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For most big ideas in recorded intellectual history one can answer the question: “What became of the person who originated it?” But late last year I tried to answer that for Moses Schönfinkel, who sowed a seed for what’s probably the single biggest idea of the past century: abstract computation and its universality.

I managed to find out quite a lot about Moses Schönfinkel. But I couldn’t figure out what became of him. Still, I kept on digging. And it turns out I was able to find out more. So here’s an update…. Continue reading

“What about Consciousness?”

For years I’ve batted it away. I’ll be talking about my discoveries in the computational universe, and computational irreducibility, and my Principle of Computational Equivalence, and people will ask “So what does this mean about consciousness?” And I’ll say “that’s a slippery topic”. And I’ll start talking about the sequence: life, intelligence, consciousness.

I’ll ask “What is the abstract definition of life?” We know about the case of life on Earth, with all its RNA and proteins and other implementation details. But how do we generalize? What is life generally? And I’ll argue that it’s really just computational sophistication, which the Principle of Computational Equivalence says happens all over the place. Then I’ll talk about intelligence. And I’ll argue it’s the same kind of thing. We know the case of human intelligence. But if we generalize, it’s just computational sophistication—and it’s ubiquitous. And so it’s perfectly reasonable to say that “the weather has a mind of its own”; it just happens to be a mind whose details and “purposes” aren’t aligned with our existing human experience. Continue reading

“[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]

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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