The big question: how long is a piece of string theory? (can we ever understand the universe?)

Sydney Morning Herald ^ | February 25, 2004

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The inherent uncertainty of mathematics means we will never fully understand our world, writes Paul Davies.

The world about us looks so bewilderingly complex, it seems impossible that humans could ever understand it completely. But dig deeper, and the richness and variety of nature are found to stem from just a handful of underlying mathematical principles. So rapid has been the advance of science in elucidating this hidden subtext of nature that many scientists, especially theoretical physicists, believe we are on the verge of formulating a "theory of everything".

When Stephen Hawking accepted the Lucasian Chair of Mathematics at Cambridge University in 1980 he chose as the title of his inaugural lecture: "Is the end in sight for theoretical physics?" What he meant was that physicists could glimpse the outlines of a final theory, in which all the laws of nature would be melded into a single, elegant mathematical scheme, perhaps so simple and compact it could be emblazoned on your T-shirt. Now Hawking has done something of a U-turn by claiming in a lecture at Cambridge last month that we will never be able to grasp in totality how the universe is put together.

The quest for a final theory began 2500 years ago. The Greek philosophers Leucippus and Democritus suggested that however complicated the world might seem to human eyes, it was fundamentally simple. If only we could look on a small enough scale of size, we would see that everything is made up of just a handful of basic building blocks, which the Greeks called atoms. It was then a matter of identifying these elementary particles, and classifying them, for all to be explained.

Today we know atoms are not the elementary particles the Greek philosophers supposed, but composite bodies with bits inside. However, this hasn't scuppered the essential idea that a bottom level of structure exists on a small enough scale. Physicists have been busy peering into the innards of atoms to expose what they hope is the definitive set of truly primitive entities from which everything in the universe is built. The best guess is that the ultimate building blocks of matter are not particles at all, but little loops of vibrating string about 20 powers of 10 smaller than an atomic nucleus.

String theory has been enormously beguiling, and occupies the attention of physicists and mathematicians. It promises to describe correctly not only the inventory of familiar particles but the forces that act between them, like electromagnetism and gravity. It could even explain the existence of space and time, too.

Though string theorists are upbeat about achieving the much sought-after theory of everything, others remain sceptical about the entire enterprise. A bone of contention has always surrounded the word "everything". Understanding the basic building blocks of physical reality wouldn't help explain how life originated, or why people fall in love. Only if these things are dismissed as insignificant embellishments on the basic scheme would the physicist's version of a final theory amount to a true theory of everything.

Then there is a deeper question of whether a finite mind can ever fully grasp all of reality. By common consent, the most secure branch of human knowledge is mathematics. It rests on rational foundations, and its results flow seamlessly from sequences of precise definitions and logical deductions. Who could doubt that 1+1=2, for example? But in the 1930s the Austrian philosopher Kurt Godel stunned mathematicians by proving beyond doubt that the grand and elaborate edifice of mathematics was built on sand. It turns out that mathematical systems rich enough to contain arithmetic are shot through with logical contradictions. Any given mathematical statement (eg, 11 is a prime number) must either be true or false, right?

Wrong! Godel showed that however elaborate mathematics becomes, there will always exist some statements (not the above ones though) that can never be proved true or false. They are fundamentally undecidable. Hence mathematics will always be incomplete and in a sense uncertain.

Because physical theories are cast in the language of mathematics, they are subject to the limitations of Godel's theorem. Many physicists have remarked that this will preclude a truly complete theory of everything. Now it seems Hawking has joined their ranks.

So does this mean physicists should give up string theory and other attempts at unifying the laws of nature, if their efforts are doomed to failure? Certainly not, for the same reason that we don't give up teaching and researching mathematics because of Godel's theorem. What these logical conundrums tell us is there are limits to what can be known using the rational method of inquiry. It means that however heroic our efforts may be at understanding the world about us, there will remain some element of mystery at the end of the universe.

Paul Davies is professor of natural philosophy at the Australian Centre for Astrobiology at Macquarie University.

[jpsb]

Is the author French?

[Virginia-American]

I don't know if it has a name; it's a consequence of there being an infinite number of theorems. I don't see how it follows that there are some that require an infinite proof.

Dr S: can you add anything?

[Doctor Stochastic]

It's been a long time since I looked at length of proofs. There's a long discussion of this (and lots of other things) here. I think that a proof using a Turing machine (and this covers all models of computation so far proposed) requires that the machine stop. There is an enumeration of all proofs though. Even if all proofs were finite, there would be infinitely many proofs in a system.

One can of course talk about infinte objects using only finitary means. For example, mathematical induction is only a schema for infinitely steps. Transcendental induction is also possible.

I'll look around on Google and in some texts and maybe have something to post later.

I still don't know how to pronounce adele. I found three spellings in different encyclopaedias: adele, adéle, and adèle; not helpful.

[Virginia-American]

...requires that the machine stop. ...

I would think so! Otherwise we're in the position of 'proving' Fermat's Last Theorem by an infinite exhaustive search.

Even if all proofs were finite, there would be infinitely many proofs in a system.

That's what I was getting at.

Re: pronouncing 'adele'
Thanks. I *thought* I'd seen it with both grave and acute accents. (Grave seems more French-like somehow).

Funny, I can *pronounce* 'restricted product of p-adic completions'...

[LibWhacker]

I don't see how it follows that there are some that require an infinite proof.

The reasoning is similar to the proof that the number of primes is infinite. You start by assuming that there is some largest prime and show that that leads to a contradiction. We can do the same thing here . . . Assume that there is some "largest proof," and you'll see this leads to an immediate contradiction.

[Virginia-American]

The number of theorems that can be proved is infinite, but each proof is finite.

If assuming that there is a longest proof leads to a contradiction, then there is no longest one. That doesn't imply that any of them is infinite; by analogy, no prime number is infinite.

[LibWhacker]

Yes, quite right there, VA. I still think that one can show that the cardinality (if I can use that term) of some theorem must be "Aleph nought," the cardinality of the positive integers. Let me think about it some more.

[longshadow]

Something I've been wanting to point out before this thread withers away a dies:

The perveyors of Gödelian Uncertainty love to cast it like a wet blanket over all of Mathematics, and by extension onto physics; but here's the rub:

They never seem to grasp that just because there exists SOME truths about arithmetic of Natural numbers that can't be proven or disproven from within the system, it doesn't mean that the theorems we DO PROVE are in any way suspect!

So long as Mathematicians, and Physicists, restrict themselves to using only those Mathematical Theorems which HAVE been proven, they can never fall victim to Gödelian Uncertainty. An unproven theorem is still an unproven theorem, regardless of whether it's unproven because it's difficult (Fermat's Last Theorem), or because it is a Gödel statement. And, as long as we don't use unproven theorems, they have no effect on anything we are doing!

In conclusion, Gödel's Incompleteness theorem poses no practical impediment at all to most of science and Mathematics, which is why I consider those who raise it in this context to be either ill-informed or disingenuous.

[Doctor Stochastic]

Good point.

[Doctor Stochastic]

Note that in your comments, you have proved that there are an infinite number of primes, not that there is an infinitely large prime.

(It's possible that) one may show that for any N, there is a proof that requires N+1 steps, but that doesn't mean that there is a proof that requires infinitely many steps. To show that, one must exhibit a statement that, for any N, cannot be proved in N or fewer steps.

[Doctor Stochastic]

I think that Hilbert's (and other people's) versions of proofs require that the proof be finite. This does not mean one cannot talk about infinite sets.

[LibWhacker]

Yep, I made two mistakes here. First, I thought it was trivial and intuitively obvious that if there were proofs of any given length N, then necessarily there were infinitely long proofs. Second, I thought that by showing there were infinitely long proofs that that would disprove the antecedent of the first, because the idea of an infinitely long proof is meaningless (wrong again!).

But in trying to prove it, I found out in doing a little research that not only are there proofs that are infinitely long, their existence contradicts nothing! Some mathematicians seem quit comfortable with the notion. This is all covered in a field called "proof theory," to which I had no exposure in grad school, and I gather is an area of mathematical logic.

It still seems very strange to me, though, and by "proof" they must be working with some generalized definition of "proof." Heck, I can even think of a specific example of a countably infinite proof now, though I have no idea if this is what they are talking about, but I think it must be. For example, take Catalan's conjecture that says 8 and 9 (2³ and 3²) are the only consecutive powers in Z⁺:

Proof
1 and 2 aren't consecutive powers
2 and 3 aren't consecutive powers
3 and 4 aren't consecutive powers
4 and 5 aren't consecutive powers
5 and 6 aren't consecutive powers
6 and 7 aren't consecutive powers
7 and 8 aren't consecutive powers
8 and 9 ARE consecutive powers
9 and 10 aren't consecutive powers
10 and 11 aren't consecutive powers
11 and 12 aren't consecutive powers
. . .
And so on, forever. Q.E.D.

This "proof" has a countably infinite number of lines and if such infinitely long proofs are allowed, is in fact a "proof" since it exhausts all consecutive pairs -- PROVIDING, of course, that the conjecture is true (I suppose I should've used an example whose truth is known, but I'm too lazy now to go back and change it).

I even found references to artificially expanding any finite proof so that it has an infinite number of lines, and now that I've read a little bit about proof theory, I wouldn't doubt at all that it's possible and wager that if a person tried, it wouldn't be too hard to come up with an example of that also.

Finally, I found a claim that there exist proofs that are not only infinitely long, but have infinite logical "depth;" i.e., that are hopelessly beyond man's understanding. This is bad news, imo. Never dreamed there were results like this out there. Nice talking to you guys.

[Doctor Stochastic]

This purported proof of the Catalan conjecture is generally rejected as valid because it requires an examination of infinitely many cases. In fact, the Catalan conjecture is still considered unproven (as of today.)

An example that some people consider to use infinitely many statements would be using induction. The infinitely long strings are telescoped by this idea. Example, to prove SUM(i,i=1,N)=N*(N+1)/2; first verify this is true for 1 (by direct computation); second show that the truth for N implies the truth for N+1, that is, SUM(i,i=1,N)+N=SUM(i,i=1,N+1) thus N*(N+1)/2+N = N*(N+1)/2 + 2*N/2 = (N**2+N +2N)/2 = (N+1)(N+2)/2. By direct calcualtion the formula for N implies the formula for N+1. The inductive procedure states that the result if true for all natural numbers.

The whole business is a bit hard. (At least Barbie thinks so.)