“[Despite] Considerable Effort… [It Proved] Intractable”

In the early years of the twentieth century it looked as if—if only the right approach could be found—all of mathematics might somehow systematically be solved. In 1910 Whitehead and Russell had published their monumental Principia Mathematica showing (rather awkwardly) how all sorts of mathematics could be represented in terms of logic. But Emil Post wanted to go further. In what seems now like a rather modern idea (with certain similarities to the core structure of the Wolfram Language, and very much like the string multiway systems in our Physics Project), he wanted to represent the logic expressions of Principia Mathematica as strings of characters, and then have possible operations correspond to transformations on these strings.

In the summer of 1920 it was all going rather well, and Emil Post as a freshly minted math PhD from Columbia arrived in Princeton to take up a prestigious fellowship. But there was one final problem. Having converted everything to string transformations, Post needed to have a theory of what such transformations could do. Continue reading

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Over the years I’ve studied the simplest ordinary Turing machines quite a bit, but I’ve barely looked at multiway Turing machines (also known as nondeterministic Turing machines or NDTMs). Recently, though, I realized that multiway Turing machines can be thought of as “maximally minimal” models both of concurrent computing and of the way we think about quantum mechanics in our Physics Project. So now this piece is my attempt to “do the obvious explorations” of multiway Turing machines. And as I’ve found so often in the computational universe, even cases with some of the very simplest possible rules yield some significant surprises….

Ordinary vs. Multiway Turing Machines

An ordinary Turing machine has a rule such as

&#10005 RulePlot[TuringMachine[2506]]

that specifies a unique successor for each configuration of the system (here shown going down the page starting from an initial condition consisting of a blank tape):

&#10005 RulePlot[TuringMachine[2506], {{1, 6}, Table[0, 10]}, 20, Mesh -> True, Frame -> False]

Continue Reading

It All Started with Feynman Diagrams

Any serious calculation in particle physics takes a lot of algebra. Maybe it doesn’t need to. But with the methods based on Feynman diagrams that we know so far, it does. And in fact it was these kinds of calculations that first led me to use computers for symbolic computation. That was in 1976, which by now is a long time ago. But actually the idea of doing Feynman diagram calculations by computer is even older. Continue reading

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!

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December 7, 1920

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

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

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

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

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…

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

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