Posted on 02/24/2004 6:38:39 AM PST by dead
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.
I've never claimed that human reasoning is a consistent, formal mathematical system with only finitistic methods of proof. That's the precondition for containing Godel statements.
Show me. ... don't bother with in-practice arguments like the limited human lifespan.
What, in principle, is an upper bound on the longest proof that could be written down and reasoned over? I can see several ways of getting at it. Because the universal expansion is accelerating, the amount of energy available for the calculation is bounded. There is some irreducible enery cost to preparing/erasing a bit of information. The amount of that needed will be at least proportional to the length of the proof. That gives an upper bound on the length of proof accessible to us.
Whatever that number is, there are theorems whose smallest proof is longer.
No, the nature of Gödel statements is that their truth or falsity is not provable by the set of axioms of the formal system in which they appear.
Yes, that's true. But the truth or falsity of any statement is only relative to some formal system. I was responding to the statement in your post which I took to imply that Godel statements could be said to be true or false irrespective of any formal system. IOW that mathematical statements can have some kind of objective truth.
No mathematical statement is undecideable in an absolute sense; one can always add an otherwise undecideable statement as an axiom (or its negation or some set that of statements that imply it or its negation) and presto, it is then decidable.
That's what's up with CH. Evidently it would be bad form to just add CH so they'll cook up something not quite so blatant instead.
In other words, there are proofs that require an infinite number of steps? That's the immediate implication.
I guess it'd help me if I understood whether we are talking about proofs in general or only about algorithmic "proofs." It's true that there are brute force proofs which take a huge number of steps to check every case, but of course that doesn't imply that more "elegant" solutions don't exist. The four-color theorem may be a case in point.
Close?
Your getting the flavor of it.....
Perfect. This article is pure nonsense. You have nailed it.
Correct me if I'm wrong here, but the Parallel Postulate isn't a Gödel statement, because geometry is exempt from Gödel's Incompleteness Theorem (because the axioms of Geometry don't incorporate the axioms for arithmetic on Natural numbers).
It is worthy of keeping in mind, it seems to me, that the essence of the Gödel's Incompleteness Theorem is that "there are some 'truths' about arithmetic of natural numbers we can't prove or disprove using only the axioms for arithmetic of Natural numbers..." The Parallel Postulate isn't a "truth" about arithmetic of Natural Numbers.
It also occurs to me that just because Gödel proved all formal systems that include the axioms of arithmetic of natural numbers will contain undecideable statements, it does not follow that ALL undecideable Mathematical statements are Gödelian in nature, the Euclidean Postulate being one counterexample.
As well you AND your flea-bitten friend, Plato the Platy, should!
We can 'understand' nothing about our 'world'. We can discover the cognitive and perceptual processes and limitations of ourselves. This is what our 'world' is. If it isn't prewired (into the brain and nervous system)we can't do it.
Just a minor quibble here: in a manifold, EVERY point has a "local Euclidean" region around it, not just some points. That is to say it is everywhere locally Euclidean, but globally non-Euclidean.
That's not proof; that's "decideability" by fiat. It reminds of a quote:
"Some men think the world is round, others think it flat. It is a matter capable of question. But if it is flat, will the King's command make it round, and if it is round, will the King's command flatten it?" - "A Man for All Seasons" -Robert Bolt.
That's what's up with CH. Evidently it would be bad form to just add CH so they'll cook up something not quite so blatant instead.
Let me guess, you don't like Mathematicians, do you?
;-)
I don't know. All I said was that the number of steps is unbounded.
I guess it'd help me if I understood whether we are talking about proofs in general or only about algorithmic "proofs."
All proofs in a formal system.
Holy cow, this is really amazing to me, VA. My inclination is to say that any proof of any theorem that never gets to the punch line is no proof at all. Surely such an important theorem has a name. Please tell me what it is so I can go look it up. Thanks. I mean, it's embarrassing; I have an advanced degree in mathematics and I've never heard of it! :-(
And a book of Vogon poetry.
-PJ
It seems to me that, if at the quantum level particles and waves are interchangeable (the duality as Hawking puts it), and then if particle physics is really string theory, then tell me what a wave would look like as a string counterpart, if a point on a wave is a particle counterpart.
-PJ
What I said was any given natural number. Unbounded and infinite are not the same thing.
I do understand that you're talking about any given natural number. Surely you understand that if there are theorems that require more steps (to prove) than any given natural number, then it follows that there are theorems that require an infinite number of steps. Of course, infinite and unbounded aren't the same in the case of convergence! But here you clearly have divergence.
Kindly just give me the name of the theorem you're quoting, please. I'd really like to take a look at it and can't find it on Google using the keywords and phrases I'd expect to find in any description or discussion of it. Thx.
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