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.
Then why did he use it? What is it with the crappy math articles lately? Now we're going to have ignoramuses running around saying mathematicians can't prove 11 is prime and smirking at everything else they say as well.
Whatever they are.
bump
Any given mathematical statement (eg, 11 is a prime number) must either be true or false, right?The "primeness" of "11" is no more in doubt than any other proven theorem in Mathematics. The author is very misleading in his presentation of the example, though he then correctly states the reality in the subsequent paragraph, though he leaves out an important caveat: Gödel showed that there will exist some statements derivable in a mathematical system which can't be decided based on the axioms of that system. There is nothing, however, to prevent you from proving the statement true by appealing to axioms outside of the system you are working in.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.
But the point remains: any statement which is proveable by standard Mathematical proof techniques is NOT a "Gödel statement" and hence is NOT undecideable. That some statements are undecideable cast no doubt on the ones we can, and do eventually prove. There is no evidence that reality will turn out to be based on Gödelian statements, so it may not even be an issue for physicists.
That's true, but the original statement in question will none-the-less have been proven, and thus is no longer a Gödel statement, though there will now be new ones as you correctly point out.
"A manifold is a topological space which is locally Euclidean ..." source: http://mathworld.wolfram.com/Manifold.html
"locally Euclidean" simply means that on short distance scales ("local") it exhibits the topological characteristics of flat Euclidean Geometry.
A topological space is the most primitive (least complex) mathematical structure:
In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff Axioms.1. To each point x there corresponds at least one neighborhood U(x), and U(x) contains x.
2. If U(x) and V(x) are neighborhoods of the same point x, then there exists a neighborhood W(x) of x such that W(x) is a subset of the intersection of U(x) and V(x).
3. If y is a point in U(x), then there exists a neighborhood U(y) of y such that U(y) is a subset of U(x).
4. For distinct points x and y, there exist two disjoint neighborhoods U(x) and U(y).
source: http://mathworld.wolfram.com/TopologicalSpace.html
I'm afraid there's no good way to state this in laymans' terms without losing the precision of the axioms.
Well, no. He didn't say that. He gave the primality of 11 as an example of a mathematical truth, and then went on to say that the truth of some mathematical statements is undecidable.
I do disagree with Davies, however, that Gödel's theorem has anything to do with physics. Gödel's theorem only applies to certain types of formal systems. It is by no means clear that there exists no formal system appropriate for describing physics that is free from undecidable propositions. Furthermore, if even if all appropriate systems suffer this blind spot, it isn't clear that it would conceal anything important about the universe.
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