these possibilities. Okay, so then the question is, so, okay, so that's sort of a picture of what's going on. Now you say, okay, well, quantum mechanics has all these features of, you know, all this mathematical structure and so on. How do you get that mathematical structure? Okay, a couple of things to say. So quantum mechanics is actually, in a sense, two different theories glued together. Quantum mechanics is the theory of how quantum amplitudes work that more or less give you the probabilities of things happening. And it's the theory of quantum measurement, which is the theory of how we actually conclude definite things, because the mathematics just gives you these quantum amplitudes, which are more or less probabilities of things happening, but yet we actually observe definite things in the world. Quantum measurement has always been a bit mysterious. It's always been something where people just say, well, the mathematics says this, but then you do a measurement and there are philosophical arguments about what the measurement is. But it's not something where there's a theory of the measurement. - Somebody on Reddit also asked, please ask Stephen to tell his story of the double slit experiment. - Okay, yeah, I can. - Does that make sense? - Oh yeah, it makes sense. Absolutely makes sense. - Why, is this like a good way to discuss? - A little bit. Let me explain a couple of things first. So the structure of quantum mechanics is mathematically quite complicated. One of the features, let's see, well, how to describe this. Okay, so first point is there's this multi-way graph of all these different paths of things that can happen in the world. And the important point is that these, you can have branchings and you can have mergings. Okay, so this property turns out causal invariance is the statement that the number of mergings is equal to the number of branchings. So in other words, every time there's a branch, eventually there will also be a merge. In other words, every time there were two possibilities for what might've happened, eventually those will merge. - Beautiful concept, by the way. But yeah, yeah, yeah. - So that idea, okay. So then, so that's one thing, and that's closely related to the sort of objectivity in quantum mechanics. The fact that we believe definite things happen, it's because although there are all these different paths, in some sense, because of causal invariance, they all imply the same thing. I'm cheating a little bit in saying that, but that's roughly the essence of what's going on. Okay, next thing to think about is we have this multi-way graph. It has all these different possible things that are happening. Now we ask, this multi-way graph is sort of evolving with time. Over time, it's branching, it's merging, it's doing all these things, okay? Question we can ask is, if we slice it at a particular time, what do we see? And that slice represents, in a sense, something to do with the state of the universe at a particular time. So in other words, we've got this multi-way graph of all these possibilities, and then we're asking, okay, we take this slice. This slice represents, okay, each of these different paths corresponds to a different quantum possibility for what's happening. When we take the slice, we're saying, what are the set of quantum possibilities that exist at a particular time? - And when you say slice, you slice the graph, and then there's a bunch of leaves. - A bunch of leaves. - And those represent the state of things. - Right, but then, okay, so the important thing that you are quickly picking up on is that what matters is kind of how these leaves are related to each other. So a good way to tell how leaves are related is just to say, on the step before, did they have a common ancestor? So two leaves might be, they might have just branched from one thing, or they might be far away, way far apart in this graph, where to get to a common ancestor, maybe you have to go all the way back to the beginning of the graph, all the way back to the beginning of the-- - So there's some kind of measure of distance. - Right, but what you get is by making the slice, we call it branchial space, the space of branches. And in this branchial space, you have a graph that represents the relationships between these quantum states in branchial space. You have this notion of distance in branchial space. Okay, so-- - It's connected to quantum entanglement? - Yes, yes. It's basically, the distance in branchial space is kind of an entanglement distance. So this-- - That's a very nice model. - Right, it is very nice. It's very beautiful. I mean, it's so clean. I mean, it's really, you know, and it tells one, okay, so anyway, so then this branchial space has this sort of map of the entanglements between quantum states. So in physical space, we have, so you know, you can say, take, let's say the causal graph, and we can slice that at a particular time, and then we get this map of how things are laid out in physical space. When we do the same kind of thing, there's a thing called the multi-way causal graph, which is the analog of a causal graph for the multi-way system. We slice that, we get essentially the relationships between things, not in physical space, but in the space of quantum states. It's like which quantum state is similar to which other quantum state. Okay, so now I think next thing to say is just to mention how quantum measurement works. So quantum measurement has to do with reference frames in branchial space. So, okay, so measurement in physical space, it matters whether how we assign spatial position and how we define coordinates in space and time. And that's how we make measurements in ordinary space. So we're making a measurement based on us sitting still here, are we traveling at half the speed of light and making measurements that way? These are different reference frames in which we're making our measurements. And the relationship between different events and different points in space and time will be different depending on what reference frame we're in. Okay, so then we have this idea of quantum observation frames, which are the analog of reference frames, but in branchial space. And so what happens is what we realize is that a quantum measurement is, the observer is sort of arbitrarily determining this reference frame. The observer is saying, I'm going to understand the world by saying that space and time are coordinatized this way. I'm gonna understand the world by saying that quantum states and time are coordinatized in this way. And essentially what happens is that, you know, the process of quantum measurement is a process of deciding how you slice up this multi-way system in these quantum observation frames. So in a sense, the observer, the way the observer enters is by their choice of these quantum observation frames. And what happens is that the observer, because, okay, this is again, another stack of other concepts, but anyway, because the observer is computationally bounded, there is a limit to the type of quantum observation frames that they can construct. - Interesting, okay. So there's, okay, so there's some constraints, some limit on- - On the choice of observation frames. - Right, and by the way, I just want to mention that there's a, I mean, it's bizarre, but there's a hierarchy of these things. So in thermodynamics, the fact that we believe entropy increases, we believe things get more disordered is a consequence of the fact that we can't track each individual molecule. If we could track every single molecule, we could run every movie in reverse, so to speak, and we would, you know, that we would not see that things are getting more disordered but it's because we are computationally bounded. We can only look at these big blobs of what all these molecules collectively do that we think that things are, that we describe it in terms of entropy increasing and so on. And it's the same phenomenon, basically, also a consequence of computational irreducibility that causes us to basically be forced to conclude that definite things happen in the world, even though there's this quantum, you know, this set of all these different quantum processes that are going on. So, I mean, I'm skipping a little bit, but that's a rough picture. - And in the evolution of the Wolfram Physics Project, where do you feel you stand on some of the puzzles that are along the way? See, you're skipping along a bunch of, you're skipping a bunch of stuff. - Oh, it's amazing how much these things are unraveling. I mean, you know, these things, look, it used to be the case that I would agree with Dick Feynman, nobody understands quantum mechanics, including me, okay? I'm getting to the point where I think I actually understand quantum mechanics. My exercise, okay, is can I explain quantum mechanics for real at the level of kind of middle school type explanation? And I'm getting closer, it's getting there. I'm not quite there, I've tried it a few times, and I realized that there are things that where I have to start talking about elaborate mathematical concepts and so on. But I think, and you've got to realize that it's not self-evident that we can explain, you know, at an intuitively graspable level, something which, you know, about the way the universe works. The universe wasn't built for our understanding, so to speak. But I think then, okay, so another important idea is this idea of branchial space, which I mentioned, this sort of space of quantum states. It is, okay, so I mentioned Einstein's equations describing, you know, the effect of mass and energy on trajectories of particles, on geodesics. The curvature of physical space is associated with the presence of energy, according to Einstein's equations, okay? So it turns out that, rather amazingly, the same thing is true in branchial space. So it turns out the presence of energy, or more accurately, Lagrangian density, which is a kind of relativistic, invariant version of energy, the presence of that causes, essentially, deflection of geodesics in this branchial space, okay? So you might say, so what? Well, it turns out that the sort of, the best formulation we have of quantum mechanics, this Feynman path integral, is a thing that describes quantum processes in terms of mathematics that can be interpreted as, well, in quantum mechanics, the big thing is you get these quantum amplitudes, which are complex numbers that represent, when you combine them together, represent probabilities of things happening. And so the big story has been, how do you derive these quantum amplitudes? And people think these quantum amplitudes, they have a complex number, has a real part and an imaginary part. You can also think of it as a magnitude and a phase. And people have sort of thought these quantum amplitudes have magnitude and phase, and you compute those together. Turns out that the magnitude and the phase come from completely different places. The magnitude comes, okay, so what do you, how do you compute things in quantum mechanics? Roughly, I'm telling you, I'm getting there to be able to do this at a middle school level, but I'm not there yet. Roughly what happens is you're asking, does this state in quantum mechanics evolve to this other state in quantum mechanics? And you can think about that like a particle traveling or something traveling through physical space, but instead it's traveling through branchial space. And so what's happening is, does this quantum state evolve to this other quantum state? It's like saying, does this object move from this place in space to this other place in space? Okay, now the way that you, these quantum amplitudes characterize kind of to what extent the thing will successfully reach some particular point in branchial space. Just like in physical space, you could say, oh, it had a certain velocity and it went in this direction. In branchial space, there's a similar kind of concept. - Is there a nice way to visualize, for me now, mentally branchial space? - It's just, you have this hypergraph, sorry, you have this multi-way graph. It's this big branching thing, branching and merging thing. - But I mean, like moving through that space. I'm just trying to understand what that looks like. (both laugh) - You know, that space is probably exponential dimensional, which makes it, again, another can of worms in understanding what's going on. That space, as in ordinary space, this hypergraph, the spatial hypergraph, limits to something which is like a manifold, like something like three-dimensional space. Almost certainly, the multi-way graph limits to a Hilbert space, which is something that, I mean, it's just a weird exponential dimensional space. And by the way, you can ask, I mean, there are much weirder things that go on. For example, one of the things I've been interested in is the expansion of the universe in branchial space. So we know the universe is expanding in physical space, but the universe is probably also expanding in branchial space. - Expanding in branchial space. - So that means the number of quantum states of the universe is increasing with time. - The diameter of the thing is growing. - Right, so that means that the, and by the way, this is related to whether quantum computing can ever work. And-- - Why? - Okay, so let me explain why. So let's talk about, okay, so first of all, just to finish the thought about quantum amplitudes, the incredibly beautiful thing, but I'm just very excited about this. The Feynman path integral is this formula. It says that the amplitude, the quantum amplitude is e to the i s over h bar, where s is this thing called the action. And it, okay, so that can be thought of as representing a deflection of the angle of this path in the multi-way graph. So it's a deflection of a geodesic in the multi-way path that is caused by this thing called the action, which is essentially associated with the energy, okay? And so this is a deflection of a path in branchial space that is described by this path integral, which is the thing that is the mathematical essence of quantum mechanics. Turns out that deflection is, the deflection of geodesics in branchial space follows the exact same mathematical setup as the deflection of geodesics in physical space, except the deflection of geodesics in physical space is described with Einstein's equations. The deflection of geodesics in branchial space is defined by the Feynman path integral, and they are the same. In other words, they are mathematically the same. So that means that general relativity is a story of essentially motion in physical space. Quantum mechanics is a story of essentially motion in branchial space. And the underlying equation for those two things, although it's presented differently because one's interested in different things in branchial space than physical space, but the underlying equation is the same. So in other words, it's just, you know, these two theories, which are those two sort of pillars of 20th century physics, which have seemed to be off in different directions are actually facets of the exact same theory. And this, I mean-- - That's exciting to see where that evolves, and exciting that that just is there. - Right, I mean, to me, you know, look, having spent some part of my early life, you know, working in the context of these theories of, you know, 20th century physics, it's, they just, they seem so different. And the fact that they're really the same is just really amazing. Actually, you mentioned double-slit experiment, okay? So the double-slit experiment is an interference phenomenon where you say there are, you know, you can have a photon or an electron, and you say there are these two slits, it could have gone through either one, but there is this interference pattern where there's destructive interference, where you might've said in classical physics, oh, well, if there are two slits, then there's a better chance that it gets through one or the other of them. But in quantum mechanics, there's this phenomenon of destructive interference that means that even though there are two slits, two can lead to nothing, as opposed to two leading to more than, for example, one slit. And what happens in this model, and we've just been understanding this in the last few weeks, actually, is that the, what essentially happens is that the double-slit experiment is a story of the interface between branchial space and physical space. And what's essentially happening is that the destructive interference is the result of the two possible paths associated with photons going through those two slits, winding up at opposite ends of branchial space. And so they don't, and so that's why there's sort of nothing there when you look at it, is because these two different sort of branches couldn't get merged together to produce something that you can measure in physical space. - Is there a lot to be understood about branchial space? I guess, mathematically speaking. - Yes, it's a very beautiful mathematical thing, and it's very, I mean, by the way, this whole theory is just amazingly rich in terms of the mathematics that it says should exist, okay? So for example, calculus, you know, is the story of infinitesimal change in integer-dimensional space, one-dimensional, two-dimensional, three-dimensional space. We need a theory of infinitesimal change in fractional-dimensional and dynamic-dimensional space. No such theory exists. - So there's tools in mathematics that are needed here. - Right. - And this is the motivation for that, actually. - Right, and it's, you know, there are indications, and we can do computer experiments, and we can see how it's gonna come out, but we need to, you know, the actual mathematics doesn't exist. And in branchial space, it's actually even worse. There's even more sort of layers of mathematics that are, you know, we can see how it works, roughly, by doing computer experiments, but to really understand it, we need more sort of mathematical sophistication. - So quantum computers. - Okay, so the basic idea of quantum computers, the promise of quantum computers is quantum mechanics does things in parallel, and so you can sort of intrinsically do computations in parallel, and somehow that can be much more efficient than just doing them one after another. And, you know, I actually worked on quantum computing a bit with Dick Feynman back in 1980, one, two, three, that kind of timeframe, and we-- - A fascinating image. You and Feynman work on quantum computers. - Well, we tried to work, the big thing we tried to do was invent a randomness chip that would generate randomness at a high speed using quantum mechanics. And the discovery that that wasn't really possible was part of the story of, we never really wrote anything about it. I think maybe he wrote some stuff, but we didn't write stuff about what we figured out about sort of the fact that it really seemed like the measurement process in quantum mechanics was a serious damper on what was possible to do in sort of, you know, the possible advantages of quantum mechanics for computing. But anyway, so the sort of the promise of quantum computing is let's say you're trying to, you know, factor an integer. Well, you can, instead of, you know, when you factor an integer, you might say, well, does this factor work? Does this factor work? Does this factor work? In ordinary computing, it seems like we pretty much just have to try all these different factors, you know, kind of one after another. But in quantum mechanics, you might have the idea, oh, you can just sort of have the physics, try all of them in parallel, okay? And the, you know, and there's this algorithm, Shor's algorithm, which allows you, according to the formalism of quantum mechanics, to do everything in parallel and to do it much faster than you can on a classical computer. Okay, the only little footnote is, you have to figure out what the answer is. You have to measure the results. So the quantum mechanics internally has figured out all these different branches, but then you have to pull all these branches together to say, and the classical answer is this, okay? The standard theory of quantum mechanics does not tell you how to do that. It tells you how the branching works, but it doesn't tell you the process of corralling all these things together. And that process, which intuitively you can see is going to be kind of tricky, but our model actually does tell you how that process of pulling things together works. And the answer seems to be, we're not absolutely sure. We've only got to two times three so far, which is kind of in this factorization in quantum computers, but we can, what seems to be the case is that the advantage you get from the parallelization from quantum mechanics is lost from the amount that you have to spend pulling together all those parallel threads to get to a classical answer at the end. Now, that phenomenon is not unrelated to various decoherence phenomena that are seen in practical quantum computers and so on. I mean, I should say as a very practical point, I mean, it's like, should people stop bothering to do quantum computing research? No, because what they're really doing is they're trying to use physics to get to a new level of what's possible in computing. And that's a completely valid activity. Whether you can really put, whether you can say, oh, you can solve an NP complete problem, you can reduce exponential time to polynomial time, we're not sure. And I'm suspecting the answer is no, but that's not relevant to the practical speed-ups you can get by using different kinds of technologies, different kinds of physics to do basic computing. - So you're saying, I mean, some of the models you're playing with, the indication is that to get all the sheep back together and to corral everything together to get the actual solution to the algorithm is-- - You lose all the-- - You lose all the-- - By the way, I mean, so again, this question, do we actually know what we're talking about about quantum computing and so on? So again, we're doing proof by compilation. So we have a quantum computing framework in Wolfram language, which is a standard quantum computing framework that represents things in terms of the standard formalism of quantum mechanics. And we have a compiler that simply compiles the representation of quantum gates into multi-way systems. So, and in fact, the message that I got was from somebody who's working on the project who has managed to compile one of the sort of a core formalism based on category theory and core quantum formalism into multi-way systems. So this-- - When you say multi-way system, these multi-way graphs? - Yes, yes. - So you're, okay, that's awesome. And then you can do all kinds of experiments on that multi-way graph. - Right, but the point is that what we're saying is the thing, we've got this representation of let's say Shor's algorithm in terms of standard quantum gates. And it's just a pure matter of sort of computation to just say that is equivalent. We will get the same result as running this multi-way system. - Can you do complexity analysis on that multi-way system? - Well, that's what we've been trying to do, yes. We're getting there. We haven't done that yet. I mean, there's a pretty good indication of how that's gonna work out. And we've done, as I say, our computer experiments, we've unimpressively gotten to about two times three in terms of factorization, which is kind of about how far people have got with physical quantum computers as well. But yes, we will be able to, we definitely will be able to do complexity analysis and we will be able to know. So the one remaining hope for quantum computing really, really working at this formal level of quantum brand exponential stuff being done in polynomial time and so on, the one hope, which is very bizarre, is that you can kind of piggyback on the expansion of Brownshield space. So here's how that might work. So you think, you know, energy conservation, standard thing in high school physics, energy is conserved, right? But now you imagine, you think about energy in the context of cosmology, in the context of the whole universe. It's a much more complicated story. The expansion of the universe kind of violates energy conservation. And so, for example, if you imagine you've got two galaxies, they're receding from each other very quickly. They've got two big central black holes. You connect a spring between these two central black holes. Not easy to do in practice, but let's imagine you could do it. Now that spring is being pulled apart. It's getting more potential energy in the spring as a result of the expansion of the universe. So in a sense, you are piggybacking on the expansion that exists in the universe and the sort of violation of energy conservation that's associated with that cosmological expansion to essentially get energy. You're essentially building a perpetual motion machine by using the expansion of the universe. And that is a physical version of that. It is conceivable that the same thing could be done in branchial space to essentially mine the expansion of the universe in branchial space as a way to get the sort of quantum computing for free, so to speak, just from the expansion of the universe in branchial space. Now, the physical space version is kind of absurd and involves springs between black holes and so on. It's conceivable that the branchial space version is not as absurd and that it's actually something you can reach with physical things you can build in labs and so on. We don't know yet. - Okay, so like you were saying, the branchial space might be expanding and there might be something that could be exploited. - Right, in the same kind of way that you can exploit that expansion of the universe in principle in physical space. - You just have like a glimmer of hope. - Right, I think that the, look, I think the real answer is going to be that for practical purposes, the official brand that says you can do exponential things in polynomial time is probably not gonna work. - For people curious to kind of learn more, so this is more like, this is not middle school. We're gonna go to elementary school for a second. Maybe middle school, let's go to middle school. So if I were to try to maybe write a pamphlet of like Wolfram Physics Project for dummies, aka for me, or maybe make a video on the basics, but not just the basics of the physics project, but the basics plus the most beautiful central ideas, how would you go about doing that? Could you help me out a little bit? - Yeah, yeah, I mean, as a really practical matter, we have this kind of visual summary picture that we made, which I think is a pretty good, when I've tried to explain this to people, and it's a pretty good place to start, is you got this rule, you apply the rule, you're building up this big hypergraph, you've got all these possibilities, you're kind of thinking about that in terms of quantum mechanics. I mean, that's a decent place to start. - So basically, the things we've talked about, which is space represented as a hypergraph, transformation of that space is kind of time. - Yes. - And then-- - Structure of that space, the curvature of that space is gravity. That can be explained without going anywhere near quantum mechanics. I would say that's actually easier to explain than special relativity. - Oh, so going into general, so going to curvature? - Yeah, I mean, special relativity, I think, is it's a little bit elaborate to explain. And honestly, you only care about it if you know about special relativity, if you know how special relativity is ordinarily derived and so on. - So general relativity is easier? - Is easier, yes. - And then what about quantum, what's the easiest way to reveal-- - I think the basic point is just this fact that there are all these different branches, that there's this kind of map of how the branches work, and that, I mean, I think actually the recent things that we have about the double slit experiment are pretty good, 'cause you can actually see this, you can see how the double slit phenomenon arises from just features of these graphs. Now, having said that, there is a little bit of sleight of hand there, because the true story of the way that double slit thing works depends on the co-ordinatization of branchial space that, for example, in our internal team, there is still a vigorous battle going on about how that works. And what's becoming clear is, I mean, what's becoming clear is that it's mathematically really quite interesting. I mean, that is that there's a, you know, it involves essentially putting space-filling curves, you basically have a thing which is naturally two-dimensional, and you're sort of mapping it into one dimension with a space-filling curve, and it's like, why is it this space-filling curve and not another space-filling curve? And that becomes a story about Riemann surfaces and things, and it's quite elaborate. But there's a more, a little bit sleight of hand way of doing it where it's, you know, it's surprisingly direct. - So a question that might be difficult to answer, but for several levels of people, could you give me advice on how we can learn more? Specifically, there is people that are completely outside and just curious and are captivated by the beauty of hypergraphs, actually. So people that just want to explore, play around with this. Second level is people from, say, people like me, who somehow got a PhD in computer science, but are not physicists, but fundamentally, the work you're doing is of computational nature, so it feels very accessible. - Yes. - So what can a person like that do to learn enough physics or not to be able to, one, explore the beauty of it, and two, the final level of contribute something of a level of even publishable, you know, like strong, interesting ideas. So at all those layers, complete beginner, kind of a CS person, and a CS person that wants to publish. - Right, I mean, I think that, you know, I've written a bunch of stuff. A person called Jonathan Gorod, who's been a key person working on this project, has also written a bunch of stuff. And some other people started writing things, too. - And he's a physicist. - Physicist, well, I would say a mathematical physicist. - Mathematical physicist. - Pretty mathematically sophisticated. He regularly out-mathematicizes me. - Yeah, strong mathematical physicist, yeah. I looked at some of the papers. - Right, but so, I mean, you know, I wrote this kind of original announcement blog post about this project, which people seem to have found. I've been really happy, actually, that people who, you know, people seem to have grokked key points from that. Much deeper key points people seem to have grokked than I thought they would grokk. - And that's a kind of a long blog post that explains some of the things we talked about, like the hypergraph and the basic rules. - Yes, yes. - And I don't, does it, I forget, it doesn't have any quantum mechanics in it. - Oh, yeah, yeah, it does. It does have quantum mechanics, yes, it does. But we know a little bit more since that blog post that probably clarifies, but that blog post does a pretty decent job. And, you know, talking about things like, again, something we didn't mention, the fact that the uncertainty principle is a consequence of curvature in Brown-Shield space. - How much physics should a person know to be able to understand the beauty of this framework and to contribute something novel? - Okay, so I think that those are different questions. So, I mean, I think that the why does this work, why does this make any sense, to really know that, you have to know a fair amount of physics, okay? And for example, have a decent-- - When you say why does this work, you're referring to the connection between this model and-- - General relativity, for example. - And general relativity. - Like you have to understand something about general relativity. - There's also a side of this where just, as the pure mathematical framework is fascinating. - Yes. - If you throw the physics out completely. - Right, then it's quite accessible to, I mean, you know, I wrote this sort of long technical introduction to the project, which seems to have been very accessible to people who are, you know, who understand computation and formal abstract ideas, but are not specialists in physics or other kinds of things. I mean, the thing with the physics part of it is, you know, there's both a way of thinking and a literally a mathematical formalism. I mean, it's like, you know, to know that we get the Einstein equations, to know we get the energy momentum tensor, you kind of have to know what the energy momentum tensor is, and that's physics. I mean, that's kind of graduate level physics, basically. And so that, you know, making that final connection is, requires some depth of physics knowledge. - I mean, that's the unfortunate thing, the difference in machine learning and physics in the 21st century. Is it really out of reach of a year or two worth of study? - No, you could get it in a year or two, but you can't get it in a month. - Right. - I mean, it-- - But it doesn't require necessarily like 15 years. - No, it does not. And in fact, a lot of what has happened with this project makes a lot of this stuff much more accessible. There are things where it has been quite difficult to explain what's going on, and it requires much more, you know, having the concreteness of being able to do simulations, knowing that this thing that you might've thought was just an analogy is really actually what's going on, makes one feel much more secure about just sort of saying, this is how this works. And I think it will be, you know, I'm hoping the textbooks of the future, the physics textbooks of the future, there will be a certain compression. There will be things that used to be very much more elaborate, because for example, even doing continuous mathematics versus this discrete mathematics, that, you know, to know how things work in continuous mathematics, you have to be talking about stuff and waving your hands about things. Whereas with the discrete version, it's just like, here is a picture, this is how it works. And there's no, oh, did we get the limit right? Did this, you know, did this thing that is of, you know, zero, you know, measure zero object, you know, interact with this thing in the right way. You don't have to have that whole discussion. It's just like, here's a picture, you know, this is what it does. And, you know, you can, then it takes more effort to say, what does it do in the limit when the picture gets very big? - But you can do experiments to build up an intuition actually. - Yes, right. And you can get sort of core intuition for what's going on. Now, in terms of contributing to this, the, you know, I would say that the study of the computational universe and how all these programs work in the computational universe, there's just an unbelievable amount to do there. And it is very close to the surface. That is, you know, high school kids, you can do experiments. It's not, you know, and you can discover things. I mean, you know, we, you can discover stuff about, I don't know, like this thing about expansion of branchial space. That's an absolutely accessible thing to look at. Now, you know, the main issue with doing these things is not, there isn't a lot of technical depth difficulty there. The actual doing of the experiments, you know, all the code is all on our website to do all these things. The real thing is sort of the judgment of what's the right experiment to do. How do you interpret what you see? That's the part that, you know, people will do amazing things with, and that's the part that, but it isn't like you have to have done 10 years of study to get to the point where you can do the experiments. You don't. - That's the cool thing. You can do experiments day one, basically. It's that, that's the amazing thing about, and you've actually put the tools out there. It's beautiful, it's mysterious. There's still, I would say, maybe you can correct me. It feels like there's a huge number of low-hanging fruit on the mathematical side, at least, not the physics side, perhaps. - No, look, on the, okay, on the physics side, we are, we're definitely in harvesting mode, you know. - Of which fruit, the low-hanging ones or? - The low-hanging ones, yeah, right. I mean, basically, here's the thing. There's a certain list of, you know, here are the effects in quantum mechanics, here are the effects in general relativity. It's just like industrial harvesting. It's like, can we get this one, this one, this one, this one, this one? And the thing that's really, you know, interesting and satisfying, and it's like, you know, is one climbing the right mountain? Does one have the right model? The thing that's just amazing is, you know, we keep on like, are we gonna get this one? How hard is this one? It's like, oh, you know, it looks really hard, it looks really hard. Oh, actually, we can get it. And-- - And you're continually surprised. I mean, it seems like, I've been following your progress. It's kind of exciting, all the, in harvesting mode, all the things you're picking up along the way. - Right, right, no, I mean, it's the thing that is, I keep on thinking it's gonna be more difficult than it is. Now, that's a, you know, that's a, who knows what, I mean, the one thing, so the thing that's been, was a big thing that I think we're pretty close to, I mean, I can give you a little bit of the roadmap, it's sort of interesting to see, is like, what are particles? What are things like electrons? How do they really work? - Are you close to get, like, what's, are you close to trying to understand, like, the atom, the electrons, neutrons, protons, like just particles? - Okay, so this is, this is the stack. So the first thing we want to understand is the quantization of spin. So particles, they kind of spin, they have a certain angular momentum. That angular momentum, even though the masses of particles are all over the place, you know, the electron has a mass of 0.511 MeV, but, you know, the proton is 938 MeV, et cetera, et cetera, et cetera, they're all kind of random numbers. The spins of all these particles are either integers or half integers. And that's a fact that was discovered in the 1920s, I guess. I think that we are close to understanding why spin is quantized. And that's a, and it appears to be a quite elaborate mathematical story about homotopy groups in crystal space and all kinds of things. But bottom line is that seems within reach. And that's a big deal because that's a very core feature of understanding how particles work in quantum mechanics. Another core feature is this difference between particles that obey the exclusion principle and sort of stay apart, that leads to the stability of matter and things like that, and particles that love to get together and be in the same state, things like photons, that, and that's what leads to phenomena like lasers, where you can get sort of coherently everything in the same state. That difference is the particles of integer spin or bosons like to get together in the same state, the particles of half integers spin, of fermions, like electrons, that they tend to stay apart. And so the question is, can we get that in our models? And just the last few days, I think we made, I mean, I think the story of, I mean, it's one of these things where we're really close. - Is this connected to fermions and bosons? You were talking about? - Yeah, yeah, yeah. So this was, what happens is what seems to happen, okay? It's subject to revision even the next few days. But what seems to be the case is that bosons are associated with essentially merging in multi-way graphs and fermions are associated with branching in multi-way graphs. And that essentially the exclusion principle is the fact that in branchial space, things have a certain extent in branchial space that in which things are being sort of forced apart in branchial space, whereas the case of bosons, they get, they come together in branchial space. And the real question is, can we explain the relationship between that and these things called spinners, which are the representation of half-integer spin particles that have this weird feature that usually when you go around 360 degree rotation, you get back to where you started from. But for a spinner, you don't get back to where you started from. It takes 720 degrees of rotation to get back to where you started from. And we are just, it feels like we are, we're just incredibly close to actually having that, understanding how that works. And it turns out, it looks like, my current speculation is, that it's as simple as the directed hypergraphs versus undirected hypergraphs. - Oh, interesting. - The relationship between spinners and vectors. So, which is just-- - Nice, interesting. Yeah, that would be interesting if these are all these kind of nice properties of this multigraphs of branching and rejoining. - Spinners have been very mysterious. And if that's what they turn out to be, there's gonna be an easy explanation of what's going on. - Yeah, if it's directed versus undirected. - It's just, and that's why there's only two different cases. - Why are spinners important in quantum mechanics? Can you just give a-- - Yeah, so spinners are important because they are the representation of electrons which have half an integer spin. They are the wave functions of electrons are spinners. Just like the wave functions of photons are vectors, the wave functions of electrons are spinners. And they have this property that when you rotate by 360 degrees, they come back to minus one of themselves and take 720 degrees to get back to the original value. And they are a consequence of, we usually think of rotation in space as being when you have this notion of rotational invariance and rotational invariance, as we ordinarily experience it, doesn't have the feature. If you go through 360 degrees, you go back to where you started from, but that's not true for electrons. And so that's why understanding how that works is important. - Yeah, I've been playing with Mobius strip quite a bit lately just for fun. - Yes, yes. - It has the same kind of funky properties. - Yes, right, exactly. You can have this, the so-called belt trick, which is this way of taking an extended object and you can see properties like spinners with that kind of extended object that-- - Yeah, it would be very cool if it somehow connects the directive versus undirected. - I think that's what it's gonna be. I think it's gonna be as simple as that, but we'll see. I mean, this is the thing that, this is the big sort of bizarre surprise is that, because I learned physics as probably, let's say a fifth generation in the sense that, if you go back to the 1920s and so on, there were the people who were originating quantum mechanics and so on. Maybe it's a little less than that. Maybe I was like a third generation or something. I don't know, but the people from whom I learned physics were the people who have been students of the students of the people who originated the current understanding of physics. And we're now at probably the seventh generation of physicists or something from the early days of 20th century physics. And whenever a field gets that many generations deep, it seems the foundations seem quite inaccessible and it seems like you can't possibly understand that. We've gone through seven academic generations and that's been this thing that's been difficult to understand for that long. It just can't be that simple. - But in a sense, maybe that journey takes you to a simple explanation that was there all along. That's the whole-- - Right, right, right. I mean, and the thing for me personally, the thing that's been quite interesting is, I didn't expect this project to work in this way. But I had this sort of weird piece of personal history that I used to be a physicist and I used to do all this stuff. And I know the standard canon of physics. I knew it very well. But then I'd been working on this kind of computational paradigm for basically 40 years. And the fact that I'm sort of now coming back to trying to apply that in physics, it kind of felt like that journey was necessary. - When did you first try to play with a hypergraph? - So what happened is-- - Is this recent? - Yeah, so what I had was, okay, so this is again, one always feels dumb after the fact. It's obvious after the fact. But so back in the early 1990s, I realized that using graphs as a sort of underlying thing underneath space and time was going to be a useful thing to do. I figured out about multi-way systems. I figured out the things about general relativity. I'd figured out by the end of the 1990s. But I always felt there was a certain inelegance because I was using these graphs and there were certain constraints on these graphs that seemed like they were kind of awkward. It was kind of like you can pick, it's like you couldn't pick any rule. It was like pick any number, but the number has to be prime. It was kind of like you couldn't, it was kind of an awkward special constraint. I had these trivalent graphs, graphs with just three connections from every node. Okay, so, but I discovered a bunch of stuff with that. I thought it was kind of inelegant. And the other piece of sort of personal history is obviously I spent my life as a computational language designer. And so the story of computational language design is a story of how do you take all these random ideas in the world and kind of grind them down into something that is computationally as simple as possible. And so, I've been very interested in kind of simple computational frameworks for representing things and have ridiculous amounts of experience in trying to do that. - And actually all of those trajectories of your life kind of came together. So you make it sound like you could have come up with everything you're working on now decades ago, but in reality. - Look, two things slowed me down. I mean, one thing that slowed me down was I couldn't figure out how to make it elegant. And that turns out hypergraphs were the key to that. And that I figured out about less than two years ago now. And the other, I mean, I think, so that was sort of a key thing. Well, okay, so the real embarrassment of this project is that the final structure that we have that is the foundation for this project is basically a kind of an idealized version, a formalized version of the exact same structure that I've used to build computational languages for more than 40 years. But it took me, but I didn't realize that. - And there yet may be other, so we're focused on physics now, but I mean, that's what the new kind of science is about, same kind of stuff. And this, in terms of mathematically, the beauty of it, so there could be entire other kind of objects that are useful for, like we're not talking about machine learning, for example. Maybe there's other variants of the hypergraph that are very useful for reasoning about-- - Well, we'll see whether the multi-way graph for a machine learning system is interesting, okay? (Luke laughs) - Let's leave it at that. That's conversation number three. - That's, we're not gonna go there right now. - One of the things you've mentioned is the space of all possible rules that we kind of discussed a little bit, that there could be, I guess, the set of possible rules is infinite. - Right, well, so here's the big, sort of one of the conundrums that I'm kind of trying to deal with, is let's say we think we found the rule for the universe. And we say, here it is, write it down. It's a little tiny thing. And then we say, gosh, that's really weird. Why did we get that one? And then we're in this whole situation because let's say it's fairly simple. How did we come up the winners, getting one of the simple possible universe rules? Why didn't we get some incredibly complicated rule? Why did we get one of the simpler ones? And that's the thing which, in the history of science, the whole sort of story of Copernicus and so on was, we used to think the Earth was the center of the universe, but now we find out it's not. And we're actually just in some random corner of some random galaxy out in this big universe. There's nothing special about us. So if we get universe number 317 out of all the infinite number of possibilities, how do we get something that small and simple? So I was very confused by this. And it's like, what are we going to say about this? How are we going to explain this? And I thought it might be one of these things where you just, you can get it to the threshold, and then you find out its rule number such and such, and you just have no idea why it's like that. Okay, so then I realized it's actually more bizarre than that, okay? So we talked about multi-way graphs. We talked about this idea that you take these underlying transformation rules on these hypergraphs, and you apply them wherever the rule can apply, you apply it. And that makes this whole multi-way graph of possibilities. Okay, so let's go a little bit weirder. Let's say that at every place, not only do you apply a particular rule in all possible ways it can apply, but you apply all possible rules in all possible ways they can apply. As you say, that's just crazy. That's way too complicated. You're never going to be able to conclude anything. Okay, however, turns out that- - Don't tell me there's some kind of invariance. - Yeah, yeah. (laughing) So what happens is- - Oh man, that would be amazing. - Right, so this thing that you get, there's this kind of ruleal multi-way graph, this multi-way graph that is a branching of rules as well as a branching of possible applications of rules. This thing has causal invariance. It's an inevitable feature that it shows causal invariance. And that means that you can take different reference frames, different ways of slicing this thing, and they will all in some sense be equivalent. If you make the right translation, they will be equivalent. So, okay, so the basic point here is- - If that's true, that would be beautiful. - It is true, and it is beautiful. - So it's not just an intuition, there is some- - No, no, no, there's real mathematics behind this. And it is, okay, so here's how it comes out. - That's amazing. - Right, so by the way, I mean, the mathematics it's connected to is the mathematics of higher category theory and groupoids and things like this, which I've always been afraid of, but now I'm finally wrapping my arms around it. But it's also related to, it also relates to computational complexity theory. It's also deeply related to the P versus NP problem and other things like this. Again, it seems completely bizarre that these things are connected, but here's why it's connected. This space of all possible, okay, so a Turing machine, very simple model of computation. You just got this tape where you write down ones and zeros or something on the tape, and you have this rule that says, you change the number, you move the head on the tape, et cetera. You have a definite rule for doing that. A deterministic Turing machine just does that deterministically. Given the configuration of the tape, it will always do the same thing. A non-deterministic Turing machine can have different choices that it makes at every step. And so, you know this stuff, you probably teach this stuff. So a non-deterministic Turing machine has the set of branching possibilities, which is in fact one of these multi-way graphs. And in fact, if you say, imagine the extremely non-deterministic Turing machine, the Turing machine that can just do, that takes any possible rule at each step. That is this real multi-way graph. The set of possible histories of that extreme non-deterministic Turing machine is a really a multi-way graph. - What term are you using, Rulio? - Rulio. - Rulio, I like it. - It's a weird word. Yeah, it's a weird word. - Rulio. - Multi-way graph. Okay, so this, so that. - I'm trying to think of the space of rules. So these are basic transformations. - So in a Turing machine, it's like it says, move left, move, you know, if it's a one, if it's a black square under the head, move left and right a green square. That's a rule. That's a very basic rule, but I'm trying to see the rules on the hypergraphs, how rich of the programs can they be? Or do they all ultimately just map into something simple? - Yeah, they're all, I mean, hypergraphs, that's another layer of complexity on this whole thing. You can think about these in transformations of hypergraphs, but Turing machines are a little bit simpler. - You should stick up with Turing machines, okay. - Right, they're a little bit simpler. So if you look at these extreme non-deterministic Turing machines, you're mapping out all the possible non-deterministic paths that the Turing machine can follow. And if you ask the question, can you reach, okay, so a deterministic Turing machine follows a single path. The non-deterministic Turing machine fills out this whole sort of ball of possibilities. And so then the P versus NP problem ends up being questions about, and we haven't completely figured out all the details of this, but it's basically has to do with questions about the growth of that ball relative to what happens with individual paths. And so on. So essentially there's a geometrization of the P versus NP problem that comes out of this. That's a sideshow, okay. The main event here is the statement that you can look at this multi-way graph where the branches correspond, not just to different applications of a single rule, but to different applications, to applications of different rules, okay. And that then that when you say, I'm going to be an observer embedded in that system, and I'm going to try and make sense of what's going on in the system. And to do that, I essentially am picking a reference frame. And that turns out to be, well, okay. So the way this comes out essentially is the reference frame you pick is the rule that you infer is what's going on in the universe. Even though all possible rules are being run, although all those possible rules are in a sense giving the same answer because of causal invariance. But what you see could be completely different. If you pick different reference frames, you essentially have a different description language for describing the universe. Okay, so what does this really mean in practice? So imagine there's us. We think about the universe in terms of space and time, and we have various kinds of description models and so on. Now let's imagine the friendly aliens, for example, right? How do they describe their universe? Well, our description of the universe probably is affected by the fact that we are about the size we are, a meter-ish tall, so to speak. We have brain processing speeds of about the speeds we have. We're not the size of planets, for example, where the speed of light really would matter. In our everyday life, the speed of light doesn't really matter. Everything can be, the fact that the speed of light is finite is irrelevant. It could as well be infinite. We wouldn't make any difference. It affects the ping times on the internet. That's about the level of how we notice the speed of light. In our sort of everyday existence, we don't really notice it. And so we have a way of describing the universe that's based on our sensory, our senses, these days, also on the mathematics we've constructed and so on. But the realization is it's not the only way to do it. There will be completely, utterly incoherent descriptions of the universe which correspond to different reference frames in this sort of ruleal space. - In the ruleal space, that's fascinating. So we have some kind of reference frame in this ruleal space. - Right. - And from that-- - That's why we are attributing this rule to the universe. So in other words, when we say, "Why is it this rule and not another?" The answer is just shine the light back on us, so to speak. It's because of the reference frame that we've picked in our way of understanding what's happening in this sort of space of all possible rules and so on. - But also in the space from this reference frame, because of the ruleal, the invariance, that simple, that the rule on which the universe, with which you can run the universe, might as well be simple. - Yes, yes. Okay, so here's another point. So this is, again, these are a little bit mind twisting in some ways, but the, okay, another thing that's sort of we know from computation is this idea of computation universality. The fact that given that we have a program that runs on one kind of computer, we can as well, you know, we can convert it to run on any other kind of computer. We can emulate one kind of computer with another. So that might lead you to say, "Well, you think you have the rule for the universe, but you might as well be running it on a Turing machine, because we know we can emulate any computational rule on any kind of machine." And that's essentially the same thing that's being said here. That is that what we're doing is we're saying these different interpretations of physics correspond to essentially running physics on different underlying, you know, thinking about the physics as running in different, with different underlying rules, as if different underlying computers were running them. And, but because of computation universality, or more accurately, because of this principle of computational equivalence thing of mine, there's, they are, these things are ultimately equivalent. So the only thing that is the ultimate fact about the universe, the ultimate fact that doesn't depend on any of these, you know, we don't have to talk about specific rules, et cetera, et cetera, et cetera. The ultimate fact is the universe is computational, and it is the things that happen in the universe are the kinds of computations that the principle of computational equivalence says should happen. Now, that might sound like, hey, you're not really saying anything there, but you are, because you can, you could in principle have a hypercomputer that things that take an ordinary computer an infinite time to do, the hypercomputer can just say, "Oh, I know the answer. It's this, immediately." What this is saying is the universe is not a hypercomputer. It's not simpler than an ordinary Turing machine type computer. It's exactly like an ordinary Turing machine type computer. And so that's in the end, the sort of net net conclusion is that's the thing that is the sort of the hard immovable fact about the universe. That's sort of the fundamental principle of the universe is that it is computational and not hypercomputational and not sort of infracomputational. It is this level of computational ability and it kind of has, and that's sort of the core fact. But now, this idea that you can have these different kind of ruleal reference frames, these different description languages for the universe, it makes me, I used to think, okay, imagine the aliens, imagine the extraterrestrial intelligence thing, at least they experienced the same physics. And now I've realized it isn't true. - They could have a different ruleal frame. That's fascinating. - They can end up with a description of the universe that is utterly, utterly incoherent with ours. And that's also interesting in terms of how we think about, well, intelligence, the nature of intelligence and so on. I'm fond of the quote, "The weather has a mind of its own," because these are sort of computationally, that system is computationally equivalent to the system that is our brains and so on. And what's different is we don't have a way to understand what the weather is trying to do, so to speak. We have a story about what's happening in our brains. We don't have a sort of connection to what's happening there. - So we actually, it's funny, last time we talked, maybe over a year ago, we talked about how it was more based on your work with Arrival. We talked about how would we communicate with alien intelligences. Can you maybe comment on how we might, how the Wolfram Physics Project changed your view of how we might be able to communicate with alien intelligence? Like if they showed up, is it possible that because of our comprehension of the physics of the world might be completely different, we would just not be able to communicate at all? - Here's the thing. Intelligence is everywhere. The fact, this idea that there's this notion of, oh, there's gonna be this amazing extraterrestrial intelligence and it's gonna be this unique thing, it's just not true. It's the same thing. You know, I think people will realize this about the time when people decide that artificial intelligences are kind of just natural things that are like human intelligences. They'll realize that extraterrestrial intelligences or intelligences associated with physical systems and so on, it's all the same kind of thing. - It's ultimately computation. It's all the same. It's all just computation. And the issue is, are you sort of inside it? Are you thinking about it? Do you have sort of a story you're telling yourself about it? And, you know, the weather could have a story it's telling itself about what it's doing. We just, it's utterly incoherent with the stories that we tell ourselves based on how our brains work. - I mean, ultimately it must be a question whether we can align. - Exactly, exactly. - Align with the kind of intelligence. - Right, right, right. - There's a systematic way of doing it. - Right, so the question is in the space of all possible intelligences, what's the, how do you think about the distance between description languages for one intelligence versus another? And needless to say, I have thought about this. And, you know, I don't have a great answer yet, but I think that's a thing where there will be things that can be said and there'll be things that where you can sort of start to characterize, you know, what is the translation distance between this, you know, version of the universe or this, you know, kind of set of computational rules and this other one. In fact, okay, so this is a, you know, there's this idea of algorithmic information theory. There's this question of sort of what is the, when you have some something, what is the sort of shortest description you can make of it where that description could be saying, run this program to get the thing, right? So I'm pretty sure that there will be a physicalization of the idea of algorithmic information and that, okay, this is again, a little bit bizarre, but so I mentioned that there's the speed of light, maximum speed of information transmission in physical space. There's a maximum speed of information transmission in branchial space, which is a maximum entanglement speed. There's a maximum speed of information transmission in ruleal space, which has to do with a maximum speed of translation between different description languages. And again, I'm not fully wrapped my brain around this one. - Yeah, that one just blows my mind to think about that, but that starts getting closer to the, yeah, the-- - It's kind of a physicalization, right. And it's also a physicalization of algorithmic information. And I think there's probably a connection between, I mean, there's probably a connection between the notion of energy and some of these things, which again, I hadn't seen all this coming. I've always been a little bit resistant to the idea of connecting physical energy to things in computation theory, but I think that's probably coming. - And that's what essentially at the core with the physics project is that you're connecting information theory with physics. - Yeah, it's computation. - Computation with our physical universe. - Yeah, right. I mean, the fact that our physical universe is, right, that we can think of it as a computation and that we can have discussions like, the theory of the physical universe is the same kind of a theory as the P versus NP problem and so on, is really, I think that's really interesting. And the fact that, well, okay, so this kind of brings me to one more thing that I have to in terms of this sort of unification of different ideas, which is metamathematics. - Yeah, let's talk about that. You mentioned that earlier. What the heck is mathematics and-- - Okay, so here's what, here's, okay. So what is mathematics? Mathematics, sort of at a lowest level, one thinks of mathematics as you have certain axioms, you say, you know, you say things like X plus Y is the same as Y plus X. That's an axiom about addition. And then you say, we've got these axioms and from these axioms, we derive all these theorems that fill up the literature of mathematics. The activity of mathematicians is to derive all these theorems. Actually, the axioms of mathematics are very small. You can fit, you know, when I did my new kind of science book, I fit all of the standard axioms of mathematics on basically a page and a half. Not much stuff. It's like a very simple rule from which all of mathematics arises. The way it works, though, is a little different from the way things work in sort of a computation because in mathematics, what you're interested in is a proof and the proof says from here, you can use from this expression, for example, you can use these axioms to get to this other expression. So that proves these two things are equal. Okay, so we can begin to see how this is going to work. What's gonna happen is there are paths in metamathematical space. So what happens is each, two different ways to look at it. You can just look at it as mathematical expressions or you can look at it as mathematical statements, postulates or something, but either way, you think of these things and they are connected by these axioms. So in other words, you have some fact, you, or you have some expression, you apply this axiom, you get some other expression. And in general, given some expression, there may be many possible different expressions you can get. You basically build up a multi-way graph and a proof is a path through the multi-way graph that goes from one thing to another thing. The path tells you how did you get from one thing to the other thing? It's the story of how you got from this to that. The theorem is the thing at one end is equal to the thing at the other end. The proof is the path you go down to get from one thing to the other. - You mentioned that Gadel's incompleteness theorem is the natural, it fits naturally there. How does it fit? - Yeah, so what happens there is that the Gadel's theorem is basically saying that there are paths of infinite length. That is, that there's no upper bound. If you know these two things, you say, I'm trying to get from here to here, how long do I have to go? You say, well, I've looked at all the paths of length 10. Somebody says, that's not good enough. That path might be of length a billion. And there's no upper bound on how long that path is. And that's what leads to the incompleteness theorem. So, I mean, the thing that is kind of an emerging idea is you can start asking, what's the analog of Einstein's equations in metamathematical space? What's the analog of a black hole in metamathematical space? - What's the hope, so yeah, it's fascinating to model all the mathematics in this way. - Well, so here's what it is. This is mathematics in bulk. So human mathematicians have made a few million theorems. They published a few million theorems. But imagine the infinite future of mathematics. Apply something to mathematics that mathematics likes to apply to other things. Take a limit. What is the limit of the infinite future of mathematics? What does it look like? What is the continuum limit of mathematics? What is the, as you just fill in more and more and more theorems, what does it look like? What does it do? How does, what kinds of conclusions can you make? So for example, one thing I've just been doing is taking Euclid. So Euclid, very impressive. He had 10 axioms. He derived 465 theorems, okay? His book, you know, that was the sort of defining book of mathematics for 2000 years. So you can actually map out, I actually did this 20 years ago, but I've done it more seriously now. You can map out the theorem dependency of those 465 theorems. So from the axioms, you grow this graph, it's actually a multi-way graph, of how all these theorems get proved from other theorems. And so you can ask questions about, you know, well, you can ask things like, what's the hardest theorem in Euclid? The answer is, the hardest theorem is that there are five platonic solids. That turns out to be the hardest theorem in Euclid. That's actually his last theorem in all his books. That's the final-- - What's the hardness, the distance you have to travel? - Yeah, let's say it's 33 steps from the, the longest path in the graph is 33 steps. So that's the, there's a 33 step path you have to follow to go from the axioms, according to Euclid's proofs, to the statement there are five platonic solids. So, okay, so then-- - Great. - Then the question is, in, what does it mean, if you have this map, okay, so, in a sense, this meta-mathematical space is the infrastructural space of all possible theorems that you could prove in mathematics. That's the geometry of meta-mathematics. There's also the geography of mathematics. That is, where did people choose to live in space? And that's what, for example, exploring the sort of empirical meta-mathematics of Euclid is doing that. - Each individual, like, human mathematician, you can embed them into that space. I mean, they kind of live-- - Yeah, well, they represent a path in space. - The little path. - The things they do. - Maybe a set of paths. - Right. - So, like, a set of axioms that are chosen. - Right, so, for example, here's an example of a thing that I realized. So, one of the surprising things about, well, there are two surprising facts about math. One is that it's hard, and the other is that it's doable. Okay, so first question is, why is math hard? You know, you've got these axioms, they're very small. Why can't you just solve every problem in math easily? - Yeah, it's just logic. - Right, yeah, well, logic happens to be a particular special case that does have certain simplicity to it. But general mathematics, even arithmetic, already doesn't have the simplicity that logic has. - So, why is it hard? - Because of computational irreducibility. - Right. - Because what happens is, to know what's true, and this is this whole story about the path you have to follow, and how long is the path, and Gödel's theorem is the statement that the path is not a bounded length, but the fact that the path is not always compressible to something tiny is a story of computational irreducibility. So, that's why math is hard. Now, the next question is, why is math doable? Because it might be the case that most things you care about don't have finite length paths. Most things you care about might be things where you get lost in the sea of computational irreducibility and worse, undecidability. That is, there's just no finite length path that gets you there. Why is mathematics doable? Gödel proved his incompleteness theorem in 1931. Most working mathematicians don't really care about it. They just go ahead and do mathematics, even though it could be that the questions they're asking are undecidable. It could have been that Fermat's last theorem is undecidable. It turned out it had a proof. It's a long, complicated proof. The twin prime conjecture might be undecidable. The Riemann hypothesis might be undecidable. These things might be, the axioms of mathematics might not be strong enough to reach those statements. It might be the case that depending on what axioms you choose, you can either say that's true or that's not true. - And by the way, Fermat's last theorem, it could be a shorter path. - Absolutely. Yeah, so the notion of geodesics in metamathematical space is a notion of shortest proofs in metamathematical space. And that's a, you know, human mathematicians do not find shortest paths, nor do automated theorem provers. But the fact, and by the way, I mean, this stuff is so bizarrely connected. I mean, if you're into automated theorem proving, there are these so-called critical pair lemmas in automated theorem proving. Those are precisely the branch pairs in our, that in multi-way graphs. Let me just finish on the why mathematics is doable. - Oh yes, the second part. So we know why it's hard. Why is it doable? - Right, why do we not just get lost in undecidability all the time? - Yeah. - So, and here's another fact, is in doing computer experiments and doing experimental mathematics, you do get lost in that way. When you just say, I'm picking a random integer equation, how do I, does it have a solution or not? And you just pick it at random without any human sort of path getting there. Often, it's really, really hard. It's really hard to answer those questions. We just pick them at random from the space of possibilities. But what I think is happening is, and that's a case where you just fell off into this ocean of sort of irreducibility and so on. What's happening is, human mathematics is a story of building a path. You started off, you're always building out on this path where you are proving things. You've got this proof trajectory, and you're basically, human mathematics is the sort of the exploration of the world along this proof trajectory, so to speak. You're not just parachuting in from anywhere, you're following Lewis and Clark or whatever. You're actually going, doing the path. And the fact that you are constrained to go along that path is the reason you don't end up with, every so often, you'll see a little piece of undecidability and you'll avoid that part of the path. But that's basically the story of why human mathematics has seemed to be doable. It's a story of exploring these paths that are by their nature, they have been constructed to be paths that can be followed. And so you can follow them further. Now, why is this relevant to anything? So, okay, so here's my belief. The fact that human mathematics works that way is, I think there's some sort of connections between the way that observers work in physics and the way that the axiom systems of mathematics are set up to make mathematics be doable in that kind of way. And so, in other words, in particular, I think there is an analog of causal invariance, which I think is, and this is again, it's sort of the upper reaches of mathematics and stuff that it's a thing, there's this thing called homotopy type theory, which is an abstract, it's came out of category theory, and it's sort of an abstraction of mathematics. Mathematics itself is an abstraction, but it's an abstraction of the abstraction of mathematics. And there is the thing called the univalence axiom, which is a sort of a key axiom in that set of ideas. And I'm pretty sure the univalence axiom is equivalent to causal invariance. - What was the term used again? - Univalence. - Is that something for somebody like me accessible? Or is this? - There's a statement of it that's fairly accessible. I mean, the statement of it is, basically it says things which are equivalent can be considered to be identical. - In which space? - Yeah, it's in higher category. - Okay, in category theory. - Okay, so it's a, but I mean, the thing just to give a sketch of how that works, so category theory is an attempt to idealize, it's an attempt to sort of have a formal theory of mathematics that is at a sort of higher level than mathematics. It's where you just think about these mathematical objects and these categories of objects and these morphisms, these connections between categories. Okay, so it turns out the morphisms and categories, at least weak categories, are very much like the paths in our hypergraphs and things. And it turns out, again, this is where it all gets crazy. I mean, it's the fact that these things are connected is just bizarre. So category theory, our causal graphs are like second order category theory. And it turns out you can take the limits of infinite order category theory. So just give roughly the idea. This is a roughly explainable idea. So a mathematical proof will be a path that says you can get from this thing to this other thing. And here's the path that you get from this thing to this other thing. But in general, there may be many paths, many proofs that get you many different paths that all successfully go from this thing to this other thing, okay? Now you can define a higher order proof, which is a proof of the equivalence of those proofs. Okay, so you're saying there's a-- - Path between those proofs, essentially. - Yes, a path between the paths, okay? And so you do that. That's the sort of second order thing. That path between the paths is essentially related to our causal graphs. Then you take the limit-- - Ah, wow, okay. - Path between path between path between path, the infinite limit, that infinite limit turns out to be our Rullio multiway system. - Yeah, the Rullio multiway system, that's a fascinating thing, both in the physics world and as you're saying now. - Yeah, yeah. - That's fascinating. I'm not sure I've loaded it in completely, but-- - Well, I'm not sure I have either. And it may be one of these things where in another five years or something, it's like, this was obvious, but I didn't see it. No, but the thing which is sort of interesting to me is that there's sort of an upper reach of mathematics, of the abstraction of mathematics. This thing, there's this mathematician called Grothendieck who's generally viewed as being sort of one of the most abstract, sort of creator of the most abstract mathematics of 1970s-ish timeframe. And one of the things that he constructed was this thing he called the infinity groupoid. And he has this sort of hypothesis about the inevitable appearance of geometry from essentially logic in the structure of this thing. Well, it turns out this Rullio multiway system is the infinity groupoid. So it's this limiting object, and this is an instance of that limiting object. So what to me is, I mean, again, I've been always afraid of this kind of mathematics because it seemed incomprehensibly abstract to me. But what I'm sort of excited about with this is that we've sort of concretified the way that you can reach this kind of mathematics, which makes it, well, both seem more relevant and also the fact that that, you know, I don't yet know exactly what mileage we're gonna get from using the sort of the apparatus that's been built in those areas of mathematics to analyze what we're doing. But the thing that's-- - So both ways, using the mathematics to understand what you're doing and using, what you're doing computationally to understand that mathematics. - Right, so for example, the understanding of metamathematical space, one of the reasons I really want to do that is because I want to understand quantum mechanics better. And that, what you see, you know, we live that kind of the multi-way graph of mathematics because we actually know this is a theorem we've heard of. This is another one we've heard of. We can actually say these are actual things in the world that we relate to, which we can't really do as readily for the physics case. And so it's kind of a way to help my intuition. It's also, you know, there are bizarre things like what's the analog of Einstein's equations in metamathematical space? What's the analog of a black hole? You know, it turns out it looks like, not completely sure yet, but there's this notion of non-constructive proofs in mathematics. And I think those relate to, well, actually, they relate to things related to event horizons. So the fact that you can take ideas from physics, like event horizons-- - And map them into the same kind of space-- - It's really-- - Do you think you might stumble upon some breakthrough ideas in theorem proving, like from the other direction? - Yeah, yeah, yeah. No, I mean, what's really nice is that we are using, so this absolutely directly maps to theorem proving. So paths and multi-way graphs, that's what a theorem prover is trying to do. - But I also mean like automated theorem proving. - Yeah, yeah, yeah. That's what, right, so the finding of paths, the finding of shortest paths or finding of paths at all is what automated theorem provers do. And actually what we've been doing, so we've actually been using automated theorem proving both in the physics project to prove things and using that as a way to understand multi-way graphs. And because what an automated theorem prover is doing is it's trying to find a path through a multi-way graph. And it's critical pair lemmas are precisely little stubs of branch pairs going off into branchial space. And that's, I mean, it's really weird. We have these visualizations in Wolfram language of proof graphs from our automated theorem proving system. - And they look reminiscent of-- - Well, it's just bizarre because we made these up a few years ago and they have these little triangle things. And they are, we didn't quite get it right. We didn't quite get the analogy perfectly right, but it's very close. Just to say in terms of how these things are connected. So there's another bizarre connection that I have to mention because, which again, we don't fully know, but it's a connection to something else you might not have thought was in the slightest bit connected, which is distributed blockchain like things. Now you might figure out that that's, you would figure out that that's connected because it's a story of distributed computing. And the issue, with a blockchain, you're saying there's gonna be this one ledger that globally says, this is what happened in the world. But that's a bad deal if you've got all these different transactions that are happening. And this transaction in country A doesn't have to be reconciled with the transaction in country B, at least not for a while. And that story is just like what happens with our causal graphs. That whole reconciliation thing is just like what happens with light cones and all this kind of thing. - Yeah, so that's where the causal invariance comes into play. I mean, that's, you know, most of your conversations are about physics, but it's kind of funny that this probably and possibly might have even bigger impact and revolutionary ideas in totally other disciplines. - Right, so the question is, why is that happening? Right, and the reason it's happening, I've thought about this, obviously, because I like to think about these meta questions of, you know, what's happening is this model that we have is an incredibly minimal model. And once you have an incredibly minimal model, and this happened with cellular automata as well, cellular automata are an incredibly minimal model. And so it's inevitable that it gets used, sort of an upstream thing that gets used in lots of different places. And it's like, you know, the fact that it gets used, you know, cellular automata are sort of a minimal model of like, say, road traffic flow or something. And they're also a minimal model of something in, you know, chemistry. And they're also a minimal model of something in epidemiology, right? It's because they're such a simple model that they can, that they apply to all these different things. Similarly, this model that we have of the physics project is another, cellular automata are a minimal model of parallel, of basically of parallel computation where you've defined space and time. These models are minimal models where you have not defined space and time. And they have been very hard to understand in the past. But the, I think the, perhaps the most important breakthrough there is the realization that these are models of physics. And therefore that you can use everything that's been developed in physics to get intuition about how things like that work. And that's why you can potentially use ideas from physics to get intuition about how to do parallel computing. And because the underlying model is the same. And, but we have all of this achievement in physics. I mean, you know, you might say, oh, you've come up with the fundamental theory of physics. That throws out what people have done in physics before. Well, it doesn't, but also the real power is to use what's been done before in physics to apply it in these other places. - Yes, absolutely. - This kind of brings up, I know you probably don't particularly love commenting on the work of others, but let me bring up a couple of personalities just 'cause it's fun and people are curious about it. So there's Sabine Hassenfelder. I don't know if you're familiar with her. She wrote this book that I need to read, but it based, I forget what the title is, but it's "Beauty Leads Us Astray in Physics" is a subtitle, something like that. Which so much about what we're talking about now, like this simplification is to us humans seems to be beautiful. Like there's a certain intuition with physicists, with people that a simple theory, like this reducibility, pockets of reducibility is the ultimate goal. And I think what she tries to argue is, no, we just need to come up with theories that are just really good at predicting physical phenomena. It's okay to have a bunch of disparate theories as opposed to trying to chase this beautiful theory of everything is the ultimate beautiful theory, a simple one. What's your response to that? - Well, so what you're quoting, so I don't know that Sabine Hossenfelder's, exactly what she said, but I mean- - You might be misquoting the title of her book. - Okay, well, let me respond to what you were describing, which may or may not have anything to do with what Sabine Hossenfelder says or thinks. - Sorry, Sabine. - The, right. - Sorry for misquoting. - But I mean, the question is, is beauty a guide to whether something is correct? Which is kind of also the story of Occam's razor. If you've got a bunch of different explanations of things, is the thing that is the simplest explanation likely to be the correct explanation? And there are situations where that's true and there are situations where it isn't true. Sometimes in human systems, it is true because people have kind of, in evolutionary systems, sometimes it's true because it's sort of been kicked to the point where it's minimized. But in physics, does Occam's razor work? Is there a simple, quote, "beautiful explanation for things," or is it a big mess? We don't intrinsically know. I think that the, I wouldn't, before I worked on the project in recent times, I would have said, "We do not know how complicated "the rule for the universe will be." And I would have said, the one thing we know, which is a fundamental fact about science, that's the thing that makes science possible, is that there is order in the universe. I mean, early theologians would have used that as an argument for the existence of God, because it's like, why is there order in the universe? Why doesn't every single particle in the universe just do its own thing? Something must be making there be order in the universe. We, in the sort of early theology point of view, that's, the role of God is to do that, so to speak. In our, we might say it's the role of a formal theory to do that. And then the question is, but how simple should that theory be? And should that theory be one that, where I think the point is, if it's simple, it's almost inevitably somewhat beautiful in the sense that, because all the stuff that we see has to fit into this little tiny theory. And the way it does that has to be, it depends on your notion of beauty, but I mean, for me, the sort of the surprising connectivity of it is, at least in my aesthetic, that's something that responds to my aesthetic. - But the question is, I mean, you're a fascinating person in the sense that you're at once talking about computational, the fundamental computational reducibility of the universe, and on the other hand, trying to come up with a theory of everything, which simply describes the simple origins of that computational reducibility. I mean, both of those things are kind of, it's paralyzing to think that we can't make any sense of the universe in the general case, but it's hopeful to think, like, one, we can think of a rule that generates this whole complexity, and two, we can find pockets of reducibility that are powerful for our everyday life to do different kinds of predictions. I suppose Sabine wants to find, focus on the finding of small pockets of reducibility versus the theory of everything. - You know, it's a funny thing because, you know, a bunch of people have started working on this, you know, physics project, people who are, you know, physicists, basically, and it is really a fascinating sociological phenomenon because what, you know, when I was working on this before in the 1990s, you know, wrote it up, put it, it's 100 pages of this 1200 page book that I wrote, "New Kind of Science," it's, you know, 100 pages of that is about physics. But I saw it at that time, not as a pinnacle achievement, but rather as a use case, so to speak. I mean, my main point was this new kind of science, and it's like, you can apply it to biology, you can apply it to, you know, other kinds of physics, you can apply it to fundamental physics. It's just an application, so to speak. It's not the core thing. But then, you know, one of the things that was interesting with that book was, you know, book comes out, lots of people think it's pretty interesting and lots of people start using what it has in different kinds of fields. The one field where there was sort of a heavy pitchforking was from my friends, the fundamental physics people, which was, it's like, no, this can't possibly be right. And, you know, it's like, you know, if what you're doing is right, it'll overturn 50 years of what we've been doing. And it's like, no, it won't, was what I was saying. And it's like, but, you know, for a while, when I started, you know, I was going to go on back in 2002, well, 2004, actually, I was going to go on working on this project. And I actually stopped, partly because it's like, why am I, you know, this is like, I've been in business a long time, right? I'm building a product for a target market that doesn't want the product. And it's like-- - Why work, yeah, yeah, why work against, swim against the current or whatever. - Right, but you see what's happened, which is sort of interesting, is that a couple of things happened. And it was like, you know, it was like, I don't want to do this project because I can do so many other things, which I'm really interested in, where, you know, people say, great, thanks for those tools, thanks for those ideas, et cetera. Whereas, you know, if you're dealing with kind of a, you know, sort of a structure where people are saying, no, no, we don't want this new stuff, we don't need any new stuff, we're really fine with what we're doing. - Yeah, there's like literally, like, I don't know, millions of people who are thankful for Wolfram Alpha, a bunch of people wrote to me how thankful, they are a different crowd than the theoretical physics community, perhaps. - Yeah, well, right, but you know, the theoretical physics community pretty much uniformly uses Wolfram language and Mathematica, right? And so it's kind of like, you know, and that's, but the thing is, what happens, you know, this is what happens, mature fields do not, you know, it's like, we're doing what we're doing, we have the methods that we have, and we're just fine here. Now, what's happened in the last 18 years or so, I think there's a couple of things have happened. First of all, the hope that, you know, string theory or whatever would deliver the fundamental theory of physics, that hope has disappeared. That the, another thing that's happened is the, the sort of the interest in computation around physics has been greatly enhanced by the whole quantum information, quantum computing story. People, you know, the idea there might be something sort of computational related to physics has somehow grown. And I think, you know, it's sort of interesting, I mean, right now, if we say, you know, it's like, if you're like, who else is trying to come up with the fundamental theory of physics? It's like, there aren't professional, no professional physicists. - Oh, no professional. - No professional physicists. - What are your, I mean, you've talked with him, but just as a matter of personalities, 'cause it's a beautiful story, what are your thoughts about Eric Weinstein's work? - You know, I think his, I mean, he did a PhD thesis in mathematical physics at Harvard. - He's a mathematical physicist. - And, you know, it's, it seems like it's kind of, you know, it's in that framework and it's kind of like, I'm not sure how much further it's got than his PhD thesis, which was 20 years ago or something. And I think that, you know, it's a fairly specific piece of mathematical physics that's quite nice. And- - What trajectory do you hope it takes? I mean- - Well, I think in his particular case, I mean, from what I understand, which is not everything at all, but, you know, I think I know the rough tradition, at least, that he's operating in is sort of theory of gauge theories. - Gauge theories, yeah. - Local gauge invariance and so on. Okay, we are very close to understanding how local gauge invariance works in our models and it's very beautiful and it's very, and, you know, does some of the mathematical structure that he's enthusiastic about fit? Quite possibly, yes. - So there might be a possibility of trying to understand how those things fit, how gauge theory fits. - Yeah, might very well. I mean, the question is, you know, so there are a couple of things one might try to get in the world. So for example, it's like, can we get three dimensions of space? We haven't managed to get that yet. Gauge theory, the standard model of particle physics says that it's SU3 cross SU2 cross U1. Those are the designations of these Lie groups. It doesn't, but anyway, so those are sort of representations of symmetries of the theory. And so, you know, it is conceivable that it is generically true. Okay, so all those are subgroups of a group called E8, which is a weird, exceptional Lie group. Okay, it is conceivable, I don't know whether it's the case, that that will be generic in these models, that it will be generic, that the gauge invariance of the model has this property, just as things like general relativity, which corresponds to a thing called general covariance, which is another gauge-like invariance. It could conceivably be the case that the kind of local gauge invariance that we see in particle physics is somehow generic. And that would be a, you know, the thing that's really cool, I think, you know, sociologically, although this hasn't really hit yet, is that all of these different things, all these different things people have been working on in these, in some cases, quite abstruse areas of mathematical physics, an awful lot of them seem to tie into what we're doing. And, you know, it might not be that way. - Yeah, absolutely. That's a beautiful thing of the theory. I mean, but the reason I, the reason Erick Weinstein is important is to the point that you mentioned before, which is it's strange that the theory of everything is not at the core of the passion, the dream, the focus, the funding of the physics community. - It's too hard. It's too hard, and people gave up. I mean, basically what happened is, ancient Greece, people thought we're nearly there. You know, the world is made of platonic solids. It's, you know, water is a tetrahedron or something. We're almost there, okay? Long period of time where people were like, no, we don't know how it works. You know, time of Newton, you know, we're almost there. Everything is gravitation. You know, time of Faraday and Maxwell, we're almost there. Everything is fields. Everything is the ether. You know, then-- - And the whole time we're making big progress, though. - Oh, yes, absolutely. But the fundamental theory of physics is almost a footnote because it's like, it's the machine code. It's like, we're operating in the high-level languages. - Yeah. - You know, that's what we really care about. That's what's relevant for our everyday physics. - You talked about different centuries, and the 21st century will be everything is computation. - Yes. - If that takes us all the way, we don't know, but it might take us pretty far. - Yes, right, that's right. But I think the point is that it's like, you know, if you're doing biology, you might say, "How can you not be really interested "in the origin of life and the definition of life?" Well, it's irrelevant. You know, you're studying the properties of some virus. It doesn't matter, you know, where, you know, you're operating at some much higher level. And it's the same, what's happened with physics is, I was sort of surprised, actually. I was sort of mapping out this history of people's efforts to understand the fundamental theory of physics. And it's remarkable how little has been done on this question. And it's, you know, because, you know, there've been times when there's been bursts of enthusiasm. Oh, we're almost there. And then it decays, and people just say, "Oh, it's too hard, but it's not relevant anyway." And I think that the thing that, you know, so the question of, you know, one question is why does anybody, why should anybody care, right? Why should anybody care what the fundamental theory of physics is? I think it's intellectually interesting, but what will be the sort of, what will be the impact of this? - What, I mean, this is the key question. What do you think will happen if we figure out the fundamental theory of physics? - Right. - Outside of the intellectual curiosity of us. - Okay, so here's my best guess, okay? So if you look at the history of science, I think a very interesting analogy is Copernicus. Okay, so what did Copernicus do? There'd been this Ptolemaic system for working out the motion of planets. It did pretty well. It used epicycles, et cetera, et cetera, et cetera. It had all this computational ways of working out what planets will be. When we work out where planets are today, we're basically using epicycles. But Copernicus had this different way of formulating things in which he said, you know, "And the Earth is going around the sun." And that had a consequence. The consequence was you can use this mathematical theory to conclude something which is absolutely not what we can tell from common sense, right? So it's like, trust the mathematics, trust the science. Okay, now fast forward 400 years, and now we're in this pandemic, and it's kind of like everybody thinks the science will figure out everything. It's like, from the science, we can just figure out what to do. We can figure out everything. That was before Copernicus. Nobody would have thought if the science says something that doesn't agree with our everyday experience, where we just have to compute the science and then figure out what to do. People say that's completely crazy. - And so your sense is, once we figure out the framework of computation that can basically do any, understand the fabric of reality, we'll be able to derive totally counterintuitive things. - No, the point, I think, is the following, that right now, I talk about computational irreducibility. People, I was very proud that I managed to get the term computational irreducibility into the congressional record last year. - That's right, by the way. That's a whole 'nother topic we could talk about. Fascinating. - Different topic. - Different day. - Different topic. But in any case, so computational irreducibility is one of these sort of concepts that I think is important in understanding lots of things in the world. But the question is, it's only important if you believe the world is fundamentally computational. But if you know the fundamental theory of physics and it's fundamentally computational, then you've rooted the whole thing. That is, you know the world is computational. And while you can discuss whether, it's not the case that people would say, well, you have this whole computational irreducibility, all these features of computation, we don't care about those because after all, the world isn't computational, you might say. But if you know, base, base, base thing, physics is computational, then you know that that stuff is, you know, that that's kind of the grounding for that stuff. Just as in a sense, Copernicus was the grounding for the idea that you could figure out something with math and science that was not what you would intuitively think from your senses. So now we've got to this point where, for example, we say, you know, once we have the idea that computation is the foundational thing that explains our whole universe, then we have to say, well, what does it mean for other things? Like it means there's computational irreducibility. That means science is limited in certain ways. That means this, that means that. But the fact that we have that grounding means that, you know, and I think for example, for Copernicus, for instance, the implications of his work on the set of mathematics of astronomy were cool, but they involved a very small number of people. The implications of his work for sort of the philosophy of how you think about things were vast and involved, you know, everybody more or less. - But do you think, so that's actually the way scientists and people see the world around us. So it has a huge impact in that sense. Do you think it might have an impact more directly to engineering derivations from physics, like propulsion systems, our ability to colonize the world? Like for example, okay, this is like sci-fi, but if you understand the computational nature, say, of the different forces of physics, you know, there's a notion of being able to, you know, warp gravity, things like this. - Yeah, can we make warp drive? - Warp drive, yeah. So like, would we be able to, will it, will, you know, will like Elon Musk start paying attention? Like it's awfully costly to launch these rockets. Do you think we'll be able to, yeah, create warp drive? - Right, you know, I set myself some homework. I agreed to give a talk at some NASA workshop in a few weeks about faster than light travel. So I haven't figured it out yet, but no, but-- - You got two weeks. - Yeah, right. - But do you think that kind of understanding of fundamental theory of physics can lead to those engineering breakthroughs? - Okay, I think it's far away, but I'm not certain. I mean, you know, this is the thing that, I set myself an exercise when gravity waves, gravitational waves were discovered, right? I set myself the exercise of what would black hole technology look like? In other words, right now, you know, black holes are far away. They're, you know, how on earth can we do things with them? But just imagine that we could get, you know, pet black holes right in our backyard. You know, what kind of technology could we build with them? I got a certain distance, not that far. But I think in, you know, so there are ideas. You know, I have this, one of the weirder ideas is things I'm calling space tunnels, which are higher dimensional pieces of space time, where basically you can, you know, in our three-dimensional space, there might be a five-dimensional, you know, region, which actually will appear as a white hole at one end and a black hole at the other end. You know, who knows whether they exist? And then the question's another one, okay, this is another crazy one, is the thing that I'm calling a vacuum cleaner, okay? So I mentioned that, you know, there's all this activity in the universe, which is maintaining the structure of space. And that leads to a certain energy density, effectively, in space. And so the question, in fact, dark energy is a story of essentially negative mass produced by the absence of energy you thought would be there, so to speak. And we don't know exactly how it works in either our model or the physical universe, but this notion of a vacuum cleaner is a thing where, you know, you have all these things that are maintaining the structure of space, but what if you could clean out some of that stuff that's maintaining the structure of space and make a simpler vacuum somewhere? - Yeah. - You know, what would that do? - Totally different kind of vacuum. - Right, and that would lead to negative energy density, which would need to, so gravity is usually a purely attractive force, but negative mass would lead to repulsive gravity and lead to all kinds of weird things. Now, can it be done in our universe? You know, my immediate thought is no, but, you know, the fact is that, okay, so-- - Well, once you understand the fact, 'cause you're saying, like, at this level of abstraction, you can reach to the lower levels and mess with it. - Yes. - Once you understand the levels, I think you can start to-- - I know, and I'm, you know, I have to say that this reminds me of people telling one years ago that, you know, you'll never transmit data over a copper wire at more than 1,000, you know, 1,000 baud or something, right? And this is, why did that not happen? You know, why do we have these much, much faster data transmission? Because we've understood many more of the details of what's actually going on. And it's the same exact story here. And it's the same, you know, I think that this, as I say, I think one of the features of sort of, one of the things about our time that will seem incredibly naive in the future is the belief that, you know, things like heat is just random motion of molecules, that it's just, just throw up your hands, it's just random, we can't say anything about it. That will seem naive. - Yeah, at the heat death of the universe, those particles would be laughing at us humans thinking that life is not beautiful. - Right, we'll have a whole civilization, you know. - Humans used to think they're special with their little brains. - Well, right, but also, and they used to think that this would just be random and uninteresting. But that's, but so this question about whether you can, you know, mess with the underlying structure and how you find a way to mess with the underlying structure, that's a, you know, I have to say, you know, my immediate thing is, boy, that seems really hard. But then, and, you know, possibly computational irreducibility will bite you, but then there's always some path of computational reducibility. And that path of computational reducibility is the engineering invention that has to be made. - Those little pockets can have huge engineering impact. - Right, and I think that that's right. And I mean, we live in, you know, we make use of so many of those pockets. And the fact is, you know, I, you know, this is, yes, it's a, you know, it's one of these things where, where, you know, I'm a person who likes to figure out ideas and so on, and there's sort of tests of my level of imagination, so to speak. And so a couple of places where there's sort of serious humility in terms of my level of imagination. One is this thing about different reference frames for understanding the universe, where like, imagine the physics of the aliens, what will it be like? And I'm like, that's really hard. I don't know, you know? And I mean, I think- - But once you have the framework in place, you can at least reason about the things you don't know. - Yes. - Or maybe can't know, or like it's too hard for you to know. But then the mathematics can, that's exactly it. Allow you to reach beyond what you can reason about. - Right, so I'm, you know, I'm trying to not have, you know, if you think back to Alan Turing, for example, and, you know, when he invented Turing machines, you know, and imagining what computers would end up doing, so to speak. - Yeah. - You know, and it's- - It's very difficult. - It's difficult, right. And it's, I mean, this thing- - He made a few reasonable predictions, but most of it he couldn't predict, possibly. - By the time, by 1950, he was making reasonable predictions about something. - But not the '30s, yeah. - Right, not when he first, you know, conceptualized, you know, and he conceptualized universal computing for a very specific mathematical reason that wasn't as general. But yes, it's a good sort of exercise in humility to realize that it's kind of like, it's really hard to figure these things out. The engineering of the universe, if we know how the universe works, how can we engineer it? - That's such a beautiful vision. That's such a beautiful- - By the way, I have to mention one more thing, which is the ultimate question from physics is, okay, so we have this abstract model of the universe. Why does the universe exist at all, right? So, you know, we might say there is a formal model that if you run this model, you get the universe, or the model gives you, you know, a model of the universe, right? You run this mathematical thing, and the mathematics unfolds in the way that corresponds to the universe. But the question is, why was that actualized? Why does the actual universe actually exist? And so this is another one of these humility, and it's like, can you figure this out? I have a guess, okay, about the answer to that. And my guess is somewhat unsatisfying, but my guess is that it's a little bit similar to Godel's second incompleteness theorem, which is the statement that from within as an axiomatic theory like Peano arithmetic, you cannot from within that theory prove the consistency of the theory. So my guess is that for entities within the universe, there is no finite determination that can be made of the statement the universe exists is essentially undecidable to any entity that is embedded in the universe. - Within that universe, how does that make you feel? Does that put you at peace that it's impossible, or is it really ultimately frustrating? - Well, I think it just says that it's not a kind of question that, you know, there are things that it is reasonable. I mean, there's kinds of, you know, you can talk about hypercomputation as well. You can say, imagine there was a hypercomputer, here's what it would do. So, okay, great, it would be lovely to have a hypercomputer, but unfortunately we can't make it in the universe. Like, it would be lovely to answer this, but unfortunately we can't do it in the universe. And, you know, this is all we have, so to speak. And I think it's really just a statement. It's sort of, in the end, it'll be a kind of a logical, logically inevitable statement, I think. I think it will be something where it is, as you understand what it means to have a sort of predicate of existence and what it means to have these kinds of things, it will sort of be inevitable that this has to be the case, that from within that universe, you can't establish the reason for its existence, so to speak. You can't prove that it exists and so on. - And nevertheless, because of computational reducibility, the future is ultimately not predictable, full of mystery, and that's what makes life worth living. - Right, I mean, right. And, you know, it's funny for me, because as a pure sort of human being doing what I do, it's, you know, I'm, you know, I like, I'm interested in people. I like sort of, you know, the whole human experience, so to speak. And yet, it's a little bit weird when I'm thinking, you know, it's all hypergraphs down there, and it's all just-- - Hypergraphs all the way down. - Right. - It's like turtles all the way down. - Yeah, yeah, right. And it's kind of, you know, it's, to me, it is a funny thing, because every so often I get this, you know, as I'm thinking about, I think we've really gotten, you know, we've really figured out kind of the essence of how physics works, and I'm like thinking to myself, you know, here's this physical thing, and I'm like, you know, this feels like a very definite thing. How can it be the case that this is just some ruleal reference frame of, you know, this infinite creature that is so abstract and so on? And I kind of, it is a, it's a funny sort of feeling that, you know, we are, we're sort of, it's like, in the end, it's just sort of-- - We're just like a little speck. - Be happy we're just humans type thing. And it's kind of like, but we're making, we make things as, it's not like we're just a tiny speck. We are, in a sense, we are more important by virtue of the fact that, in a sense, it's not like there's, there is no ultimate, you know, it's like we're important because, because, you know, we're here, so to speak, and we're not, it's not like there's a thing where we're saying, you know, we are just but one sort of intelligence out of all these other intelligences, and so, you know, ultimately, there'll be the super intelligence, which is all of these put together, and it'll be very different from us. No, it's actually gonna be equivalent to us. And the thing that makes us sort of special is just the details of us, so to speak. It's not something where we can say, oh, there's this other thing, you know, just you think humans are cool, just wait until you've seen this. You know, it's gonna be much more impressive. Well, no, it's all going to be kind of computationally equivalent. And the thing that, you know, it's not gonna be, oh, this thing is amazingly much more impressive and amazingly much more meaningful, let's say. No, we're it. I mean, that's the-- - And the symbolism of this particular moment, so this has been one of the favorite conversations I've ever had, Stephen. It's a huge honor to talk to you, to talk about a topic like this for four plus hours on the fundamental theory of physics, and yet we're just two finite descendants of apes that have to end this conversation because darkness have come upon us. - Right, and we're gonna get bitten by mosquitoes and all kinds of terrible things. - The symbolism of that, we're talking about the most basic fabric of reality and having to end because of the fact that things end. It's tragic and beautiful, Stephen. Thank you so much, huge honor. I can't wait to see what you do in the next couple of days and next week, a month. We're all watching with excitement. Thank you so much. - Thanks. - Thanks for listening to this conversation with Stephen Wolfram, and thank you to our sponsors, Simply Safe, Sun Basket, and Masterclass. Please check out our sponsors in the description to get a discount and to support this podcast. If you enjoy this thing, subscribe on YouTube, review it with Five Stars on Apple Podcast, follow on Spotify, support on Patreon, or connect with me on Twitter @LexFriedman. And now, let me leave you with some words from Richard Feynman. Physics isn't the most important thing. Love is. Thank you for listening. I hope to see you next time. (upbeat music) (upbeat music) Thanks for watching.