P: So what is mathematics for you?
C: Oh, I think mathematics is a lot of fun. You see, for me it's like music, it's a music that unfortunately most people can't hear.
But it's more than that, it's not just pretty. For me, mathematics is an adventure, it's like trying to climb a mountain; I'm an explorer at heart. You go off somewhere where no one else has gone, and you try to figure out what's happening. You try to climb up so you can look around and discover what's going on.
And I like going on weekends or holidays up into the mountains and trying to climb things. I like doing it in the winter with snowshoes and ski poles. It's very, very beautiful.
In a way, mathematical research is like mountain climbing. You are trying to get somewhere higher, where no one else has been yet, where the air is very pure, where the sky is a beautiful dark blue, where you get those incredible views, where you think you're closer to God, you understand! And it's a great adventure!
P: To be a great mathematician, does one have to be mad?
C: [Laughs] Well, I don't think it's necessary, but it helps! A normal human being, a healthy human being, someone who is a good father, a good husband, a good member of the community, that's a wonderful person, but it gets in the way! You have to be obsessed with the mathematical ideas, you have to go on all day long thinking about them, all night long thinking about them, you have to lie awake at night in bed thinking about them, and this really creates a problem!
Let me give you an example. Toward the end of his life, Einstein got a letter from the wife of his friend Michele Besso, who stayed in Switzerland. And she said, ``You know, you and Michele were friends in Bern, and Michele is so talented, how come he never accomplished anything?'' And Einstein said, ``But of course, it's because he's a good man!'' You see, Einstein considered that his two marriages had been failures. And you look at Michele Besso with his wife, and they look so in love!
Having mathematical talent is wonderful, but it's also a bit of a curse, it tends to take over your life. Now I didn't want it to take over my life, I think I tried to have a normal human life, but it did sort of take over my life, I guess.
P: But there's also the pleasure, you mentioned that the pleasure you have doing math is like the pleasure of being with a beautiful woman.
C: Well, they're different obviously. But when I was an adolescent, when I was a teenager, yes, I felt that there was something sensual about a beautiful mathematical idea.
Talking about sensuality, that's a feeling I have very much when I'm visiting Brazil, and when I visited Rio just before Carnival and spent a lot of time at the beach and dancing in the streets in Carnival in Rio in 1970. Brazil is very sensual, and it made quite an impression on me. That was a time when I had one of my best ideas, it was the week before Carnival in Rio in 1970.
P: You were inspired by the great romantic mathematicians that you read about when you were young, for instance, Evariste Galois. Tell us about it.
C: Well, he's a genius, but his life is a tragedy. Typically what mathematicians joke to each other when they're in their late twenties, is ``You know, when Galois was my age, he had already done immortal work and he had been dead for five years!'' And they say this to each other as a stimulus, saying ``You've got to get to work!''
You read these stories of these very young mathematicians, and I read them as a child, and I said to myself as a joke, ``If I don't have a great idea by the time I'm eighteen, forget it, I obviously shouldn't continue with math!''
But the funny thing is, I did have an idea when I was fifteen.
P: And what was that idea?
C: The question that I started with was the question of what is randomness, what is lack of structure. What does it mean to say that something does not obey a law, does not have any pattern, does not have any structure?
The idea of randomness I got by studying physics; I was fascinated by physics as a child. I originally wanted to be a theoretical physicist or maybe an astronomer. And I took this idea with me into mathematics, and it's a foreign idea, one that physicists like, but that logicians don't like. So I'm sort of persona non grata, I think it's fair to say, in part of the logic community. But physicists tend to find my work interesting because I took an idea, randomness, from physics and found it in the foundations of mathematics, an unexpected place to find it.
P:
What was the insight that you had in Rio in 1970 during Carnival?
C:
Ah, just before Carnival. During Carnival I was too busy dancing
in the street and looking at those luscious carioca women.
[Women from Rio de Janeiro.]
The idea was that most things satisfy my definition of randomness,
most numbers are random in my sense, but you can never prove it.
You see, I had come up with a definition of randomness, and then
I realized that the main significance of this idea was that it
showed that there were limits to what reasoning could achieve.
P:
The way that math is taught in schools, do you think that kills
in the bud mathematical genius like yours?
C:
You know, schools are trying to do something different, schools
are trying to teach people how to survive in a complicated
technological world. What I did, was I marched off in a
different direction from the rest of the human race, trying to
create a new field of mathematics. Now if everybody tried to
do that, it would be a disaster! So I think that for most
people normal schools are probably just about right.
But if you get somebody who is very creative and bright, I would
ask the school system to please not destroy the personality of
such a child, give them a chance. Sometimes a child like that
is just a rebel or an eccentric, but sometimes this is how you
get creative people.
P:
If you could design how math is taught, how would that be?
C:
What I would like to do is to take very bright kids, and give
them fundamental ideas. I would teach them Einstein's theory
of gravity, curved space-time, I would teach them quantum
mechanics, the uncertainty principle, I would teach them
Gödel's incompleteness theorem. I would skip everything
and go to the frontiers.
That's what I wanted to do as a child. I was always going
through piles of books trying to get to the interesting stuff
and teach myself that. Because in the normal school system
you take years and years and years to get to the interesting
things, and that way everyone dies of boredom. So that was
one of the reasons I was studying so much on my own.
P:
Tell us about the dream you had as a child, it's more
like a nightmare, about the future.
C:
When I was young, I used to have vivid dreams, and I would
remember the dreams the next morning, which
doesn't happen to me anymore. And I would try to control
my dreams, I guess there's a name for this, it's lucid
dreaming, when you realize that you're dreaming and you try
to control it. Like for example you want to fly.
So a dream that I had several times, I remember, was being in
the future. I don't know how I knew it was the future. And
I was in a library. And I was desperate to find out what
humanity had discovered, where science had gone. So I'd go,
I'd pick up a book, and I'd start to read it. And at first
it looked like a book, there are words, and I can read the
words, but they don't make any sense! And that was very
disconcerting!
Another thing I would do, I think, perhaps, was to go and
look and see
if my name appeared anywhere. You know, that's a typical
thing people do.
P:
You sound like a character out of Jorge Luis Borges.
[C. laughs.] Do you feel some affinity?
C:
Oh yes, I love the Borges stories. I think some of them
are very philosophical and very mathematical. They have
a very European flavor, don't they?
P:
Any of his ideas have helped you or made you think about
something?
C:
Well, I don't know if they've helped me. But Borges I think
likes paradox, and my own work deals with paradox.
When you talk about things that are unknowable, how can you
talk about something that's unknowable? How can you know
anything about something that's unknowable? So there's a
paradox involved in everything I'm doing.
What I'm working on is reason trying to discover its own
limitations. And that's a paradox also, because you're
criticizing the tool you're using.
So I've always enjoyed the stories of Borges, I've always
enjoyed Magritte's paintings, Escher's drawings, but I think
Magritte even more, because there's something paradoxical
in those sensual paintings, something that appeals to me
as a mathematician. And I showed you my little
Kenneth
Snelson sculpture, which also, I feel, has a certain
mathematical beauty.
P:
So what you did discover is that mathematics is actually random?
C:
I didn't really prove that mathematics is random; I came up with
a definition of randomness which has this strange property:
the most interesting thing about it is that you can never prove
that something satisfies this definition---even though most
things do. And this was my first step, this was the idea I had
in Rio.
Then, some years later, I realized that there was an area of
mathematics that I could construct, or I could discover, where in
fact mathematical truth had no structure, was completely random,
in that area. So this is a part of mathematics which is a
black hole, where individual questions cannot be answered.
You can make statistical statements about the answers. The
answers will be one thing or another 50% of the time.
P:
Like tossing a coin?
C:
It's exactly like tossing a coin, but you can't answer individual questions.
P:
So would you say that God plays dice?
C:
Well, I found an area of mathematics, or I constructed an area of
mathematics, where in fact God does play dice, where mathematical
truth is accidental, where things are true for no reason. This is
in pure mathematics itself. So in this part of mathematics, in fact,
things are maximally random, things have absolutely no structure,
mathematical truth is completely accidental, it's a worst case.
It's sort of a nightmare for the rational mind.
But one shouldn't say that mathematics is dead or finished; on the
contrary, mathematics is alive and thriving, and in a way what our---Gödel's,
Turing's and my---work shows is that a static view of
mathematics does not work, mathematics is in constant evolution.
P:
What do mathematicians and artists have in common?
C:
Well, I think they have a lot in common. I think at the deepest
level mathematical creativity is very, very similar to artistic
creativity. You have to be passionate about it, it's irrational,
you use your intuition, you have to be inspired, it's totally
irrational.
[This shouldn't be a surprise, in as much as Tor Nørretranders
has pointed out in his book
The User Illusion that the
subconscious, irrational mind has much, much greater information-processing
capability---and therefore in many ways is more intelligent---than
the conscious, rational mind, which is a narrow funnel.]
After you create a new field of mathematics, after you get the idea,
then you have to verify it, and that part is rational and systematic.
But creating a new field, you have to be obsessed with it.
And also there's a lot more in common.
You see, mathematical ideas have to fit together beautifully.
The field that I created, I like to call it algorithmic information
theory, it went through several different versions. And the early
versions of my theory were good work, they were pioneering work,
but they were too ugly, I felt something was wrong.
So I changed the field, and I changed the definitions, the concepts
weren't quite right. And when I changed things, all of a sudden
they fit together better. So that's an aesthetic criterion, that's
like a painting...
When you create a new field of mathematics you have a certain freedom
in how you can do it. You can change the rules of the game. And
if the ideas don't fit together beautifully, something is wrong, you see.
So the early pioneering work tends to be a bit ugly, because pioneering
work is hard, but the only permanent mathematics is beautiful mathematics.
P:
You looked into the problem of how to predict if a computer program
is going to halt. And you discovered the Ω number. Can you
explain easily what is the Ω number?
C:
I'm very proud of this number; some people are nice enough
to call it ``Chaitin's number''. I call it the Ω number.
This number is the probability that a computer program will eventually
halt. So a computer is a machine, and you start it running, and
you let it run forever, until... It either goes on running
forever, or it comes to a stop and the program says, ``I'm finished.''
But the amazing thing is that if you ask what is the probability that
a program chosen at random halts---you look at all possible programs---and
if you write this number out in binary, this number is
maximally unknowable. Its individual bits look like the
results of independent tosses of a fair coin, individual outcomes
of the game of ``heads or tails''. There is no mathematical
structure.
So it has a simple physical interpretation, this Ω number of mine,
but if you want to calculate its value, digit by digit, or bit by bit
if you write it in binary, you can't, it's sort of a worst case.
The digits, if you write it in decimal, of this number---it's a number
between zero and one, you know, you have a decimal point, and then
you have a lot of digits going on forever. And the problem is
if you try to calculate this number, the digits have no structure,
no pattern, they look completely random. So it's a way to have
God play dice in pure mathematics.
P:
You wrote that information, complexity, randomness
are the spirit of the times, the math of the third millennium?
C:
I think this word information is very suggestive, it's
a very sexy word. And it's part of the computer revolution, it's
part of the idea of software, it's part of the revolution in biology
with DNA, which is biological information, in a molecule, in physical
form.