1 | Mathematics and Physics Have the Same Foundations

One of the many surprising (and to me, unexpected) implications of our Physics Project is its suggestion of a very deep correspondence between the foundations of physics and mathematics. We might have imagined that physics would have certain laws, and mathematics would have certain theories, and that while they might be historically related, there wouldn’t be any fundamental formal correspondence between them.

But what our Physics Project suggests is that underneath everything we physically experience there is a single very general abstract structure—that we call the ruliad—and that our physical laws arise in an inexorable way from the particular samples we take of this structure. We can think of the ruliad as the entangled limit of all possible computations—or in effect a representation of all possible formal processes. And this then leads us to the idea that perhaps the ruliad might underlie not only physics but also mathematics—and that everything in mathematics, like everything in physics, might just be the result of sampling the ruliad. Continue reading

The March of Innovation Continues

Just a few weeks ago it was 1/3 of a century since Mathematica 1.0 was released. Today I’m excited to announce the latest results of our long-running R&D pipeline: Version 13 of Wolfram Language and Mathematica. (Yes, the 1, 3 theme—complete with the fact that it’s the 13th of the month today—is amusing, if coincidental.)

It’s 207 days—or a little over 6 months—since we released Version 12.3. And I’m pleased to say that in that short time an impressive amount of R&D has come to fruition: not only a total of 117 completely new functions, but also many hundreds of updated and upgraded functions, several thousand bug fixes and small enhancements, and a host of new ideas to make the system ever easier and smoother to use.

Every day, every week, every month for the past third of a century we’ve been pushing hard to add more to the vast integrated framework that is Mathematica and the Wolfram Language. And now we can see the results of all those individual ideas and projects and pieces of work: a steady drumbeat of innovation sustained now over the course of more than a third of a century:

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The Entangled Limit of Everything

I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible computational rules in all possible ways. It’s yet another surprising construct that’s arisen from our Physics Project. And it’s one that I think has extremely deep implications—both in science and beyond.

In many ways, the ruliad is a strange and profoundly abstract thing. But it’s something very universal—a kind of ultimate limit of all abstraction and generalization. And it encapsulates not only all formal possibilities but also everything about our physical universe—and everything we experience can be thought of as sampling that part of the ruliad that corresponds to our particular way of perceiving and interpreting the universe.

We’re going to be able to say many things about the ruliad without engaging in all its technical details. (And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) But to ground things here, let’s start with a slightly technical discussion of what the ruliad is. Continue reading

Mathematica 1.0 was launched on June 23, 1988. So (depending a little on how you do the computation) today is its one-third-century anniversary. And it’s wonderful to see how the tower of ideas and technology that we’ve worked so hard on for so long has grown in that third of a century—and how tall it’s become and how rapidly it still goes on growing.

In the past few years, I’ve come to have an ever-greater appreciation for just how unique what we’ve ended up building is, and just how fortunate our original choices of foundations and principles were. And even after a third of a century, what we have still seems like an artifact from the future—indeed ever more so with each passing year as it continues to grow and develop.

In the long view of intellectual history, this past one-third century will be seen as the time when the computational paradigm first took serious root, and when all its implications for “computational X” began to grow. And personally I feel very fortunate to have lived at the right time in history to have been able to be deeply involved with this and for what we have built to have made such a contribution to it. Continue reading

A Minimal Example of Multicomputation

Multicomputation is an important new paradigm, but one that can be quite difficult to understand. Here my goal is to discuss a minimal example: multiway systems based on numbers. Many general multicomputational phenomena will show up here in simple forms (though others will not). And the involvement of numbers will often allow us to make immediate use of traditional mathematical methods.

A multiway system can be described as taking each of its states and repeatedly replacing it according to some rule or rules with a collection of states, merging any states produced that are identical. In our Physics Project, the states are combinations of relations between elements, represented by hypergraphs. We’ve also often considered string substitution systems, in which the states are strings of characters. But here I’ll consider the case in which the states are numbers, and for now just single integers.

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This is the first of a series of pieces I’m planning in connection with the upcoming 20th anniversary of the publication of A New Kind of Science.

“There’s a Whole New Field to Build…”

For me the story began nearly 50 years ago—with what I saw as a great and fundamental mystery of science. We see all sorts of complexity in nature and elsewhere. But where does it come from? How is it made? There are so many examples. Snowflakes. Galaxies. Lifeforms. Turbulence. Do they all work differently? Or is there some common underlying cause? Some essential “phenomenon of complexity”?

It was 1980 when I began to seriously work on these questions. And at first I did so in the main scientific paradigm I knew: models based on mathematics and mathematical equations. I studied the approaches people had tried to use. Nonequilibrium thermodynamics. Synergetics. Nonlinear dynamics. Cybernetics. General systems theory. I imagined that the key question was: “Starting from disorder and randomness, how could spontaneous self-organization occur, to produce the complexity we see?” For somehow I assumed that complexity must be created as a kind of filtering of ubiquitous thermodynamic-like randomness in the world.

At first I didn’t get very far. I could write down equations and do math. But there wasn’t any real complexity in sight. But in a quirk of history that I now realize had tremendous significance, I had just spent a couple of years creating a big computer system that was ultimately a direct forerunner of our modern Wolfram Language. So for me it was obvious: if I couldn’t figure out things myself with math, I should use a computer. Continue reading

The Path to a New Paradigm

One might have thought it was already exciting enough for our Physics Project to be showing a path to a fundamental theory of physics and a fundamental description of how our physical universe works. But what I’ve increasingly been realizing is that actually it’s showing us something even bigger and deeper: a whole fundamentally new paradigm for making models and in general for doing theoretical science. And I fully expect that this new paradigm will give us ways to address a remarkable range of longstanding central problems in all sorts of areas of science—as well as suggesting whole new areas and new directions to pursue. Continue reading

Hiding in Plain Sight for a Century?

On December 7, 1920, Moses Schönfinkel introduced the S and K combinators—and in doing so provided the first explicit example of a system capable of what we now call universal computation. A hundred years later—as I prepared to celebrate the centenary of combinators—I decided it was time to try using modern computational methods to see what we could now learn about combinators. And in doing this, I got a surprise.

It’s already remarkable that S and K yield universal computation. But from my explorations I began to think that something even more remarkable might be true, and that in fact S alone might be sufficient to achieve universal computation. Or in other words, that just applying the rule

S f g x → f[x][g[x]]

over and over again might be all that’s needed to do any computation that can be done.

I don’t know for sure that this is true, though I’ve amassed empirical evidence that seems to point in this direction. And today I’m announcing a prize of $20,000 (yes, the “20” goes with the 1920 invention of combinators, and the 2020 making of my conjecture) for proving—or disproving—that the S combinator alone can support universal computation. Continue reading

Based on a talk at Numerous Numerosity: An interdisciplinary meeting on the notions of cardinality, ordinality and arithmetic across the sciences.

Everyone Has to Have Numbers… Don’t They?

The aliens arrive in a starship. Surely, one might think, to have all that technology they must have the idea of numbers. Or maybe one finds an uncontacted tribe deep in the jungle. Surely they too must have the idea of numbers. To us numbers seem so natural—and “obvious”—that it’s hard to imagine everyone wouldn’t have them. But if one digs a little deeper, it’s not so clear.

It’s said that there are human languages that have words for “one”, “a pair” and “many”, but no words for specific larger numbers. In our modern technological world that seems unthinkable. But imagine you’re out in the jungle, with your dogs. Each dog has particular characteristics, and most likely a particular name. Why should you ever think about them collectively, as all “just dogs”, amenable to being counted? Continue reading

Look What We Made in Five Months!

It’s hard to believe we’ve been doing this for 35 years, building a taller and taller tower of ideas and technology that allow us to reach ever further. In earlier times we used to release the results of efforts only every few years. But in recent times we’ve started doing incremental (“.1”) releases that deliver our latest R&D achievements—both fully fleshed out, and partly as “coming attractions”—much more frequently.

We released Version 12.2 on December 16, 2020. And today, just five months later, we’re releasing Version 12.3. There are some breakthroughs and major new directions in 12.3. But much of what’s in 12.3 is just about making Wolfram Language and Mathematica better, smoother and more convenient to use. Things are faster. More “But what about ___?” cases are handled. Big frameworks are more completely filled out. And there are lots of new conveniences.

There are also the first pieces of what will become large-scale structures in the future. Early functions—already highly useful in their own right—that will in future releases be pieces of major systemwide frameworks. Continue reading