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.
"There is a set of numbers 'smaller' than the reals and 'bigger' than the natural numbers" (aka the Continuum Hypothesis) is undecidable but is easily grasped by most with a short explanation of what 'bigger' and 'smaller' mean in this context.
There is an important assumption that is often overlooked: the methods of proof in the system must be "finitistic." IOW if you allow something like transfinite induction then you can prove completeness and consistency.
Of course then the question is, is a method like transfinite induction intuitively obvious?
It seems intuitively clear to me if you make the assumption that people (i.e. beings capable of formulating number theory) are a natural consequence of physical law.
I'll grant you what's not clear is whether the incompleteness theorem has anything interesting to do with physics.
Actually section 1 (esp. 1.1) of that page is pretty short and easy-to-read and the rest is not material to understanding CH.
It's not clear to me at all. Number systems, formal as they are, were arrived at via human reasoning, which is not a formal system. Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it, including the truth or falisity of formally undecidable Gödel statements. The price paid for this is inconsistency: some wrong statements will seem to be true, and vice-versa. I don't think I need to make a case that human reasoning is inconsistent.
Nothing more than the fact that the precise mathematical statement of the axioms gives a rigorous definition of a topological space. The moment one tries to water that down to a layman-level understanding, one sacrifices rigor (or "precision") for understandability.
Sorry about the headache....
Caution must be execised to avoid confusing "undecided" hypotheses from "undecideable" hypotheses; the CH is undecided -- that is, we don't know if it is true or not, and it also makes no difference to standard set theory whether it is true or false (IOW, standard set theory is independent of the CH), but no one to my knowledge has demonstrated the CH can NEVER be decided. IOW, no one has shown the CH is a Gödel statement.
IMO, this is a very, very horrible way to get the idea across. For starters, it's flat out false. It makes the claim in an even more positive fashion than I put it, and is just going to end up confusing a lot of people, despite his rather insufficient attempt later to correct himself. I do agree with everything else you've said, Physicist -- no argument there -- but I guess I just don't think it's a good idea to make false statments, as he's done here. It only adds to the confusion, skepticism and ignorance. We can almost guarantee there are going to be people running around after reading this article claiming that math is a joke, very likely including NEA types who think there are no "right" answers anyway.
You're most welcome. But I hope you were talking notes, because there will be a "pop" quiz next period!
;-)
I think you're misreading it, but let's be explicit: what is wrong about that statement, in your reading?
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