The Basic Setup

Whether one’s dealing with biology, economics, politics or a host of other fields, it’s common to encounter situations that can be modeled as involving two agents that repeatedly compete with each other. One imagines that at each step each agent can take one of a certain set of actions, and that then—in a classic game theory way—each agent (or “player”) gets a certain fixed “payoff” based on the action they and their opponent take. But how do the agents decide what action to take? We imagine that each agent has a certain fixed procedure—or “strategy”—for making its decisions. And we imagine that the input to each of those decisions is the sequence of past actions that the agent and its opponent have taken.

There’s been lots of work done over the course of nearly a century on particular choices of strategies. But something I’ve long been curious about is what happens if one systematically considers all possible strategies. And if we think of strategies as programs this becomes a question to which we can immediately apply ruliological methods. Which is what I’m going to do here. Continue reading

Foundation Models Need a Foundation Tool

LLMs don’t—and can’t—do everything. What they do is very impressive—and useful. It’s broad. And in many ways it’s human-like. But it’s not precise. And in the end it’s not about deep computation.

So how can we supplement LLM foundation models? We need a foundation tool: a tool that’s broad and general and does what LLMs themselves don’t: provides deep computation and precise knowledge.

And, conveniently enough, that’s exactly what I’ve been building for the past 40 years! My goal with Wolfram Language has always been to make everything we can about the world computable. To bring together in a coherent and unified way the algorithms, the methods and the data to do precise computation whenever it’s possible. It’s been a huge undertaking, but I think it’s fair to say it’s been a hugely successful one—that’s fueled countless discoveries and inventions (including my own) across a remarkable range of areas of science, technology and beyond. Continue reading

The Wolfram Institute recently received a grant from the Templeton World Charity Foundation for “Computational Metaphysics”. I wrote this piece in part as a launching point for discussions with experts in traditional philosophy.

Moving Metaphysics from Philosophy to Science

“What ultimately is there?” has always been seen as a fundamental—if thorny—question for philosophy, or perhaps theology. But despite a couple of millennia of discussion, I think it’s fair to say that only modest progress has been made with it. But maybe, just maybe, this is the moment where that’s going to change—and on the basis of surprising new ideas and new results from our latest efforts in science, it’s finally going to be possible to make real progress, and in the end to build what amounts to a formal, scientific approach to metaphysics.

It all centers around the ultimate foundational construct that I call the ruliad—and how observers like us, embedded within it, must perceive it. And it’s a story of how—for observers like us—fundamental concepts like space, time, mathematics, laws of nature, and indeed, objective reality, must inevitably emerge. Continue reading

Empirical Theoretical Computer Science

“Could there be a faster program for that?” It’s a fundamental type of question in theoretical computer science. But except in special cases, such a question has proved fiendishly difficult to answer. And, for example, in half a century, almost no progress has been made even on the rather coarse (though very famous) P vs. NP question—essentially of whether for any nondeterministic program there will always be a deterministic one that is as fast. From a purely theoretical point of view, it’s never been very clear how to even start addressing such a question. But what if one were to look at the question empirically, say in effect just by enumerating possible programs and explicitly seeing how fast they are, etc.?

One might imagine that any programs one could realistically enumerate would be too small to be interesting. But what I discovered in the early 1980s is that this is absolutely not the case—and that in fact it’s very common for programs even small enough to be easily enumerated to show extremely rich and complex behavior. With this intuition I already in the 1990s began some empirical exploration of things like the fastest ways to compute functions with Turing machines. But now—particularly with the concept of the ruliad—we have a framework for thinking more systematically about the space of possible programs, and so I’ve decided to look again at what can be discovered by ruliological investigations of the computational universe about questions of computational complexity theory that have arisen in theoretical computer science—including the P vs. NP question. Continue reading

Ruliology is taking off! And more and more people are talking about it. But what is ruliology? Since I invented the term, I decided I should write something to explain it. But then I realized: I actually already wrote something back in 2021 when I first invented the term. What I wrote back then was part of something longer. But here now is the part that explains ruliology:

If one sets up a system to follow a particular set of simple rules, what will the system do? Or, put another way, how do all those simple programs out there in the computational universe of possible programs behave?

These are pure, abstract questions of basic science. They’re questions one’s led to ask when one’s operating in the computational paradigm that I describe in A New Kind of Science. But at some level they’re questions about the specific science of what abstract rules (that we can describe as programs) do. Continue reading

Scaling Up Your Computations

Let’s say you’ve done a computation in Wolfram Language. And now you want to scale it up. Maybe 1000x or more. Well, today we’ve released an extremely streamlined way to do that. Just wrap the scaled up computation in RemoteBatchSubmit and off it’ll go to our new Wolfram Compute Services system. Then—in a minute, an hour, a day, or whatever—it’ll let you know it’s finished, and you can get its results.

For decades I’ve often needed to do big, crunchy calculations (usually for science). With large volumes of data, millions of cases, rampant computational irreducibility, etc. I probably have more compute lying around my house than most people—these days about 200 cores worth. But many nights I’ll leave all of that compute running, all night—and I still want much more. Well, as of today, there’s an easy solution—for everyone: just seamlessly send your computation off to Wolfram Compute Services to be done, at basically any scale.

For nearly 20 years we’ve had built-in functions like ParallelMap and ParallelTable in Wolfram Language that make it immediate to parallelize subcomputations. But for this to really let you scale up, you have to have the compute. Which now—thanks to our new Wolfram Compute Services—everyone can immediately get. Continue reading

Towards a Theory of Bulk Orchestration

It’s a key feature of living systems, perhaps even in some ways the key feature: that even right down to a molecular scale, things are orchestrated. Molecules (or at least large ones) don’t just move around randomly, like in a liquid or a gel. Instead, what molecular biology has discovered is that there are endless active mechanisms that in effect orchestrate what even individual molecules in living systems do. But what is the result of all that orchestration? And could there perhaps be a general characterization of what happens in systems that exhibit such “bulk orchestration”? I’ve been wondering about these questions for some time. But finally now I think I may have the beginnings of some answers.

The central idea is to consider the effect that “being adapted for an overall purpose” has on the underlying operation of a system. At the outset, one might imagine that there’d be no general answer to this, and that it would always depend on the specifics of the system and the purpose. But what we’ll discover is that there is in fact typically a certain universality in what happens. Its ultimate origin is the Principle of Computational Equivalence and certain universal features of the phenomenon of computational irreducibility that it implies. But the point is that so long as a purpose is somehow “computationally simple” then—more or less regardless of what in detail the purpose is—a system that achieves it will show certain features in its behavior. Continue reading

What Are Lambdas?

It’s a story of pure, abstract computation. In fact, historically, one of the very first. But even though it’s something I for one have used in practice for nearly half a century, it’s not something that in all my years of exploring simple computational systems and ruliology I’ve ever specifically studied. And, yes, it involves some fiddly technical details. But it’ll turn out that lambdas—like so many systems I’ve explored—have a rich ruliology, made particularly significant by their connection to practical computing.

In Wolfram Language it’s the Function function. Back when Alonzo Church first discussed it in the 1930s he called it λ (lambda). The idea is to have something that serves as a “pure function”—which can be applied to an argument to give a value. For example, in the Wolfram Language one might have:

Continue reading

Theories of the World

Several often arrive in a single day. Sometimes they’re marked “urgent”. Sometimes they’re long. Sometimes they’re short. Sometimes they’re humble. Sometimes they’re conspiratorial. And sometimes, these days, they’re written “in collaboration with” an AI. But there’s a common theme: they’re all emails that present some kind of fundamental theory invented by their authors (or perhaps their AI) about how our universe works.

At some level it’s encouraging to see how many people find it interesting to think about fundamental questions in science. But at some level it’s also to me very frustrating. All that effort being spent. And so much of it so wide of the mark. Most of the time it’s based on at best high-school physics—missing everything that was learned in twentieth-century physics. Sometimes it’s easy to tell that what’s being said just can’t be right; often things are too vague or tangled for one to be able to say much.

Most physicists term people who send such theories “crackpots”, and either discard their missives or send back derisive responses. I’ve never felt like that was the right thing to do. Somehow I’ve always felt as if there has to be a way to channel that interest and effort into something that would be constructive and fulfilling for all concerned. And maybe, just maybe, I now have at least one idea in that direction. Continue reading

This Is a Big Release

Version 14.2 launched on January 23 of this year. Now, today, just over six months later, we’re launching Version 14.3. And despite its modest .x designation, it’s a big release, with lots of important new and updated functionality, particularly in core areas of the system.

I’m particularly pleased to be able to report that in this release we’re delivering an unusually large number of long-requested features. Why didn’t they come sooner? Well, they were hard—at least to build to our standards. But now they’re here, ready for everyone to use.

Those who’ve been following our livestreamed software design reviews (42 hours of them since Version 14.2) may get some sense of the effort we put into getting the design of things right. And in fact we’ve been consistently putting in that kind of effort now for nearly four decades—ever since we started developing Version 1.0. And the result is something that I think is completely unique in the software world—a system that is consistent and coherent through and through, and that has maintained compatibility for 37 years. Continue reading