God not only plays dice in physics but also in pure mathematics. Mathematical truth is sometimes nothing more than a perfect coin toss
G. J. Chaitin
The notion of randomness obsesses
physicists today. To what extent can we predict the future? Does it depend
on our own limitations? Or is it in principle impossible to predict the future?
The question of predictability has a long history in physics. In the early
19th century, the
classical deterministic laws of
Isaac Newton led
Pierre Simon de Laplace
to believe that the future of the Universe could be
Then quantum mechanics came along. This is the theory that
is fundamental to our understanding of the nature of matter. It describes
very small objects, such as electrons and other fundamental particles. One
of the controversial features of quantum mechanics was that it introduced
probability and randomness at a fundamental level to physics. This greatly
upset the great physicist
Albert Einstein, who said that
God did not play dice.
Then surprisingly, the modern study of
nonlinear dynamics showed us that even
the classical physics of Newton had randomness and unpredictability at its
core. The theory of chaos, as the series of articles in New Scientist last
year described, has revealed how the notion of randomness and unpredictability
is beginning to look like a unifying principle.
It seems that the same principle even extends to mathematics.
I can show that there are theorems connected with number theory that cannot
be proved because when we ask the appropriate questions, we obtain results
that are equivalent to the random toss of a coin.
My results would have shocked many 19th-century mathematicians,
who believed that mathematical truths could always be proved. For example,
in 1900, the mathematician,
David Hilbert, gave a famous lecture
in which he proposed a list of 23 problems as a challenge to the new century.
His sixth problem had to do with establishing the fundamental universal truths,
or axioms, of physics. One of the points in this question concerned probability
theory. To Hilbert, probability was simply a practical tool that came from
physics; it helped to describe the real world when there was only a limited
amount of information available.
Another question he discussed was his tenth problem, which
was connected with solving so-called 'diophantine' equations, named after
the Greek mathematician Diophantus. These are algebraic equations
involving only whole numbers, or integers. Hilbert asked: 'Is there a way
of deciding whether or not an algebraic equation has a solution in whole
Little did Hilbert imagine that these two questions are subtly
related. This was because Hilbert had assumed something that was so basic
to his thinking that he did not even formulate it as a question in his talk.
That was the idea that every mathematical problem has a solution. We may
not be bright enough or we may not have worked long enough on the problem
but, in principle, it should be possible to solve it - or so Hilbert thought.
For him, it was a black or white situation.
It seems now that Hilbert was on shaky ground. In fact, there
is a connection between Hilbert's sixth question dealing with probability
theory and his tenth problem of solving algebraic equations in whole numbers
that leads to a surprising and rather chilling result. That is: randomness
lurks at the heart of that most traditional branch of pure mathematics, number
Clear, simple mathematical questions do not always have clear
answers. In elementary number theory, questions involving diophantine equations
can give answers that are completely random and look grey, rather than black
or white. The answer is random because the only way to prove it is to postulate
each answer as an additional independent axiom. Einstein would be horrified
to discover that not only does God play dice in quantum and classical physics
but also in pure mathematics.
Where does this surprising conclusion come from? We have to
go back to Hilbert. He said that when you set up a formal system of axioms
there should be a mechanical procedure to decide whether a mathematical proof
is correct or not, and the axioms should be consistent and complete. If the
system of axioms is consistent, it means that you cannot prove both a result
and its contrary. If the system is complete, then you can also prove any
assertion to be true or false. It follows that a mechanical procedure would
ensure that all mathematical assertions can be decided mechanically.
There is a colourful way to explain how this mechanical procedure works: the so-called 'British Museum algorithm'. What you do - it cannot be done in practice because it would take forever - is to use the axiom system, set in the formal language of mathematics, to run through all possible proofs, in order of their size and lexicographic order. You check which proofs are correct - which ones follow the rules and are accepted as valid. In principle, if the set of axioms is consistent and complete, you can decide whether any theorem is true or false. Such a procedure means that a mathematician no longer needs ingenuity or inspiration to prove theorems. Mathematics becomes mechanical.
Why mathematics is not complete
Of course, mathematics is not like that. Kurt Godel, the Austrian logician, and Alan Turing, the father of the computer, showed that it is impossible to obtain both a consistent and complete axiomatic theory of mathematics and a mechanical procedure for deciding whether an arbitrary mathematical assertion is true or false, or is provable or not.
Godel was the first to devise the ingenious proof, couched
in number theory, of what is called the
theorem (see 'The incompleteness of arithmetic', New Scientist, 5 November
1987). But I think that the Turing version of the theorem is more fundamental
and easier to understand. Turing used the language of the computer - the
instructions, or program, that a computer needs to work out problems. He
showed that there is no mechanical procedure for deciding whether an arbitrary
program will ever finish its computation and halt.
To show that the so-called halting problem can never be solved,
we set the program running on a Turing
machine, which is a mathematical idealisation of a digital computer with
no time limit. (The program must be self-contained with all its data wrapped
up inside the program.) Then we simply ask: 'Will the program go on forever,
or at some point will it say 'I'm finished' and halt?'
Turing showed that there is no set of instructions that you
can give the computer, no algorithm, that will decide if a program will ever
halt. Godel's incompleteness theorem follows immediately because if there
is no mechanical procedure for deciding the halting problem, then there is
no complete set of underlying axioms either. If there were, they would provide
a mechanical procedure for running through all possible proofs to show whether
programs halt - although it would take a long time, of course.
To obtain my result about randomness in mathematics, I simply
take Turing's result and just change the wording. What I get is a sort of
a mathematical pun. Although the halting problem is unsolvable, we can look
at the probability of whether a randomly chosen program will halt. We start
with a thought experiment using a general purpose computer that, given enough
time, can do the work of any computer - the universal Turing machine.
Instead of asking whether or not a specific program halts,
we look at the ensemble of all possible computer programs. We assign to each
computer program a probability that it will be chosen. Each bit of information
in the random program is chosen by tossing a coin, an independent toss for
each bit, so that a program containing so many bits of information, say,
N bits, will have a probability of 2-N. We can now ask what is the total
probability that those programs will halt. This halting probability, call
it V, wraps up Turing's question of whether a program halts into one number
between 0 and 1. If the program never halts, V is 0; if it always halts,
V is 1.
In the same way that computers express numbers in binary notation,
we can describe V in terms of a string of 1s and 0s. Can we determine whether
the Nth bit in the string is a 0 or a 1? In other words, can we compute V?
Not at all. In fact, I can show that the sequence of 0s and 1s is random
using what is called algorithmic information theory. This theory ascribes
a degree of order in a set of information or data according to whether there
is an algorithm that will compress the data into a briefer form.
For example, a regular string of 1s and 0s describing some data
such as 0101010101 . . . which continues for 1000 digits can be encapsulated
in a shorter instruction 'repeat 01 500 times'. A completely random string
of digits cannot be reduced to a shorter program at all. It is said to be
My analysis shows that the halting probability is algorithmically
random. It cannot be compressed into a shorter program. To get N bits of
the number out of a computer, you need to put in a program at least N bits
long. Each of the N bits of V is an irreducible independent mathematical
fact, as random as tossing a coin. For example, there are as many 0s in V
as 1s. And knowing all the even bits does not help us to know any of the
My result that the halting probability is random corresponds
to Turing's assertion that the halting problem is undecidable. It has turned
out to provide a good way to give an example of randomness in number theory,
the bedrock of mathematics. The key was a dramatic development about five
years ago. James Jones of the University of Calgary in Canada and Yuri
Matijasevic of the Steklov Institute of Mathematics in Leningrad discovered
a theorem proved by Edouard Lucas in France a century ago. The theorem provides
a particularly natural way to translate a universal Turing machine into a
universal diophantine equation that is equivalent to a general purpose
I thought it would be fun to write it down. So with the help
of a large computer I wrote down a universal-Turing-machine equation. It
had 17 000 variables and went on for 200 pages. The equation is of a type
that is referred to as 'exponential diophantine'. All the variables and constants
in it are non-negative integers, 0, 1, 2, 3, 4, 5, and so on. It is called
'exponential' because it contains numbers raised to an integer power. In
normal diophantine equations the power has to be a constant. In this equation,
the power can be a variable. So in addition to having X3, it also contains
To convert the assertion that the halting probability V is
random into an assertion about the randomness of solutions in arithmetic,
I need only to make a few minor changes in this 200-page universal-Turing-machine
diophantine equation. The
result, my equation exhibiting randomness, is also 200 pages long. The equation
has a single parameter, the variable N. For any particular value of this
parameter, I ask the question: 'Does my equation have a finite or infinite
number of whole-number solutions?' Answering this question turns out to be
equivalent to calculating the halting probability. The answer 'encodes' in
arithmetical language whether the Nth bit of V is a 0 or a 1. If the Nth
bit of V is a 0, then my equation for that particular value of N has a finite
number of solutions. If the Nth bit of the halting probability V is a 1,
then this equation for that value of the parameter N has an infinite number
of solutions. Just as the Nth bit of V is random - an independent, irreducible
fact like tossing a coin - so is deciding whether the number of solutions
of my equation is finite or infinite. We can never know.
To find out whether the number of solutions is finite or infinite
in particular cases, say, for k values of the parameter N, we would have
to postulate the k answers as k additional independent axioms. We would have
to put in k bits of information into our system of axioms, so we would be
no further forward. This is another way of saying that the k bits of information
are irreducible mathematical facts.
I have found an extreme form of randomness, of irreducibility,
in pure mathematics - in a part of elementary number theory associated with
the name of Diophantus and which goes back 2000 years to classical Greek
mathematics. Hilbert believed that mathematical truth was black or white,
that something was either true or false. I think that my work makes things
look grey, and that mathematicians are joining the company of their theoretical
physics colleagues. I do not think that this is necessarily bad. We have
seen that in classical and quantum physics, randomness and unpredictability
are fundamental. I believe that these concepts are also found at the very
heart of pure mathematics.