Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People

by Stephen Wolfram

There was no book like it. It was a new paradigm, both in presentation and in the informal style of explanation it employed. Papers slowly started appearing (quite often with Mandelbrot as co-author) that connected it to different fields, such as biology and social science. Results were mixed and often controversial. Did the paths traced by animals or graphs of stock prices really have nested structures or follow exact power laws? Even Mandelbrot himself began to water down his message, introducing concepts like “multifractals”.

He campaigned for the Nobel Prize in physics; later it was economics. I used to ask him why he cared so much. I pointed out that really great science—like fractals—tends to be too original for there to be prizes defined for it. But he would slough off my comments and recite some other evidence for the greatness of his achievements.

In 2002, my book A New Kind of Science—in which I argued that many phenomena across science are the complex results of relatively simple, program-like rules—appeared.

For a while he wouldn’t suggest anything. But then one day he said to me: “You should call it Mathematica.”

With one slight exception, perhaps of at least curiosity interest to Mathematica aficionados: he suggested that cells in Mathematica notebook documents (and now CDFs) should be indicated not by simple vertical lines—but instead by brackets with little serifs at their ends. And as it happens, that idea opened the way to thinking of hierarchies of cells, and ultimately to many features of symbolic documents.

And as a curious footnote to history (which I learned years later), one batch of NeXTs bought for the purpose of running Mathematica went to CERN in Geneva, Switzerland—where they ended up having no less distinction than being the computers on which the web was first developed.

My direct interactions with Steve Jobs decreased during the decade that I was for all practical purposes a hermit working on A New Kind of Science. For most of that time, though, I used a NeXT computer in almost every waking hour—and in fact my main discoveries were made on it. And when the book was finished, Steve asked for a pre-release copy, which I duly sent.

For many decades, Marvin was perhaps the world’s greatest energy source for artificial intelligence research. He was a fount of ideas, which he fed to his long sequence of students at MIT. And though the details changed, he always kept true to his goal of figuring out how thinking works, and how to make machines do it.

It didn’t take long for a proof of universality to be submitted, and Marvin got quite involved in some of the technical details of validating it, noting that perhaps we should all have known something like this was possible, given the complexity that Emil Post had observed with the simple rules of what he called a tag system—back in 1921, before Marvin was even born.

Probably the most spectacular example was the Connection Machine, developed by Marvin’s student Danny Hillis and his company Thinking Machines (for which Richard Feynman and I were both consultants). It was always in the air that the Connection Machine was built to implement one of Marvin’s theories about the brain, and might be seen one day as like the “transistor of artificial intelligence”. But I, for example, ended up using its massively parallel architecture to implement cellular automaton models of fluids, and not anything AI-ish at all.

I remember a few years ago bringing up the topic of teaching programming, and how I was hoping the Wolfram Language would be relevant to it. Marvin immediately launched into talking about how programming languages are the only ones that people are expected to learn to write before they can read. He said he’d been trying to convince Seymour Papert that the best way to teach programming was to start by showing people good code. He gave the example of teaching music by giving people Eine kleine Nachtmusik, and asking them to transpose it to a different rhythm and see what bugs occur. (Marvin was a long-time enthusiast of classical music.) In just this vein, one way the Wolfram Programming Lab that we launched just last week lets people learn programming is by starting with good code, and then having them modify it.

In the mid-1800s, however, that began to change, notably with the introduction of non-Euclidean geometries and algebras other than those of ordinary numbers. And by the end of the 1800s, there was a general movement towards thinking of mathematics as abstract formalism, independent of the natural world.

In my own work leading up to A New Kind of Science, I started by studying the natural world, yet found myself increasingly being led to generalize beyond traditional mathematical constructs. But I did not wind up with logic. Instead, I began to consider all possible kinds of rules—or as I have tended to describe it (making use of modern experience), the computational universe of all possible programs. Some of these programs describe parts of the natural world. Some give us interesting fodder for technology. And some correspond to traditional formal systems like logic and mathematics. One thing to do is to look at the space of all possible axiom systems.

So how is it that Ramanujan managed in effect to predict all these deep principles of later mathematics? I think there are two basic logical possibilities. The first is that if one drills down from any sufficiently surprising result, say in number theory, one will eventually reach a deep principle in the effort to explain it. And the second possibility is that while Ramanujan did not have the wherewithal to express it directly, he had what amounts to an aesthetic sense of which seemingly random facts would turn out to fit together and have deeper significance.