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 think you're misreading it . . .
Okay, that's a good point. Let's take for example the statement
"One plus one is two."That's a given mathematical statement. But according to Davies, if I say,
"Class, that statement must either be true or false,"then I am mistaken because, according to him, you can't make that claim about ANY given mathematical statement, including one so trivial as this. At least that's how I read it, and I think that's how a lot of people will read it.
Do you think I'm making a mountain out of a molehill? Maybe so. It just struck me as sooooo wrong and misleading. Is Davies a Brit? Gotta make exceptions for those guys, lol.
I found the statement to be misleading in the sense that to a casual reader, they might think that he's saying that "11 is a prime number" and similar simple straighforward true statements) are somehow in doubt.
You know that's no true, and I know it's not true, and we both know he couldn't have meant it that way, but for lay people not familiar with Gödel, it is way too easy to misinterpret what the sentence was intended to convey.
I read it differently. Davies has gainsaid the proposition that any given mathematical statement MUST necessarily be true or false. He presumably accepts that any given statement MAY be true or false. As it turns out, most mathematical statements are in fact either true or false. Davies is simply pointing out that there are exceptions.
Davies is using sloppy language, though. Gödel's theorem doesn't say that some mathematical statements are neither true nor false. What it says is that, in some formal systems, there exist some statements whose truth or falsity cannot be proven, one way or the other, within the system itself. But even these statements may themselves actually be true or false. In Davies's defense, however, I'll say that this distinction is lost on most people.
Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the Axiom of choice).
Which has no bearing. The number system is formal and contains Godel statements. It is (assumption) the natural product of human reasoning which is (another assumption) a natural product of physical law. By composition, the number system is a natural product of physical law. Ergo, physical law contains Godel statements.
Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it ...
Not so. Human reasoning has bounded algorithmic complexity. Any truth exceeding that complexity is not open to it. So far as I know, there's no way out.
... including the truth or falisity of formally undecidable Gödel statements
Again not so. The nature of Godel statements is that they are neither true nor false but may be assumed to be either true or false.
I don't think I need to make a case that human reasoning is inconsistent.
Actually I think "unreliable" is what you're after. Inconsistency would be believing X and not-X at the same time and not thinking that's a problem.
Not true. What is true is that such systems cannot within themselves be proven to be contratiction-free. Perhaps Paul Davies doesn't fully understand current mathematics. Of course, not all of mathematics is subject to these limitations. Pressberger arithmetnc (no multiplication, just addition, but you can multiply by any given number through iterated addition; you just can talk about multipllication as such) has no such problems. Euclidean geometry (and thus the non-Euclidean geometries too) are provably consistent.
Not necessarily true; Euclidean geometry (with the Hilbert axioms) is complete and consistent.
Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove? I've never heard that claim made before in quite the way you just made it. Of course, there are a whole host of things that are undecidable in Godel's sense of it, but that's a fundamental constraint having nothing to do with our intelligence or lack of it. I've often thought that perhaps there is some "critical mass" ito intelligence, and that once a species exceeds that critical mass, it can in principle prove anything that can be proven or, alternatively, understand any valid proof. As Albert Einstein was supposed to have said, "The most incomprehensible thing about the universe is that it is comprehensible." Hope that's true. It's gonna be depressing if we're just another species of dumb apes.
Have you looked at some of Chaitin's work? The digits of his Omega number are an interesting case. His take on it is interesting: anything sufficiently complex is essentially random.
But on a more mundane level, any finite theorem proving machine (and I'm assuming people are that) will be helpless in the face of a theorem whose smallest proof exceeds the capacity of the machine.
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