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Two Philosophies of Mathematical Weirdness

Jaron Lanier

Meta Math! The Quest for Omega. Gregory Chaitin. xiv + 220 pp. Pantheon Books, 2005. $26.

The Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me about Ultimate Reality, the Meaning of Life, and How to Be Happy. Rudy Rucker. x + 564 pp. Thunder's Mouth Press, 2005. $35.

A strange thing happened to the philosophy of mathematics in the past century or so: Math-ematics increasingly revealed truths about itself that utterly confounded the expectations of philosophers. In other words, math got weird.

A century ago, math was thought to be an orderly Platonic phenomenon, imperious in its perfection. The first prominent onset of weirdness came in 1931, when Kurt Gödel showed that important systems of mathematical ideas could never be completed. In order to get his result, he indexed mathematical ideas in a way that was somewhat analogous to the way the Web is now indexed by services such as Google. That computational framework began to give mathematicians a completely new perspective.

Since Gödel, developments in mathematics have only gotten more surprising—Benoit Mandelbrot's fractals, the noncommutative geometry of Alain Connes, quantum information theory. It sometimes feels as though the determining factor in mathematical truth is that the next result will be weirder than what could have been imagined from the previous findings.

Now mathematician Gregory Chaitin is finding tremendous joy at the edges of the current state of weirdness. His work both extends Gödel's approach and makes it more accessible. Although most of Chaitin's ideas and all of his results have been presented in earlier books, Meta Math! The Quest for Omega is fresh in that it is aimed at general readers and achieves what can only be described as an unprecedented level of gushy enthusiasm for numbers.

Entering Chaitin's world feels a bit like coming across the diary of a rapturous teenager. Sentences commonly end with outrageously amplified punctuation!?!! The atmosphere is giddy. True love, first love, is clearly present; as readers, we almost feel as though we're intruding on something private and racy.

Like many teenagers, Chaitin has a genius for generating negative results. He destroys at every turn. Chaitin shows us how he constructed bridges between ideas from computer science and more traditional math, only to discover that math would forever remain full of holes in some fascinating and occasionally terrifying ways.

If there were a prize for books with real live math equations that can hold the attention of readers who lack technical training, I'd nominate this one. Chaitin does a lot of things here that ought not to work: His argument wanders, his voice shifts. And yet by opening his quirky self to us without any defenses, he draws us in strongly, and we see the equations through his eyes. He invites everyone to think about math, refusing to grant the subject even one tiny bit of elite mystique.

One of Chaitin's revelations is that the innocent-looking continuum implied by the white space in every illustration in a calculus book is made up almost entirely of numbers that are unspeakable, meaning that even God wouldn't be able to identify any of them because the task would never end. In this hopeless void are scattered occasional numbers that can be talked about, like 2 and pi. Halting probability omega (Ω), sometimes referred to as "Chaitin's number," is one of these, but it is a perverse creature. The simplest definition of Ω is that it encodes the probabilities that programs of progressive lengths chosen randomly from a comprehensive set of possible programs will yield a result (come to a halt) when those programs are run on a particular computer in a thought experiment. If you change the details of the design of the computer, you get a different value for Ω, but for a given computer it's a completely well-defined number. So Ω, even though it is only one number, is also a catalog of mathematical results. No, that fact doesn't make it easy to find out about those results; the randomness of Ω only proves how few of the results we'll ever know. Also, Ω is the most indirectly identified number, the last stop before the inexpressibility of the continuum. I cannot do justice to the ideas in this brief review. I hope merely to inspire interest: Go read what Chaitin has to say about his number!

The bleakness of Chaitin's number leads to a startling vision of the future of mathematics. Mathematical ideas that can be discovered will not be densely ordered as if in a grid to be filled in. Discoverable math will instead be a splatter—a messy, formally random structure full of holes. No systematic approach will tame its nature. Creativity and intuition will play essential roles in exploring it.

Chaitin thinks we should love this development instead of fearing it. He adores the lure of the unknown and is relieved to learn that there will be an infinite amount of it. He has little sympathy for those who feel destabilized by work such as his own or Gödel's. We should be relieved that some claustrophobia-inducing rigid system didn't turn out to work! More freedom, creativity and adventure are available to us.

Chaitin has no patience with "straights," such as that rigid fellow Isaac Newton, whose reputation Chaitin is determined to ruin once and for all. By contrast, Chaitin is extremely fond of Newton's rival, Gottfried Wilhelm Leibniz.

Chaitin has some stern words for physicists. His message seems to be, "Get over the continuum already. The numbers aren't there, so the reality can't be there either." He also admonishes computer scientists, telling them to forget about making clean code—tangled messes are a fundamental property of programs.

But he does all this with such joy! Chaitin is the most optimistic bearer of bad news in the history of science.

Just as an electronic computer has hardware...Click to Enlarge Image

Rudy Rucker's The Lifebox, the Seashell, and the Soul is reminiscent of an influential book that appeared in 1979, Douglas Hofstadter's Gödel, Escher, Bach. Both books aim to synthesize a computation-centric worldview, following in the tradition of earlier figures such as Norbert Wiener. Both offer an intimate tone, describe adventures and tell tales of eccentric friends. Both books are also physically huge, enormous in breadth and enhanced with intriguing examples and illustrations. Hofstadter's central metaphors generally reflected early research in artificial intelligence, whereas Rucker's ideas mostly radiate from the approach to cellular automata presented by Stephen Wolfram.

Rucker, in his glossary, defines a cellular automaton as "a parallel computation that's carried out in a space of cells," a space that can have one, two, three or more dimensions. Cellular automata "are characterized by updating all their cells at once, and by having each cell only accept input from immediately neighboring cells." Further explanation and examples of some simple cellular automata can be found here:

Rucker explicitly adopts Hegel's dialectical method, noting that the German philosopher was his great-great-great-grandfather. Rucker presents a sequence of computation-centric philosophical theses, each of which is a variant of the proposition that something (such as physical reality or human consciousness) is a deterministic computer defined by finitely stated rules and starting conditions. (He uses the word "lifebox" as shorthand for the thesis that everything is a computation.) The antithesis is always that there must be more to it than that. (We feel we have something—a "soul"—that's not captured by a mechanical model.) The synthesis is then a series of realizations about how rich computing can be. For example, unpredictable yet deterministic computations are found in nature—on a seashell, say. So contemplation of the qualities of computation that had seemed inadequate leads to a deepening of ideas. Rucker performs a similar synthesis for each in a series of topics, including fundamentals of physics, psychology, economics and practical philosophy. Each of these forays gives him a chance not only to tell stories but also to introduce new ideas in computer science.

Is the computational metaphor ready for the Hegelian treatment yet? It seems to me that Rucker's syntheses work better in some instances than in others. His discussion of the way inverse power laws operate in a competitive society is particularly effective and will be of interest to economists.

The general idea of computation is so broad that it doesn't always serve well as a focusing metaphor. A specific model of computation, such as the cellular automaton, has the potential to be more useful. But what should the criteria be for selecting a particular computational model as a scientific metaphor?

It turns out that being among the most powerful is an easy standard for abstract computers to meet. Almost all computational models of any interest achieve impressive abstract designations. Therefore it's not useful to fall back on abstract computational potential to justify the value of a given model of computation, because by that standard, most models are equally interesting. A great computational metaphor ought to speak eloquently to the human mind, produce useful results at scales that can be achieved by human-initiated computation in a given era, and provide some means of integrating the underlying model into empirical methods.

Has Rucker chosen his computational metaphors wisely? He uses the term "gnarly" to refer to an informal class of cellular automata defined by Wolfram to have the least trivial behavior. Gnarly cellular automata produce results that are neither periodic nor pseudorandom but instead reveal flavorful, stylish structures at varying levels of description. A one-liner to summarize the book might be, "Perhaps reality is gnarly."

Is gnarliness the right stuff to support a new kind of philosophy? I'm not convinced. Cellular automata, although they can produce amazing-looking graphics, aren't exactly spitting out mini-life forms or ecologies. Of course it may be that no one has hit upon the right rules or starting conditions, or that no one has run a gnarly cellular automaton long enough.

To this observer, however, the behaviors observed in cellular automata thus far look like patterns instead of systems. The difference is best appreciated by comparing the automata with abstract models explored in other streams of research, some of which are also discussed by Rucker. For instance, artificial life simulations have also produced dramatic graphical results. As happens with cellular automata, intriguing forms emerge from simple rules and starting conditions, but there is an important difference.

In my experience, the computer models that have generated intriguing, provocatively lifelike results, such as artificial life, have tended to be ones in which there are parts that interact with one another, such that there is a greater level of integrity or bandwidth within the parts than there is between them. Cellular automata, on the other hand, are structured out of uniform, consistent and unchanging causal connections between fixed, fine-grained background elements. My guess is that our universe will turn out to be best understood by models like those favored in artificial life, in which causal connections are bundled unevenly, so that parts are separated and can interact. In dramatic terms, causality can be best modeled between actors instead of between points on the stage.

A systemic model also makes more sense epistemologically. When models like many of those used in artificial life are built out of objects, the resultant systemic behavior is often robust in that small changes in the definition of the system do not always completely disrupt its behavior. This is a merit shared by useful established theories in physics and biology. By contrast, when you do see a picture of interest emerge in a gnarly cellular automaton, a slight change in starting conditions or rules can completely reshuffle it. That makes gnarly cellular automata as they are currently defined hard to use as potential theories about specific physical processes.

Rucker's book is biased toward ontology and away from epistemology. He is perhaps not sufficiently concerned with how a fundamentally computational worldview might eventually connect with empirical method, and the hair-trigger disruptability of gnarly cellular automata is not the only example: Rucker loves to create charts and diagrams that attempt to exhaust all possibilities in a system of ideas. For instance, in the last chapter there's a grid of Venn diagrams and logical expressions in which Rucker presents possible cosmic ontologies. The grid is generated from the combinations of three ontological objects: human thought, physical reality and computation (which Rucker points out would have been called logic in the old days). An interesting comparison is to Roger Penrose's construction of a similar set of three ontological components in his big book The Road to Reality. Penrose placed the elements into a cyclic, Klein bottle-like structure that could not be expressed within the terms Rucker uses. Penrose is thinking epistemologically, in that the world must be known through the mind, which must be known through the world, and so on. Rucker's temperament as a philosopher is to look at the world from the outside as if looking at the output of a cellular automaton program.

There is an inevitable compromise in any attempt to be intimate and Hegelian at the same time. The range of The Lifebox, the Seashell, and the Soul is vast, but the domain is personal, and examples come mostly from the author's circle of friends. Fortunately, Rucker is good at choosing friends. For instance, he performs a great service in raising the visibility of the important and colorful mathematician Bill Gosper. On the other hand, here is yet another book on cellular automata that omits from the history of their development key figures such as Alvy Ray Smith. Smith's pioneering work of the 1960s and 1970s was essential to making possible arguments such as those presented here.

These two books are near opposites even though they appear to explore similar topics. Chaitin loves negative results and is thrilled by the prospect of future generations of mathematicians finding ever weirder math. The Chaitinesque intellectual future will be eternally youthful and anarchic. Neither mathematicians nor computer scientists will settle down into a single preferred pattern of thought. Rucker, in contrast, is reaching as high as he can to try to use available computer science and math metaphors to create a new, comprehensive, multidisciplinary sensibility. The Ruckerian future is one in which new guiding explanatory ideas will connect all areas of intellectual curiosity.

One cannot live by deconstruction alone, so certainly we must have Ruckers to balance our Chaitins. At the same time, the Hegelian school of philosophy is notorious for overreach. The most valuable synthesis will perhaps come not from a single thinker but from the whole community of cybernetic thinkers, as we gradually learn to chart a middle course.

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