Platonism

Webster 1913 does not mention it explicitly, but Platonism is one of the primary streams¹ of mathematical thought. This is the notion that the geometric and algebraic forms that mathematicians like to play around with exist outside of human thought.

This would seem like common sense to most people: Physics obeys mathematical laws. (Should I add a litany of examples?)

Even the fact that objects in the physical world can only approximate these forms only add weight to the argument, as object after object approximates the same form time and time again, with different patterns of error.

But there are real objections to certain bits of mathematical Platonism.

The flimsiest criticism is that Platonism conjures notions of the ideal, say, chair, the very paragon of chairness. This is pretty silly.

Less silly is the argument that if these ideal forms have to exist somewhere, and if not the real Universe (we don't even have a Euclidean space to deal with!), then where? (Plato had his true world, but that does have an ideal chair in it). Responses of "the human mind" only make the finite constructionist's argument for him.

The shakiest part of Platonism seems to lie in some authors' presentation of its response to Godel's Incompleteness Theorem: The Platonist would argue, so they say, "Ok, such-and-such a statement of arithmetic is undecidable, but that doesn't mean it isn't true or false!" The strict Platonist would argue that the Axiom of Choice and the Continuum Hypothesis are "true" independently of the rest of set theory.

The (post-Hilbert) Formalist would argue the creation of two new finite systems, one with the undecidable proposition as a new axiom, and one with its negation as an axiom. This seems to me to be more in the spirit of Platonism than the preceding view.

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¹The others are finite constructionism and formalism.

Well, if you ask me -- and you were wise not to -- Platonism in its place is one of the best ideas we ever had, and out of its place it's one of the worst.

As Gorgonzola observes, it's real hard to accept the idea that there is an ideal chairness somehow (mysteriously) immanent in the world which is independent of the physical reality of actual chairs like them what you sits on. William "Wild Bill" Occam throws a fit when you start dreaming up stuff like that. It doesn't explain anything that can't be explained far more sensibly in some other way, but it was an attempt to explain something that does indeed need explaining: When you look at a chair which is different from other chairs that you've seen, you nevertheless recognize it as a chair. Not only that, but we can conceive of "chair" independently of any particular chair.

So what's with that?

It's pattern recognition. The word "chair" signifies a set of characteristics, and if a thing fits the description reasonably well (fuzzy logic blah blah blah), we call it a "chair", even if it's really a partially-hollowed-out tree stump. Plato saw this at work in his own mind, and he jumped to the wrong conclusion about it: He assumed that it was something external affecting his perceptions, rather than something between his ears doing the same. He described something using words, and then thought that he could learn everything there was to know about the thing merely by studying the words. He took as an axiom the bizarre notion that language can and does provide an accurate and complete model of reality.

Still, that's not so relevant: Plato's been dead for years now.

What matters is the fact that we do think this way. When we say "chair", the most that that can really mean is "this is a thing that's good for approximately one person to sit on", but we've got an awful tendency to take it more seriously than that. Part of our problem is that us monkeys on roller skates are just smart enough to screw ourselves with it: To a dog, "prey" == "that which darts across my field of vision", and that's that. If the cat sniffs your nose, it's a friend; if it darts across your field of vision, you eat it. No problem. The same organism is changing basic categories in the twinkling of an eye, but dogs are dumb as a rock and they don't care. We, on the other hand, are just smart enough to try to insist that the cat is always one thing or the other. Well, and I'm speaking figuratively here, it ain't. See "partisanism" for starters.

Categorizing is a good thing, though, where appropriate: When you learn to think of "tigers" as a general class of inimical critters, and you learn to recognize members of that class, you've got a better chance of living long enough to breed than you would if you treated each tiger as a unique and precious snowflake or something. Yeah, each tiger is unique and precious in some ways, especially to its mother, but you really should face the fact that they'll all eat you. This kind of thinking can be very handy. It's why we're still around: We can generalize about things. We can assign a common identity to things that are merely similar rather than identical. It's a blunt instrument, but a powerful one, because it provides a very simple model of reality which is flawed only in relatively subtle ways. Call it tempered ontological intonation, if you will: A necessary fudge factor.

The problem (as we can learn from anybody who's ever driven through New Jersey with a dark complexion) is that blunt instruments are just that: Blunt instruments. They're worse than useless when it comes time to slice a tomato, and in this world of ours there are tomatoes a-plenty to be sliced.

Blunt instruments like "horse" and "tiger" are great when you need a fast way to escape something that might want to eat you, but when it comes time to study taxonomy (for example), they're maddeningly clumsy. Throw in evolution (for a better example) and you're in serious trouble: At one point there was eohippus, and now there is horse. At what point did the change happen? To which individual do you point when you say, "this is horse, and its parents were eohippus"? None, none at all. It's ridiculous on the face of it. There is no place to draw that sharp dividing line which taxonomy and Plato both demand that we draw between one word (and/or that which is signified by that word; same difference in their terms) and the next. You can pretend to draw the line only if you take widely separated samples and refuse to think about all the fine gradations which necessarily lie between them.

If there are any creationists in the studio audience tonight who have a fondness for Borges, they could do worse than to refute evolution on ontological grounds: It is an axiom that an animal must be wholly a member of one category or another. An "intermediate form" is inconceivable and therefore unreal. I guarantee you, you'll find people who find that convincing. The trick is to move from the indefensible axiom to the logic quickly enough to get 'em worrying about the logic instead. That way, they'll implicitly accept the really destructive part and concentrate on biting the feathers. Once they've done that, they're sunk.

Computer programming, by the way, is fun and fascinating precisely because a computer is a funny little man-made world where Platonism does provide an accurate and complete model of reality.

Pla"to*nism (?), n. [Cf. F. Platonisme.]

1.

The doctrines or philosophy by Plato or of his followers.

Plato believed God to be an infinitely wise, just, and powerful Spirit; and also that he formed the visible universe out of preexistent amorphous matter, according to perfect patterns of ideas eternally existent in his own mind. Philosophy he considered as being a knowledge of the true nature of things, as discoverable in those eternal ideas after which all things were fashioned. In other words, it is the knowledge of what is eternal, exists necessarily, and is unchangeable; not of the temporary, the dependent, and changeable; and of course it is not obtained through the senses; neither is it the product of the understanding, which concerns itself only with the variable and transitory; nor is it the result of experience and observation; but it is the product of our reason, which, as partaking of the divine nature, has innate ideas resembling the eternal ideas of God. By contemplating these innate ideas, reasoning about them, and comparing them with their copies in the visible universe, reason can attain that true knowledge of things which is called philosophy. Plato's professed followers, the Academics, and the New Platonists, differed considerably from him, yet are called Platonists.

Murdock.

2.

An elevated rational and ethical conception of the laws and forces of the universe; sometimes, imaginative or fantastic philosophical notions.

 

© Webster 1913.