Thursday, June 23, 2005

Godel and the Nature of Mathematical Truth

/ truth / mathematical /
A discussion with philosopher and novelist Rebecca Goldstein:
"Godel mistrusted our ability to communicate. Natural language, he thought, was imprecise, and we usually don't understand each other. Godel wanted to prove a mathematical theorem that would have all the precision of mathematics—the only language with any claims to precision—but with the sweep of philosophy. He wanted a mathematical theorem that would speak to the issues of meta-mathematics. And two extraordinary things happened. One is that he actually did produce such a theorem. The other is that it was interpreted by the jazzier parts of the intellectual culture as saying, philosophically exactly the opposite of what he had been intending to say with it."

A very interesting interview relating to Godel and his quest for a vindication of his Platonism, as well as the intellectual climate in which he thrived.

Just one gripe... Goldstein, referring to the friendship between Godel and Einstein says:

"... Both of them saw their work in a certain philosophical context. They were both strong realists: —Einstein in physics, and obviously Godel in mathematics. That philosophical perspective put them at odds with many of their scientific peers..."

Well, mathematical realism (aka Platonism) and scientific realism are not quite the same thing. They are not related. Mathematical realism is a species of idealism, whereas scientific realism is strictly a (mutated) nephew of logical positivism, the antithesis of Platonism as explicitly mentioned by Goldstein herself in the linked article when she discusses Godel's differences with the Vienna circle. It's probably true furthermore, that calling Einstein a "realist" oversimplifies his somewhat eclectic and idiosyncratic philosophy of science. So the parallelism fails completely.

PS1: A book to read on Godel and Einstein and how Godel added a twist to Einstein's General Theory of Relativity is Godel Meets Einstein, by Palle Yourgrau.

PS2: Sorry about the missing umlauts (they wouldn't display properly)...

Link via Monochrom


lazopolis said...

Apart from the completely counter-intuitive use of "realism" (in a intuition-inviting article), I also don't exactly understand what in Einstein's theory of relativity is considered to be such an assault on objectivity, the existence of truth and rationality. My general feeling is that Goldstein is trying to convince us (as opposed to trying to explainto us) of her overstretched version of Godel-Einstein's relationship.

Michael Meo said...

While I can follow the writer's comment that Goedel did succeed in producing a theorem which rigorously stated something philosophic about the topic of meta-mathematics, I find myself puzzled by the claim that Goedel's theorem was interpreted in a manner opposite to the one which he intended.

The structure of mathematics in logically incomplete: theorems can always be stated which can be neither proven nor disproven in any finite system of definitions and postulates. That's Goedels' Theorem, can we not agree?

So, when the philosophically-inclined literati conclude that mathematics is logically incomplete, how are they violating the sense of GT?

Eh, what do I know?

talos said...

Michael: I think the key point is that Godel found propositions that are both unprovable (within a given axiomatic system) and *true*. The fact that they are shown to be true *despite* being unprovable is indicative of the absoluteness of truth value. The author says as much:

Gödel appropriated this ancient form of paradox in order to produce a proposition which we can see is true precisely because we can see it's unprovable. This proposition has a purely straightforward mathematical meaning but it's also a proposition that speaks about itself. : The proposition is, in effect: "This very proposition is unprovable". Is it true, or is it false? If it's false, then its negation is true. Its negation says that the proposition is provable. So, assuming the system to be consistent, if this problematic double-speaking proposition is false, its negation is true, which would mean the problematic proposition itself is thus provable. So if it's false it can't be false. If it's false it's true. Therefore it has to be true. But unprovable!

That's how he does it. That's the proposition that's both true and unprovable. And remember that it has a strictly arithmetical meaning as well. That's accomplished through the Gödel numbering. So he's shown that in any consistent formal system of arithmetic there will be true but unprovable arithmetical propositions. A formal system of arithmetic is either going to be inconsistent or incomplete.

That's the part that's missing for most popular accounts. Godel used the theorem philosophically as evidence for his platonism.

talos said...

Lazopolis: platonism is indeed called "mathematical realism" in the philosophy of mathematics... So the author didn't invent it but she missed the fact that it is used in quite a different sense than in physics.

As for Einstein... well in a lot of popular accounts you do have strings of thought of the form relativity -> everything is relative -> moral relativity or (even worse) gnosiological relativity...