The Challenge of Language Design for the Universe

What does it mean to make a model for the universe? If we just want to know what the universe does, well, then we have the universe, and we can just watch what it does. But when we talk about making a model, what we really mean is that we want to have a representation of the universe that somehow connects it to what we humans can understand. Given computational irreducibility, it’s not that we expect a model that will in any fundamental sense “predict in advance” the precise behavior of the universe down to every detail (like that I am writing this sentence now). But we do want to be able to point to the model—whose structure we understand—and then be able to say that this model corresponds to our universe.

In the previous section we said that we wanted to find a rule that we could in a sense connect with the description language that we use for the universe. But what should the description language for the rule itself be? Inevitably there is a great computational distance between the underlying rule and features of the universe that we’re used to describing. So—as I’ve said several times here in different ways—we can’t expect to use the ordinary concepts with which we describe the world (or physics) directly in the construction of the rule.

I’ve spent the better part of my life as a language designer, primarily building what’s now the?full-scale computational language?that is the?Wolfram Language. And I now view the effort to find a fundamental theory of physics as in many ways just another challenge in language design—perhaps even the ultimate such challenge.

In designing a computational language what one is really trying to do is to create a bridge between two domains: the abstract world of what is possible to do computationally, and the “mental” world of what people understand and are interested in doing. There are all sorts of computational processes that one can invent (say running randomly picked cellular automaton rules), but the challenge in language design is to figure out which ones people care about at this point in human history, and then to give people a way to describe these.

Usually in computational language design one is leveraging human natural language—or the more formal languages that have been developed in mathematics and science—to find words or their analogs to refer to particular “l(fā)umps of computation”. But at least in the way I have done it, the essence of language design is to try to find the purest primitives that can be expressed this way.

OK, so let’s talk about setting up a model for the universe. Perhaps the single most important idea in my effort to find a fundamental theory of physics is that the theory should be based on the?general computational paradigm?(and not, for example, specifically on mathematics). So when we talk about having a language in which to describe our model of the universe we can see that it has to bridge three different domains. It has to be a language that humans can understand. It has to be a language that can express computational ideas. And it has to be a language that can actually represent the underlying structure of physics.

So what should this language be like? What kinds of primitives should it contain? The?history that has led me to what I describe here?is in many ways the history of my attempts to formulate an appropriate language. Is it?trivalent graphs? Is it ordered graphs? Is it?rules applied to abstract relations?

In many ways, we are inevitably skating at the edge of what humans can understand. Maybe one day we will have built up familiar ways of talking about the concepts that are involved. But for now, we don’t have these. And in a sense what has made this project feasible now is that we’ve come so far in developing ways to express computational ideas—and that through the Wolfram Language in particular those forms of expression have become familiar, at the very least to me.

And it’s certainly satisfying to see that the basic structure of the models we’re using can be expressed?very cleanly and succinctly in the Wolfram Language. In fact, in what perhaps can be viewed as some sort of endorsement of the structure of the Wolfram Language, the models are in a sense just a quintessential example of?transformation rules for symbolic expressions, which is exactly what the?Wolfram Language is based on. But even though the structure is well represented in the Wolfram Language, the “use case” of “running the universe” is different from what the Wolfram Language is normally set up to do.

In the effort to serve what people normally want, the Wolfram Language is primarily about taking input, evaluating it by doing computation, and then generating output. But that’s not what the universe does. The universe in a sense had input at the very beginning, but now it’s just running an evaluation—and with all our different ideas of foliations and so on, we are sampling certain aspects of that ongoing evaluation.

It’s computation, but it’s computation sampled in a different way than we’ve been used to doing it. To a language designer like me, this is something interesting in its own right, with its own scientific and technological spinoffs. And perhaps it will take more ideas before we can finish the job of finding a way to represent a rule for fundamental physics.

But I’m optimistic that we actually already have pretty much all the ideas we need. And we also have a crucial piece of methodology that helps us: our ability to do explorations through?computer experiments. If we based everything on the traditional methodology of mathematics, we would in effect only be able to explore what we somehow already understood. But in running computer experiments we are in effect sampling the raw computational universe of possibilities, without being limited by our existing understanding.

Of course, as with physical experiments, it matters how we define and think about our experiments, and in effect what description language we use. But what certainly helps me, at least, is that I’ve now been doing computer experiments for more than forty years, and over that time I’ve been able to slowly refine the art and science of how best to do them.

In a way it’s very much like how we learn from our experience in the physical world. From seeing the results of many experiments, we?gradually build up intuition, which in turn lets us start creating a conceptual framework, which then informs the design of our language for describing things. One always has to keep doing experiments, though. In a sense computational irreducibility implies that there will always be surprises, and that’s certainly what I constantly find in practice, not least in this project.

Will we be able to bring together physics, computation and human understanding to deliver what we can reasonably consider to be a final, fundamental theory of physics? It is difficult to know how hard this will be. But I am extremely optimistic that we are finally on the right track, and may even have effectively already solved the fascinating problem of language design that this entails.

Let’s Go Find the Fundamental Theory!

OK, so given all this, what’s it going to take to find the fundamental theory of physics? The most important thing—about which I’m extremely excited—is that I think we’re finally on the right track. Of course, perhaps not surprisingly, it’s still technically difficult. Part of that difficulty comes directly from computational irreducibility and from the difficulty of working out the consequences of underlying rules. But part of the difficulty also comes from the very success and sophistication of existing physics.

In the end our goal must be to build a bridge that connects our models to existing knowledge about physics. And there is difficult work to do on both sides. Trying to frame the consequences of our models in terms that align with existing physics, and trying to frame the (usually mathematical) structures of existing physics in terms that align with our models.

For me, one of the most satisfying aspects of our discoveries over the past couple of months has been the extent to which they end up resonating with a huge range of existing—sometimes so far seemingly “just mathematical”—directions that have been taken in physics in recent years. It almost seems like everyone has been right all along, and it just takes adding a new substrate to see how it all fits together. There are hints of string theory, holographic principles, causal set theory, loop quantum gravity, twistor theory, and much more. And not only that, there are also modern mathematical ideas—geometric group theory, higher-order category theory, non-commutative geometry, geometric complexity theory, etc.—that seem so well aligned that one might almost think they must have been built to inform the analysis of our models.

I have to say I didn’t expect this. The ideas and methods on which our models are based are very different from what’s ever been seriously pursued in physics, or really even in mathematics. But somehow—and I think it’s a good sign all around—what’s emerged is something that aligns wonderfully with lots of recent work in physics and mathematics. The foundations and motivating ideas are different, but the methods (and sometimes even the results) often look to be quite immediately applicable.

There’s something else I didn’t expect, but that’s very important. In studying things (like?cellular automata) out in the computational universe of simple programs, I have normally found that computational irreducibility—and phenomena like?undecidability—are everywhere. Try using sophisticated methods from mathematics;?they will almost always fail. It is as if one hits the wall of irreducibility almost immediately, so there is almost nothing for our sophisticated methods, which ultimately rely on reducibility, to do.

But perhaps because they are so minimal and so structureless our models for fundamental physics don’t seem to work this way. Yes, there is computational irreducibility, and it’s surely important, both in principle and in practice. But the surprising thing is that there’s a remarkable depth of richness before one hits irreducibility. And indeed that’s where many of our recent discoveries come from. And it’s also where existing methods from physics and mathematics have the potential to make great contributions. But what’s important is that it’s realistic that they can; there’s a lot one can understand before one hits computational irreducibility. (Which is, by the way, presumably why we are fundamentally able to form a coherent view of physical reality at all.)

So how is the effort to try to find a fundamental theory of physics going to work in practice? We plan to have a centralized effort that will push forward with the project using essentially the same R&D methods that we’ve developed at?Wolfram Researchover the past three decades, and that have successfully brought us so much technology—not to mention what exists of this project so far. But we plan to do everything in a completely open way. We’ve already posted the full?suite of software tools?that we’ve developed, along with nearly a thousand?archived working notebooks?going back to the 1990s, and soon more than 400 hours of?videos of recent working sessions.

We want to make it as easy for people to get involved as possible, whether directly in our centralized effort, or in separate efforts of their own. We’ll be?livestreaming what we do, and soliciting as much interaction as possible. We’ll be running a variety of?educational programs. And we also plan to have (livestreamed) working sessions with other individuals and groups, as well as providing channels for the computational publishing of results and intermediate findings.

I have to say that for me, working on this project both now and in past years has been tremendously exciting, satisfying, and really just fun. And I’m hoping many other people will be able to share in this as the project goes forward. I think we’ve finally got a path to finding the fundamental theory of physics. Now let’s go follow that path. Let’s have a blast. And let’s try to make this the time in human history when we finally figure out how this universe of ours works!