Discovery and Invention (4.25 / 8) (#42)
by Khedak on Sat Jan 06, 2001 at 12:38:39 AM EST


In addition, if mathematical truth is Platonic, where then does it exist?
Platonic views are simplified, but in essense discovery and invention aren't that far removed. An invention is simply a discovery of something that works for a particular purpose. For example, are airplanes a discovery or an invention? Well, the principles on which they operate had to be discovered before they could be invented. And sometimes vice versa, working inventions lead to discoveries. They go hand in hand, and you seem to ask whether mathematics is an "invention" as if this is in opposition to a discovery.
Using chess as a parallel to mathematics is fair, because chess is highly mathematical. The fact is, many mathematical constructions that may look useless at the outset in fact later are shown to have parallels in the real world. You're right in that it's an important question (perhaps the most important question) of whether mathematics and the universe are inherently related, or if such relations are coincidence. Geometry and calculus have innumerable realworld applications, whereas more abstract concepts like the proof of Fermat's last theorem seem not to. Then again, there's the question of what constitutes application to the "real world." Again, Plato has his own ideas about what is real and what is not, but that's a very deep philosophical question that is closely tied to this one.
It's important to remember that mathematics has particular rules that have been selected for a reason, usually because they are logically drawn from other logical axioms. For example, (as Hofstadter points out in his book Goedel, Escher, Bach which many k5 readers seem to be familiar), when the concept of hyperbolic or parametric (as opposed to Euclidean) geometry was proposed, it was denounced because it didn't "fit the real world." However, there are applications for which these geometries are very useful. Hoftstadter seems to think that these geometries are irreconciliably different, and that the fact that more than one can be made to fit the world means that there is no "true" geometry. But many believe that in fact the aspects that these geometries share make up what is "real", since they all differ in only their most abstract axiom.
You seem to forget all the uses of mathematics when you ask what aspect of our universe mathematicians are discovering. Citing quantum mechanics as evidence of the nonreality of mathematics is extremely ironic: quantum mechanics is one of the most heavily mathematical physical sciences there is. In that light, your question needs a little more clarification. What exactly do you want of mathematicians? Do you want them to say "Although all other physical sciences, which have real world applications, rely in some form on mathematics, mathematics itself has no intrinsic truth value." While you could make such an argument, what would be the point?
In essence, you're asking people to prove that mathematics exists in the same fashion that atheists ask the faithful to prove that God exists, or nihilists ask people to prove that anything at all exists. It's easy to deny the existence of anything at all, since any time someone presents you with evidence for its existence, you can question the reality of that piece of evidence, and so forth ad nauseam. Mathematics cannot be proven to 'exist' in the Platonic sense, but the idea that mathematics reveals fundamental truths about the reality in which we live is a useful one. Simply citing Quantum Mechanics and saying "See? It's nondeterministic, therefore deterministic systems do not reflect reality" simply does not follow, since after all, logic is also a deterministic system. If you deny logic, then why bother making an argument?
It is possible that our universe is inherently paradoxical and illogical, which would seem to be the logical end to your argument (or should I say the illogical end?). This is actually one of the central tenets of Zen Buddhism. But Zen requires satori, and I'm not enlightened so I cannot say what Zen reveals about the nature of ourselves and the universe. Sufism and Jewish Kabbalah also have the notion of 'higher truth' which is not bound by the rules of logic. But they also state explicitly that discussion of such truths is useless, since this 'truth' is hidden and cannot be communicated with language (much like you cannot explain to someone who has been blind all their life what it's like to see). But I digress.
Also, I take issue with your use of the term 'arbitrary'. Arbitrary does not mean "random" as you seem to imply. It means specifically chosen by someone. So yes, the axioms of mathematics are arbitrary, but they are chosen such that they 'make sense', and so far this has been very useful for understanding our universe. As I pointed out before, you can't take quantum mechanics seriously unless you take its mathematics seriously, since (as you know) it's highly abstract and much of it only makes sense in mathematics, though its results are supported by experiment.
So, is mathematics 'real' in the Platonic sense? That's metaphysics, and really beside the point. Does mathematics currently provide insight into the nature of the universe, and will it continue to do so in the future? In my opinion, definitely. Does this connection have any 'higher meaning', such as the existence of a creator, or the concept of the harmonious interconnectedness of all things? Despite the length of this post, I must say in all honesty, I don't know. :)

