Launching Version 12.2 of Wolfram Language & Mathematica: 228 New Functions and Much More…

Yet Bigger than Ever Before

When we released Version 12.1 in March of this year, I was pleased to be able to say that with its 182 new functions it was the biggest .1 release we’d ever had. But just nine months later, we’ve got an even bigger .1 release! Version 12.2, launching today, has 228 completely new functions!

Launching Version 12.2 of Wolfram Language & Mathematica: 228 New Functions and Much More...
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Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel


December 7, 1920

Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel—click to enlarge

On Tuesday, December 7, 1920, the Göttingen Mathematics Society held its regular weekly meeting—at which a 32-year-old local mathematician named Moses Schönfinkel with no known previous mathematical publications gave a talk entitled “Elemente der Logik” (“Elements of Logic”).

A hundred years later what was presented in that talk still seems in many ways alien and futuristic—and for most people almost irreducibly abstract. But we now realize that that talk gave the first complete formalism for what is probably the single most important idea of this past century: the idea of universal computation. Continue reading

Combinators and the Story of Computation


The Abstract Representation of Things

“In principle you could use combinators,” some footnote might say. But the implication tends to be “But you probably don’t want to.” And, yes, combinators are deeply abstract—and in many ways hard to understand. But tracing their history over the hundred years since they were invented, I’ve come to realize just how critical they’ve actually been to the development of our modern conception of computation—and indeed my own contributions to it. Continue reading

Combinators: A Centennial View

Watch the livestreamed event: Combinators: A 100-Year Celebration

Combinators: A Centennial View

Ultimate Symbolic Abstraction

Before Turing machines, before lambda calculus—even before Gödel’s theorem—there were combinators. They were the very first abstract examples ever to be constructed of what we now know as universal computation—and they were first presented on December 7, 1920. In an alternative version of history our whole computing infrastructure might have been built on them. But as it is, for a century, they have remained for the most part a kind of curiosity—and a pinnacle of abstraction, and obscurity. Continue reading

Our Mission and the Opportunity of Artifacts from the Future

In preparing my keynote at our 31st annual technology conference, I tried to collect some of my thoughts about our long-term mission and how I view the opportunities it is creating…

What I’ve Spent My Life On

I’ve been fortunate to live at a time in history when there’s a transformational intellectual development: the rise of computation and the computational paradigm. And I’ve devoted my adult life to doing what I can to make computation and the computational method achieve their potential, both intellectually and in the world at large. I’ve alternated (about five times so far) between doing this with basic science and with practical technology, each time building on what I’ve been able to do before.

The basic science has shown me the immense power and potential of what’s out there in the computational universe: the capability of even simple programs to generate behavior of immense complexity, including, I now believe, the fundamental physics of our whole universe. But how can we humans harness all that power and potential? How do we use the computational universe to achieve things we want: to take our human objectives and automate achieving them?

I’ve now spent four decades in an effort to build a bridge between what’s possible with computation, and what we humans care about and think about. It’s a story of technology, but it’s also a story of big and deep ideas. And the result has been the creation of the first and only full-scale computational language—that we now call the Wolfram Language. Continue reading

Faster than Light in Our Model of Physics: Some Preliminary Thoughts

When the NASA Innovative Advanced Concepts Program asked me to keynote their annual conference I thought it would be a good excuse to spend some time on a question I’ve always wanted to explore…

Faster than Light in Our Model of Physics: Some Preliminary Thoughts

Can You Build a Warp Drive?

“So you think you have a fundamental theory of physics. Well, then tell us if warp drive is possible!” Despite the hopes and assumptions of science fiction, real physics has for at least a century almost universally assumed that no genuine effect can ever propagate through physical space any faster than light. But is this actually true? We’re now in a position to analyze this in the context of our model for fundamental physics. And I’ll say at the outset that it’s a subtle and complicated question, and I don’t know the full answer yet.

But I increasingly suspect that going faster than light is not a physical impossibility; instead, in a sense, doing it is “just” an engineering problem. But it may well be an irreducibly hard engineering problem. And one that can’t be solved with the computational resources available to us in our universe. But it’s also conceivable that there may be some clever “engineering solution”, as there have been to so many seemingly insuperable engineering problems in the past. And that in fact there is a way to “move through space” faster than light. Continue reading

The Empirical Metamathematics of Euclid and Beyond

The Empirical Metamathematics of Euclid and Beyond

Towards a Science of Metamathematics

One of the many surprising things about our Wolfram Physics Project is that it seems to have implications even beyond physics. In our effort to develop a fundamental theory of physics it seems as if the tower of ideas and formalism that we’ve ended up inventing are actually quite general, and potentially applicable to all sorts of areas.

One area about which I’ve been particularly excited of late is metamathematics—where it’s looking as if it may be possible to use our formalism to make what might be thought of as a “bulk theory of metamathematics”.

Mathematics itself is about what we establish about mathematical systems. Metamathematics is about the infrastructure of how we get there—the structure of proofs, the network of theorems, and so on. And what I’m hoping is that we’re going to be able to make an overall theory of how that has to work: a formal theory of the large-scale structure of metamathematics—that, among other things, can make statements about the general properties of “metamathematical space”. Continue reading

A Burst of Physics Progress at the 2020 Wolfram Summer School

A Burst of Physics Progress at the 2020 Wolfram Summer School

And We’re Off and Running…

We recently wrapped up the four weeks of our first-ever “Physics track” Wolfram Summer School—and the results were spectacular! More than 30 projects potentially destined to turn into academic papers—reporting all kinds of progress on the Wolfram Physics Project.

When we launched the Wolfram Physics Project just three months ago one of the things I was looking forward to was seeing other people begin to seriously contribute to the project. Well, it turns out I didn’t have to wait long! Because—despite the pandemic and everything—things are already very much off and running!

Six weeks ago we made a list of questions we thought we were ready to explore in the Wolfram Physics Project. And in the past five weeks I’m excited to say that through projects at the Summer School lots of these are already well on their way to being answered. If we ever wondered whether there was a way for physicists (and physics students) to get involved in the project, we can now give a resounding answer, “yes”. Continue reading

Exploring Rulial Space: The Case of Turing Machines

Wolfram Physics Bulletin

Informal updates and commentary on progress in the Wolfram Physics Project

Generalized Physics and the Theory of Computation

Let’s say we find a rule that reproduces physics. A big question would then be: “Why this rule, and not another?” I think there’s a very elegant potential answer to this question, that uses what we’re calling rule space relativity—and that essentially says that there isn’t just one rule: actually all possible rules are being used, but we’re basically picking a reference frame that makes us attribute what we see to some particular rule. In other words, our description of the universe is a sense of our making, and there can be many other—potentially utterly incoherent—descriptions, etc.

But so how does this work at a more formal level? This bulletin is going to explore one very simple case. And in doing so we’ll discover that what we’re exploring is potentially relevant not only for questions of “generalized physics”, but also for fundamental questions in the theory of computation. In essence, what we’ll be doing is to study the structure of spaces created by applying all possible rules, potentially, for example, allowing us to “geometrize” spaces of possible algorithms and their applications.

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Event Horizons, Singularities and Other Exotic Spacetime Phenomena

Wolfram Physics Bulletin

Informal updates and commentary on progress in the Wolfram Physics Project

The Structure and Pathologies of Spacetime

In our models, space emerges as the large-scale limit of our spatial hypergraph, while spacetime effectively emerges as the large-scale limit of the causal graph that represents causal relationships between updating events in the spatial hypergraph. An important result is that (subject to various assumptions) there is a continuum limit in which the emergent spacetime follows Einstein’s equations from general relativity.

And given this, it is natural to ask what happens in our models with some of the notable phenomena from general relativity, such as black holes, event horizons and spacetime singularities. I already discussed this to some extent in my technical introduction to our models. My purpose here is to go further, both in more completely understanding the correspondence with general relativity, and in seeing what additional or different phenomena arise in our models.

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