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Stephen Wolfram is a computer scientist, mathematician, and theoretical physicist who is the founder and CEO of Wolfram Research, a company behind Mathematica, Wolfram Alpha, Wolfram Language, and the new Wolfram Physics project. He is the author of several books including A New Kind of Science, which on a personal note was one of the most influential books in my journey in computer science and artificial intelligence.

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What kind of computation do you think the fundamental laws of physics might emerge from?

So just to clarify, so you've done a lot of fascinating work with kind of discrete kinds of computation that, you know, because cellular automata, and we'll talk about it, have this very clean structure.

It's such a nice way to demonstrate that simple rules can create immense complexity.

But what, you know, is that actually, are cellular autonomous sufficiently general to describe the kinds of computation that might create the laws of physics?

Just to give, can you give a sense of what kind of computation do you think would create

So this is a slightly complicated issue because as soon as you have universal computation, you can in principle simulate anything with anything.

And if you're asking, were you to try to find our physical universe by looking at possible programs in the computational universe of all possible programs, would the ones that correspond to our universe be small and simple enough that we might find them by searching that computational universe?

We gotta have the right basis, so to speak.

We have got to have the right language in effect for describing computation for that to be feasible.

So the thing that I've been interested in for a long time is what are the most structuralist structures that we can create with computation?

So In other words, if you say a cellular automaton has a bunch of cells that are arrayed on a grid and it's very, you know, and every cell is updated in synchrony at a particular, you know, when there's a click of a clock, so to speak, and it goes, a tick of a clock, and every cell gets updated at the same time.

That's a very specific, very rigid kind of thing.

But my guess is that when we look at physics and we look at things like space and time, that what's underneath space and time is something as structureless as possible.

That what we see, what emerges for us as physical space, for example, comes from something that is sort of arbitrarily unstructured underneath.

And so I've been for a long time interested in kind of what are the most structureless structures that we can set up?

And actually what I had thought about for ages is using graphs, networks, where essentially, so let's talk about space, for example.

Back in the early days of quantum mechanics, for example, people said, oh, for sure, space is gonna be discrete because all these other things we're finding are discrete, but that never worked out in physics.

And so space and physics today is always treated as this continuous thing, just like Euclid imagined it.

I mean, the very first thing Euclid says in his sort of common notions is, you know, a point is something which has no part.

In other words, there are points that are arbitrarily small and there's a continuum of possible positions of points.

And so for example, if we look at, I don't know, fluid like air or water, we might say, oh, it's a continuous fluid.

We can pour it, we can do all kinds of things continuously.

But actually we know, because we know the physics of it, that it consists of a bunch of discrete molecules bouncing around and only in the aggregate is it behaving like a continuum.

And so the possibility exists that that's true of space too.

People haven't managed to make that work with existing frameworks and physics, but I've been interested in whether 1 can imagine that underneath space and also underneath time is something more structureless.

And the question is, is it computational?

It could be computational, somehow fundamentally equivalent to a Turing machine, or it could be fundamentally not.

It could not be, so a Turing machine essentially deals with integers, whole numbers, at some level.

And it can do things like it can add 1 to a number.

It can also store whatever the heck it did.

But what, when 1 thinks about doing physics, or sort of idealized physics, or idealized mathematics, 1 can deal with real numbers, numbers with an infinite number of digits.

Someone can say, we can take this number and we can multiply it by itself.

Are you comfortable with infinity in this context?

Are you comfortable in the context of computation?

I think that the role of infinity is complicated.

Infinity is useful in conceptualizing things.

Almost by definition, it's not actualizable.

But do you think infinity is part of the thing that might underlie the laws of physics?

I think there are many questions that you ask about, you might ask about physics, which inevitably involve infinity.

Like when you say, is faster than light travel possible?

You could say, given the laws of physics, can you make something even arbitrarily large, even quotes infinitely large that will make faster than light travel possible?

Then you're thrown into dealing with infinity as a kind of theoretical question.

But I mean, talking about sort of what's underneath space and time and how 1 can make a computational infrastructure, 1 possibility is that you can't make a computational infrastructure in a Turing machine sense, that you really have to be dealing with precise real numbers.

You're dealing with partial differential equations, which have precise real numbers at arbitrarily closely separated points.

Could be that that's what happens, that there's sort of a continuum for everything and precise real numbers for everything.

And then the things I'm thinking about are wrong.

And that's the risk you take if you're trying to sort of do things about nature is you might just be wrong.

It's not, it's for me personally, it's kind of a strange thing because I've spent a lot of my life building technology where you can do something that nobody cares about, but you can't be sort of wrong in that sense, in the sense you build your technology and it does what it does.

But I think this question of what the sort of underlying computational infrastructure for the universe might be.

So it's sort of inevitable it's gonna be fairly abstract because if you're gonna get all these things like there are 3 dimensions of space, there are electrons, there are muons, there are quarks, there are this.

You don't get to, if the model for the universe is simple, you don't get to have sort of a line of code for each of those things.

You don't get to have sort of the muon case, the tau lepton case and so on.

All of those things have to be emergent somehow.

Right, so that means it's sort of inevitable that it's a little hard to talk about what the sort of underlying structuralist structure actually is.

Do you think human beings have the cognitive capacity to understand, if we're to discover it, to understand the kinds of simple structure from which these laws can emerge?

I think that, I mean, I'm right in the middle of this right now, so I'm telling you that this human, yeah, I mean, this human has a hard time understanding a bunch of the things that are going on, but what happens in understanding is, 1 builds waypoints.

I mean, if you said, understand modern 21st century mathematics, starting from counting, back in whenever counting was invented 50,000 years ago, whatever it was, right?

But what happens is we build waypoints that allow us to get to higher levels of understanding.

And we see the same thing happening in language.

You know, when we invent a word for something, it provides kind of a cognitive anchor, a kind of a waypoint that lets us, you know, like a podcast or something.

You could be explaining, well, it's a thing, which this works this way, that way, the other way.

But as soon as you have the word podcast and people kind of societally understand it, you start to be able to build on top of that.

And so I think, and that's kind of the story of science actually too.

I mean, science is about building these kind of way points where we find this sort of cognitive mechanism for understanding something, then we can build on top of it.

You know, we have the idea of, I don't know, differential equations, we can build on top of that.

So my hope is that if it is the case that we have to go all the way sort of from the sand to the computer and there's no way points in between, then we're toast.

So if we're, us clever apes are good enough for building those abstractions, eventually from sand we'll get to the computer, right?

than- The question is whether it is something that you asked, whether our human brains will, will quotes understand what's going on.

And that's a different question, because for that, it requires steps that are sort of from which we can construct a human understandable narrative.

And that's something that I think I am somewhat hopeful that that will be possible, Although, you know, as of literally today, if you ask me, I'm confronted with things that I don't understand very well.

So this is a small pattern in a computation trying to understand the rules under which the computation functions.

possibility under which kinds of computations such a creature can understand itself.

My guess is that within, so we didn't talk much about computational irreducibility, but it's a consequence of this principle of computational equivalence.

And it's sort of a core idea that 1 has to understand, I think, which is, the question is, you're doing a computation, you can figure out what happens in the computation just by running every step in the computation and seeing what happens.

Or you can say, let me jump ahead and figure out, you know, have something smarter that figures out what's gonna happen before it actually happens.

And a lot of traditional science has been about that act of computational reducibility.

It's like, we've got these equations and we can just solve them and we can figure out what's gonna happen.

We don't have to trace all of those steps.

We just jump ahead because we solved these equations.

Okay, so 1 of the things that is a consequence of the principle of computational equivalence is you don't always get to do that.

Many, many systems will be computationally irreducible in the sense that the only way to find out what they do is just follow each step and see what happens.

Well, if you're saying, well, we, with our brains, we're a lot smarter.

We don't have to mess around like the little cellular automaton going through and updating all those cells, we can just use the power of our brains to jump ahead.

But if the principle of computational equivalence is right, that's not gonna be correct, because it means that there's us doing our computation in our brains, there's a little cellular automaton doing its computation, and the principle of computational equivalence says, these 2 computations are fundamentally equivalent.

So that means we don't get to say, we're a lot smarter than the cellular automaton and jump ahead, because we're just doing computation that's of the same sophistication as the cellular automaton itself.

That's computational reducibility, it's fascinating.

I think that's both depressing and humbling and so on, that we're all, we and the cellular automaton are

But the question we're talking about the fundamental laws of physics is kind of the reverse question.

You're not predicting what's gonna happen.

But saying, can I understand what rules likely generated me?

I understand, but the problem is to know whether you're right, you have to have some computational reducibility because we are embedded in the universe.

If the only way to know whether we get the universe is just to run the universe, we don't get to do that because it just ran for 14.6 billion years or whatever.

So we have to hope that there are pockets of computational reducibility sufficient to be able to say, yes, I can recognize those are electrons there.

And I think that it's a feature of computational irreducibility.

It's sort of a mathematical feature that there are always an infinite collection of pockets of reducibility.

The question of whether they land in the right place and whether we can sort of build a theory based on them is unclear.

But to this point about whether we as observers in the universe built out of the same stuff as the universe can figure out the universe, so to speak, that relies on these pockets of reducibility.

Without the pockets of reducibility, it won't work, can't work.

But I think this question about how observers operate, it's 1 of the features of science over the last hundred years particularly, has been that every time we get more realistic about observers, we learn a bit more about science.

So for example, relativity was all about, observers don't get to say when, you know, what's simultaneous with what, they have to just wait for the light signal to arrive to decide what's simultaneous.

Or for example, in thermodynamics, observers don't get to say the position of every single molecule in a gas.

They can only see the kind of large-scale features and that's why the second law of thermodynamics, law of entropy increase, and so on works.

If you could see every individual molecule, you wouldn't conclude something about thermodynamics.

You would conclude, oh, these molecules just all doing these particular things, you wouldn't be able to see this aggregate fact.

So I strongly expect that, and in fact, in the theories that I have, that 1 has to be more realistic about the computation and other aspects of observers in order to actually make a correspondence between what we experience.

In fact, my little team and I have a little theory right now about how quantum mechanics may work, which is a very wonderfully bizarre idea about how the sort of thread of human consciousness relates to what we observe in the universe.

But there's several steps to explain what that's about.

What do you make of the mess of the observer at the lower level of quantum mechanics.

Sort of the textbook definition with quantum mechanics kind of says that there's 2 worlds.

1 is the world that actually is, and the other is that's observed.

Well, I think actually the ideas we've recently had might actually give away into this.

I mean, the fact is there is a, 1 of the things that's interesting, and when people look at these models that I started talking about 30 years ago now, they say, oh no, that can't possibly be right.

And You say, okay, tell me what is the essence of quantum mechanics?

What do you want me to be able to reproduce to know that I've got quantum mechanics, so to speak?

Well, and that question comes up, it comes up very operationally actually, because we've been doing a bunch of stuff with quantum computing, and there are all these companies that say, we have a quantum computer.

We say, let's connect to your API and let's actually run it.

And they're like, well, maybe you shouldn't do that yet.

And 1 of the questions that I've been curious about is, if I have 5 minutes with a quantum computer, how can I tell if it's really a quantum computer, or whether it's a simulator at the other end?

It turns out there isn't, it's like a lot of these questions about sort of what is intelligence, what's life.

It's like, are you really a quantum computer?

Is it just a simulation or is it really a quantum computer?

But that, so, you know, this whole issue about the sort of mathematical structure of quantum mechanics and the completely separate thing that is our experience in which we think definite things happen, whereas quantum mechanics doesn't say definite things ever happen.

Quantum mechanics is all about the amplitudes for different things to happen.

But yet, our thread of consciousness operates as if definite things are happening.

To linger on the point, you've kind of mentioned the structure that could underlie everything and this idea that it could perhaps have something like the structure of a graph.

Can you elaborate why your intuition is that there's a graph structure of nodes and edges and what it might represent.

Right, okay, so the question is, what is, in a sense, the most structuralist structure you can imagine, right?

So, and in fact, what I've recently realized in the last year or so, I have a new, most structureless structure.

By the way, the question itself is a beautiful 1 and a powerful 1 in itself.

So even without an answer, just the question is a really strong question.

Essentially, what is interesting about the model I have now is it's a little bit like what happened with computation.

Everything that I think of as, oh, well, maybe the model is this.

And that's quite encouraging because it's like, I could say, well, I'm gonna look at trivalent graphs with 3 edges for each node and so on.

Or I could look at this special kind of graph, or I could look at this kind of algebraic structure.

And turns out that the things I'm now looking at, everything that I've imagined that is a plausible type of structureless structure is equivalent to this.

Well, a typical way to think about it is, well, so you might have some collection of tuples, collection of, let's say numbers.

So you might have 135234, little, just collections of numbers, triples of numbers, let's say, quadruples of numbers, pairs of numbers, whatever.

And you have all these sort of floating little tuples.

And that sort of floating collection of tuples, and I told you this was abstract, represents the whole universe.

The only thing that relates them is When a symbol is the same, it's the same, so to speak.

So if you have 2 tuples and they contain the same symbol, let's say at the same position of the tuple, the first element of the tuple, then that represents a relation.

It's just, I told you it's abstract, but this is the, So

the relationship is formed by the same, some aspect of sameness.

Right, but so think about it in terms of a graph.

So a graph, bunch of nodes, let's say you number each node, okay?

A graph is a set of pairs that say, this node has an edge connecting it to this other node.

So that's the, that's an, a graph is just a, a collection of those pairs that say, this node connects to this other node.

So this is a generalization of that, in which instead of having pairs, you have arbitrary and tuples.

And now the question is, okay, so that might be, that might represent the state of the universe.

And so the answer is that what I'm looking at is transformation rules on these hypergraphs.

In other words, you say this, whenever you see a piece of this hypergraph that looks like this, turn it into a piece of hypergraph that looks like this.

So on a graph, it might be when you see the subgraph, when you see this thing with a bunch of edges hanging out in this particular way, then rewrite it as this other graph.

So the question is what, so now you say, I mean, as I say, this is quite abstract.

And 1 of the questions is, where do you do those updating?

Is it, and I suspect everything's discrete, even in time, so.

Okay, so the question is, where do you do the updates?

And the answer is, the rule is, you do them wherever they apply.

And you do them, the order in which the updates is done is not defined.

That is, you can do them, So there may be many possible orderings for these updates.

Now, the point is, imagine you're an observer in this universe.

So, and you say, did something get updated?

Well, you don't in any sense know until you yourself have been updated.

So in fact, all that you can be sensitive to is essentially the causal network of how an event over there affects an event that's in you.

Yes, you're part of it, but even to have, so the end result of that is all you're sensitive to is this causal network of what event affects what other event.

I'm not making a big statement about sort of the structure of the observer.

I'm simply saying, I'm simply making the argument that what happens, the microscopic order of these rewrites is not something that any observer, any conceivable observer in this universe can be affected by.

Because the only thing the observer can be affected by is this causal network of how the events in the observer are affected by other events that happen in the universe.

So the only thing you have to look at is the causal network.

You don't really have to look at this microscopic rewriting that's happening.

So these rewrites are happening wherever they, they happen wherever they feel like.

Causal network, is there, you said that there's not really, So the idea would be an undefined, like what gets updated, the sequence of things is undefined.

That's what you mean by the causal network, but then the

causal- No, the causal network is given that an update has happened, that's an event.

Then the question is, is that event causally related to?

Does that event, if that event didn't happen, then some future event couldn't happen yet.

And so you build up this network of what affects what.

Okay, and so what that does, so when you build up that network, that's kind of the observable aspect of the universe in some sense.

And so then you can ask questions about, you know, how robust is that observable network of what's happening in the universe?

Okay, so here's where it starts getting kind of interesting.

So for certain kinds of microscopic rewriting rules, the order of rewrites does not matter to the causal network.

And so this is, okay, mathematical logic moment, this is equivalent to the Church-Rosser property or the confluence property of rewrite rules.

And it's the same reason that if you're simplifying an algebraic expression, for example, you can say, oh, let me expand those terms out.

Doesn't matter what order you do that in, you'll always get the same answer.

And that's, it's the same fundamental phenomenon that causes for certain kinds of microscopic rewrite rules that causes the causal network to be independent of the microscopic order of rewritings.

I mean, the reason it's important is that that property, special relativity says you can look at these sort of, you can look at different reference frames.

You can have different, you can be looking at your notion of what space and what time can be different, depending on whether you're traveling at a certain speed, depending on whether you're doing this, that, and the other.

But nevertheless, the laws of physics are the same.

That's what the principle of special relativity says, is laws of physics are the same independent of your reference frame.

Well, turns out this sort of change of the microscopic rewriting order is essentially equivalent to a change of reference frame, or at least there's a sub part of how that works that's equivalent to change a reference frame.

So, somewhat surprisingly, and sort of for the first time in forever, it's possible for an underlying microscopic theory to imply special relativity, to be able to derive it.

It's not something you put in as a, this is a, it's something where this other property, causal invariance, which is also the property that implies that there's a single thread of time in the universe.

It might not be the case that that's, that is the, that's what would lead to the possibility of an observer thinking that definite stuff happens.

Otherwise you've got all these possible rewriting orders and who's to say which 1 occurred.

But with this causal invariance property, there's a notion of a definite thread of time.

It sounds like that kind of idea of time, even space would be emergent from the system.

So it's not a fundamental part of the system.

at a fundamental level, all you've got is a bunch of nodes connected by hyper edges or whatever.

And, but the thing is that it's just like imagining, imagine you're just dealing with a graph and imagine you have something like a, you know, like a honeycomb graph where you have a hexagon, a bunch of hexagons, you know, that graph at a microscopic level, it's just a bunch of nodes connected to other nodes.

But at a macroscopic level, you say that looks like a honeycomb, you know, lattice.

It looks like a two-dimensional, you know, manifold of some kind.

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Toward a Fundamental Theory of Physics (Stephen Wolfram) | AI Podcast Clips