From: Samantha Atkins (samantha@objectent.com)
Date: Mon Apr 02 2001 - 00:46:07 MDT
Perhaps my mathematical education was neither abstract nor thorough
enough but this article looks to me to be making a great todo about
relatively little. In particular I find many of its stabs at the
implications to be utterly unjustified by the rest of the article'ss
contents. It looks like sensationalism and bad science reporting.
Would someone like to take a go at a more seasoned evaluation of this
piece?
> 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.
>
Give me a break. I have done enough mathematics to know that while
mathematical exploration can be messy and intuitive it is not subject to
being dismissed as not profound for all of 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."
>
In what sense are they "facts" or of any importance if not tied to
anything else at all? If he is saying mathematical discovery is not
mechanical, this is certainly true. But it is a jump to imply it is
thus not meaningful or is accidental.
> 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.
>
This is more bad scientific poetry. A TOE is not a complete and concise
description of the entire universe except in some quite delimited
sense. Chaitin's work or the vague intuitions of it presented here do
not in the least justify a claim that a TOE is impossible.
> 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.
>
This is ridiculous. The article tells us almost nothing about such
numbers up to this point except supposed implications that we are, one
assumes, to take on faith in the mystery of higher mathematics. There
is nothing at all that shows the existence of this class of numbers as
more than an intellectual abstraction of dubious applicability in the
entire piece. On the face of it I would be surprised if you could
rigorously show the existence of such a beast without making it somewhat
calculable. It looks like a philosophical thought experiment run amok.
> 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.
>
Actually, no Godel didn't do any such thing ("blew a gaping hope in
mathematics") or at least not with what are popularly thought to be the
implications.
> 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.
>
Sure, but it certainly does not mean that all programming is meaningless
or unpredictable or that all working programs are lucky hits in a
Sargaso Sea of unintelligble binary jumble. Although in practice it
often feels that way.
> 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.
>
So what? It is a simple restatement of the halting problem.
>
> 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.
>
OK, so you can build a mapping. But it is another thing to say that
mapping implies things true in one domain are true in the other without
more evidence than just such a mapping.
> 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."
>
This is the leap of hyper-intellectual magical thinking, a mystic
correspondence between unlike things. Perhaps it is my ignorance but
this does not strike me as terribly convincing.
> 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.
What justifies going from a 200 page specialized equation made just to
get this mystical correspondence to a claim that most mathematical
situations are utterly disconnected and random? How is this more than
sensationalism?
>
> 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.
>
OK. I give up on this. I have piddled with such unproved hypothesis
myself and I certainly do not believe that they are unprovable or
something to be taken on faith.
- samantha
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