Two Philosophies of Mathematical Weirdness
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
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
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.
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: http://www.wolframscience.com/nksonline/section-2.1.
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
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.