Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

Saturday, April 12, 2008

Philosophy of Math

I was thinking today about the various philosophies of math. Ideally, I'm a formalist, but more and more, I don't think that is quite right. Various other semi-formal positions seem to be a little closer to what I believe than formalism proper. Anyway, as I was considering this, I came to the conclusion that math would still be done by the same people, in the same way no matter which philosophy turned out to be "correct" (if there is one). With this realization I came full-circle, because it is such a formalist thing to decide: "it doesn't matter what it means outside of the system, because the system works the same way no matter what."

I'm also still bemused with Gödel's second theorem: For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. That is, any mathematical system that can express addition can not be proven consistent within the system. If it contains a statement proving its consistency, it means it is not consistent. Gödel's first theorem is likewise entertaining, but the basic idea of the second is just hilarious. It says that a mathematical system saying it is consistent is like someone saying "I have never lied." Everyone has lied at some point. Thus, we know that statement to be a lie... Only it's stronger in math.

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