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.
If this seems like an unduly pessimistic expression of biological determinism and fatalism, then allow me to conclude on an upbeat note. The biological apparatus allowing us to science encompasses our perceptual faculties as well as our cognitive ones. We have been able to move beyond the innate limitations of our five senses through the instruments we have devised-everything from telescopes to microscopes, thermometers to chromatographs, radio telescopes to bubble chambers. We may be able to similarly go beyond the biological limitations of our masses of gray mush by developing computers and other artifical intelligence systems capable of doing science, rather than simply serving as assistants. How this may be done-well, that's an answer which is beyond me.
But it's still not clear to me that there are necessarily proofs out there that exceed the capacity of man's brain to understand, but I suppose that'll come out as I read some of his articles. It seems to me that he'll never be able to prove that a given theorem is beyond mankind's ability to prove. Of course, there are lots of unproven conjectures bouncing around out there, but when it comes to many of them, we just don't know if it's because we're not and never will be smart enough to prove or disprove them (I sure hope that's not the case, though it does appear to be what Chaitin is saying), or if it's because math isn't sufficiently developed to allow us to do much with them yet (another way of saying this is that the right genius hasn't come along yet and showed us the correct way to approach the problem), or if it's because it's one of Godel's undecidables.
Consider the four-color theorem. The original proof requires the use of a computer to check billions of cases; here's a link to another proof, where the author says
"The first proof needs a computer. The second can be checked by hand in a few months, or, using a computer, it can be verified in about 20 minutes."
As a general rule, given any formal system, and given any positive N, there will be theorems in it whose proof requires at least N steps. That's any N - eg 10 to the 10 to the 10 ... to the 10.
But by the same argument, human reasoning would contain Gödel statements, which is patently absurd. Gödel statements are not a general feature of mathematical systems. They only occur in certain classes of formal axiom sets, such as number theory.
Human reasoning has bounded algorithmic complexity.
Show me. Remember that we are talking about human reasoning in principle, so don't bother with in-practice arguments like the limited human lifespan.
The nature of Godel statements is that they are neither true nor false
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.
I maintain that computer programs are implementations of human reasoning, thus all of their results fall under that heading.
Church's Thesis (if true) would certainly support this claim. So far, all models of computation (for example: Turing machines, recursive function theory, Post normal forms, quantum computers, probabilistic turing machines, stack machines, etc.) all compute exactly the same set of functions. No model of computation has been found to extend this set.
Gödel's theorem? Of course not.
The balancing of my checkbook is, however, an excellent example of Heisenberg's Uncertainty Principle.
I'm not clear on what you mean by "Euclidean description"....
The simplest way I can think to say it is a manifold is a topological space which locally Euclidean. That is to say the space appears to be Euclidean if you don't stray too "far" away from the point you are examining.
I put "far" in parens because not all topological spaces have a notion of "distance" (as best I recall)....
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).
I'm going to defer to "Doctor Stochastic" on this one, if he cares to address it. I think he's already stated the CH is NOT a Gödel statement, and I'm a bit perplexed by Wolfram's quote; I think he's saying the CH is undecideable in the same sense that the Parallel Postulate is in Geometry, which is definitely not the same as the Gödelian sense of the term.
I try to stay away from such places.
UPDATE:
The reference to Gödel, cited in your reference, is to his work on the consistency of CH with set theory, not to his Incompleteness Theorem.
Basically, Gödel showed that CH is consistent with formal set theory (Zermelo-Fraenkel) plus the Axiom of Choice (AC). Cohen later showed that ~CH is ALSO consistent with set theory + AC ("ZFC").
That means that CH is "undecideable" from within the framework of ZFC.
But that doesn't mean it is undecideable in an absolute sense, and work continues apace to find axioms that will lead to an answer to the continuum problem.
I hope that clears things up.....
"All our thoughts and concepts are called up by sense-experiences and have a meaning only in reference to these sense-experiences. On the other hand, however, they are products of the spontaneous activity of our minds; they are thus in no wise logical consequences of the contents of these sense experiences. If, therefore, we wish to grasp the essence of a complex of abstract notions we must for the one part investigate the mutual relationships between the concepts and the assertions made about them; for the other, we must investigate how they are related to the experiences."
As best I can understand you question, no; I don't think that's it.
In topology, the word "manifold" refers to the subset of topological spaces that are locally Euclidean. IOW, it is a category of topological spaces that satisfy an additional constraint: they behave like Euclidean spaces if you don't stray too far -- IOW, Euclidean Geometry might work on short distance scales, but not between points in the space which are more widely separately.
The surface of the Earth is an example: locally, we can't tell that we aren't on a Euclidean plane, but if you zoom out far enough and watch ships sail over the horizon, you begin to realize it isn't really flat. Euclidean geometry doesn't work on large distance on the surface of the Earth. Shortest paths between two points are "Great Circle " routes instead of straight lines, but on small distance scales, the Great Circles become indistinguishable from straight lines connecting the two points. That's "locally Euclidean" but globally non-Euclidean.....
Does that help?
Write on one side of a 3x5 card:
The statement on the other side of this card is a lie.
and on the other side:
The statement on the other side of this card is the truth.
Thats it.
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