http://www.newscientist.com/features/features.jsp?id=ns22811
>From New Scientist magazine, 10 March 2001.
The Omega Man
He shattered mathematics with a single number. And that was just for
starters, says Marcus Chown
TWO plus two equals four: nobody would argue with that. Mathematicians
can rigorously prove sums like this, and many other things besides. The
language of maths allows them to provide neatly ordered ways to describe
everything that happens in the world around us.
Or so they once thought. Gregory Chaitin, a mathematics researcher at
IBM's T. J. Watson Research Center in Yorktown Heights, New York, has
shown that mathematicians can't actually prove very much at all. Doing
maths, he says, is just a process of discovery like every other branch
of science: it's an experimental field where mathematicians stumble upon
facts in the same way that zoologists might come across a new species of
primate.
Mathematics has always been considered free of uncertainty and able to
provide a pure foundation for other, messier fields of science. But
maths is just as messy, Chaitin says: mathematicians are simply acting
on intuition and experimenting with ideas, just like everyone else.
Zoologists think there might be something new swinging from branch to
branch in the unexplored forests of Madagascar, and mathematicians have
hunches about which part of the mathematical landscape to explore. The
subject is no more profound than that.
The reason for Chaitin's provocative statements is that he has found
that the core of mathematics is riddled with holes. Chaitin has shown
that there are an infinite number of mathematical facts but, for the
most part, they are unrelated to each other and impossible to tie
together with unifying theorems. If mathematicians find any connections
between these facts, they do so by luck. "Most of mathematics is true
for no particular reason," Chaitin says. "Maths is true by accident."
This is particularly bad news for physicists on a quest for a complete
and concise description of the Universe. Maths is the language of
physics, so Chaitin's discovery implies there can never be a reliable
"theory of everything", neatly summarising all the basic features of
reality in one set of equations. It's a bitter pill to swallow, but even
Steven Weinberg, a Nobel prizewinning physicist and author of Dreams of
a Final Theory, has swallowed it. "We will never be sure that our final
theory is mathematically consistent," he admits.
Chaitin's mathematical curse is not an abstract theorem or an
impenetrable equation: it is simply a number. This number, which Chaitin
calls Omega, is real, just as pi is real. But Omega is infinitely long
and utterly incalculable. Chaitin has found that Omega infects the whole
of mathematics, placing fundamental limits on what we can know. And
Omega is just the beginning. There are even more disturbing
numbers--Chaitin calls them Super-Omegas--that would defy calculation
even if we ever managed to work Omega out. The Omega strain of
incalculable numbers reveals that mathematics is not simply moth-eaten,
it is mostly made of gaping holes. Anarchy, not order, is at the heart
of the Universe.
Chaitin discovered Omega and its astonishing properties while wrestling
with two of the most influential mathematical discoveries of the 20th
century. In 1931, the Austrian mathematician Kurt Gödel blew a gaping
hole in mathematics: his Incompleteness Theorem showed there are some
mathematical theorems that you just can't prove. Then, five years later,
British mathematician Alan Turing built on Gödel's work.
Using a hypothetical computer that could mimic the operation of any
machine, Turing showed that there is something that can never be
computed. There are no instructions you can give a computer that will
enable it to decide in advance whether a given program will ever finish
its task and halt. To find out whether a program will eventually
halt--after a day, a week or a trillion years--you just have to run it
and wait. He called this the halting problem.
Decades later, in the 1960s, Chaitin took up where Turing left off.
Fascinated by Turing's work, he began to investigate the halting
problem. He considered all the possible programs that Turing's
hypothetical computer could run, and then looked for the probability
that a program, chosen at random from among all the possible programs,
will halt. The work took him nearly 20 years, but he eventually showed
that this "halting probability" turns Turing's question of whether a
program halts into a real number, somewhere between 0 and 1.
Chaitin named this number Omega. And he showed that, just as there are
no computable instructions for determining in advance whether a computer
will halt, there are also no instructions for determining the digits of
Omega. Omega is uncomputable.
Some numbers, like pi, can be generated by a relatively short program
which calculates its infinite number of digits one by one--how far you
go is just a matter of time and resources. Another example of a
computable number might be one that comprises 200 repeats of the
sequence 0101. The number is long, but a program for generating it only
need say: "repeat '01' 400 times".
There is no such program for Omega: in binary, it consists of an
unending, random string of 0s and 1s. "My Omega number has no pattern or
structure to it whatsoever," says Chaitin. "It's a string of 0s and 1s
in which each digit is as unrelated to its predecessor as one coin toss
is from the next."
The same process that led Turing to conclude that the halting problem is
undecidable also led Chaitin to the discovery of an unknowable number.
"It's the outstanding example of something which is unknowable in
mathematics," Chaitin says.
An unknowable number wouldn't be a problem if it never reared its head.
But once Chaitin had discovered Omega, he began to wonder whether it
might have implications in the real world. So he decided to search
mathematics for places where Omega might crop up. So far, he has only
looked properly in one place: number theory.
Number theory is the foundation of pure mathematics. It describes how to
deal with concepts such as counting, adding, and multiplying. Chaitin's
search for Omega in number theory started with "Diophantine
equations"--which involve only the simple concepts of addition,
multiplication and exponentiation (raising one number to the power of
another) of whole numbers.
Chaitin formulated a Diophantine equation that was 200 pages long and
had 17,000 variables. Given an equation like this, mathematicians would
normally search for its solutions. There could be any number of answers:
perhaps 10, 20, or even an infinite number of them. But Chaitin didn't
look for specific solutions, he simply looked to see whether there was a
finite or an infinite number of them.
He did this because he knew it was the key to unearthing Omega.
Mathematicians James Jones of the University of Calgary and Yuri
Matijasevic of the Steklov Institute of Mathematics in St Petersburg had
shown how to translate the operation of Turing's computer into a
Diophantine equation. They found that there is a relationship between
the solutions to the equation and the halting problem for the machine's
program. Specifically, if a particular program doesn't ever halt, a
particular Diophantine equation will have no solution. In effect, the
equations provide a bridge linking Turing's halting problem--and thus
Chaitin's halting probability--with simple mathematical operations, such
as the addition and multiplication of whole numbers.
Chaitin had arranged his equation so that there was one particular
variable, a parameter which he called N, that provided the key to
finding Omega. When he substituted numbers for N, analysis of the
equation would provide the digits of Omega in binary. When he put 1 in
place of N, he would ask whether there was a finite or infinite number
of whole number solutions to the resulting equation. The answer gives
the first digit of Omega: a finite number of solutions would make this
digit 0, an infinite number of solutions would make it 1. Substituting 2
for N and asking the same question about the equation's solutions would
give the second digit of Omega. Chaitin could, in theory, continue
forever. "My equation is constructed so that asking
whether it has finitely or infinitely many solutions as you vary the
parameter is the same as determining the bits of Omega," he says.
But Chaitin already knew that each digit of Omega is random and
independent. This could only mean one thing. Because finding out whether
a Diophantine equation has a finite or infinite number of solutions
generates these digits, each answer to the equation must therefore be
unknowable and independent of every other answer. In other words, the
randomness of the digits of Omega imposes limits on what can be known
from number theory--the most elementary of mathematical fields. "If
randomness is even in something as basic as number theory, where else is
it?" asks Chaitin. He thinks he knows the answer. "My hunch is it's
everywhere," he says. "Randomness is the true foundation of
mathematics."
The fact that randomness is everywhere has deep consequences, says John
Casti, a mathematician at the Santa Fe Institute in New Mexico and the
Vienna University of Technology. It means that a few bits of maths may
follow from each other, but for most mathematical situations those
connections won't exist. And if you can't make connections, you can't
solve or prove things. All a mathematician can do is aim to find the
little bits of maths that do tie together. "Chaitin's work shows that
solvable problems are like a small island in a vast sea of undecidable
propositions," Casti says.
Take the problem of perfect odd numbers. A perfect number has divisors
whose sum makes the number. For example, 6 is perfect because its
divisors are 1, 2 and 3, and their sum is 6. There are plenty of even
perfect numbers, but no one has ever found an odd number that is
perfect. And yet, no one has been able to prove that an odd number can't
be perfect. Unproved hypotheses like this and the Riemann hypothesis,
which has become the unsure foundation of many other theorems (New
Scientist, 11 November 2000, p 32) are examples
of things that should be accepted as unprovable but nonetheless true,
Chaitin suggests. In other words, there are some things that scientists
will always have to take on trust.
Unsurprisingly, mathematicians had a difficult time coming to terms with
Omega. But there is worse to come. "We can go beyond Omega," Chaitin
says. In his new book, Exploring Randomness (New Scientist, 10 January,
p 46), Chaitin has now unleashed the "Super-Omegas".
Like Omega, the Super-Omegas also owe their genesis to Turing. He
imagined a God-like computer, much more powerful than any real computer,
which could know the unknowable: whether a real computer would halt when
running a particular program, or carry on forever. He called this
fantastical machine an "oracle". And as soon as Chaitin discovered
Omega--the probability that a random computer program would eventually
halt--he realised he could also imagine an oracle that would know Omega.
This machine would have its own unknowable halting probability, Omega'.
But if one oracle knows Omega, it's easy to imagine a second-order
oracle that knows Omega'. This machine, in turn, has its own halting
probability, Omega'', which is known only by a third-order oracle, and
so on. According to Chaitin, there exists an infinite sequence of
increasingly random Omegas. "There is even an all-seeing infinitely
high-order oracle which knows all other Omegas," he says.
He kept these numbers to himself for decades, thinking they were too
bizarre to be relevant to the real world. Just as Turing looked upon his
God-like computer as a flight of fancy, Chaitin thought these
Super-Omegas were fantasy numbers emerging from fantasy machines. But
Veronica Becher of the University of Buenos Aires has shown that Chaitin
was wrong: the Super-Omegas are both real and important. Chaitin is
genuinely surprised by this discovery. "Incredibly, they actually have a
real meaning for real computers," he says.
Becher has been collaborating with Chaitin for just over a year, and is
helping to drag Super-Omegas into the real world. As a computer
scientist, she wondered whether there were links between Omega, the
higher-order Omegas and real computers.
Real computers don't just perform finite computations, doing one or a
few things, and then halt. They can also carry out infinite
computations, producing an infinite series of results. "Many computer
applications are designed to produce an infinite amount of output,"
Becher says. Examples include Web browsers such as Netscape and
operating systems such as Windows 2000.
This example gave Becher her first avenue to explore: the probability
that, over the course of an infinite computation, a machine would
produce only a finite amount of output. To do this, Becher and her
student Sergio Daicz used a technique developed by Chaitin. They took a
real computer and turned it into an approximation of an oracle. The
"fake oracle" decides that a program halts if--and only if--it halts
within time T. A real computer can handle this weakened version of the
halting problem. "Then you let T go to infinity," Chaitin says. This
allows the shortcomings of the fake to diminish as it runs for longer
and longer.
Using variations on this technique, Becher and Daicz found that the
probability that an infinite computation produces only a finite amount
of output is the same as Omega', the halting probability of the oracle.
Going further, they showed that Omega'' is equivalent to the probability
that, during an infinite computation, a computer will fail to produce an
output--for example, get no result from a computation and move on to the
next one--and that it will do this only a finite number of times.
These might seem like odd things to bother with, but Chaitin believes
this is an important step. "Becher's work makes the whole hierarchy of
Omega numbers seem much more believable," he says. Things that
Turing--and Chaitin--imagined were pure fantasy are actually very real.
Now that the Super-Omegas are being unearthed in the real world, Chaitin
is sure they will crop up all over mathematics, just like Omega. The
Super-Omegas are even more random than Omega: if mathematicians were to
get over Omega's obstacles, they would face an ever-elevated barrier as
they confronted Becher's results.
And that has knock-on effects elsewhere. Becher and Chaitin admit that
the full implications of their new discoveries have yet to become clear,
but mathematics is central to many aspects of science. Certainly any
theory of everything, as it attempts to tie together all the facts about
the Universe, would need to jump an infinite number of hurdles to prove
its worth.
The discovery of Omega has exposed gaping holes in mathematics, making
research in the field look like playing a lottery, and it has demolished
hopes of a theory of everything. Who knows what the Super-Omegas are
capable of? "This," Chaitin warns, "is just the beginning."
Further reading:
Exploring Randomness by G. J. Chaitin, Springer-Verlag (2001)
"A Century of Controversy Over the Foundations of Mathematics" by G. J.
Chaitin, Complexity, vol 5, p 12 (2000)
The Unknowable by G. J. Chaitin, Springer-Verlag (1999)
"Randomness everywhere" by C. S. Calude and G. J. Chaitin, Nature, vol
400, p 319 (1999)
http://www.cs.umaine.edu/~chaitin/
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