Keith Devens .com

Gödel's theory of numbers

At Slate, Jordan Ellenberg reviews Rebecca Goldstein's new book, Incompleteness, about Kurt Gödel. Though I'm familiar with the argument, I've never actually read Gödel's proof of the incompleteness theorem. How it's presented in this Slate article sounds disturbingly similar to the silly style of argument I wrote about before, but I'm sure it's more substantial than that.

Goldstein's book sounds interesting, and it turns out some of Gödel's philosophy is very close to mine (see my comments on numbers and materialism and those about intuition below):

One person who would not have been surprised about the relative inconsequence of Gödel's theorem is Gödel himself. He believed that mathematical objects, like numbers, were not human constructions but real things, as real as peanut butter sandwiches. Goldstein, whose training is in philosophy, is at her strongest when tracing the relation between Gödel's mathematical results and his philosophical commitments. If numbers are real things, independent of our minds, they don't care whether or not we can define them; we apprehend them through some intuitive faculty whose nature remains a mystery. From this point of view, it's not at all strange that the mathematics we do today is very much like the mathematics we'd be doing if Gödel had never knocked out the possibility of axiomatic foundations. For Gödel, axiomatic foundations, however useful, were never truly necessary in the first place.

One of the common threads in philosophy I've noticed is that you often have a tension between two equally ultimate things. The most recent instance of this phenomenon I've noticed is in the case of language. "What someone said" is somehow dependent simultaneously on the intention of the speaker, the speech act itself (words used, intonation, gestures, etc.), and the mindset of the listener (I might also add "surrounding context", though for now I'll consider that to be made up of the combination of speaker intention and listener mindset). How do these all ultimately interrelate? I doubt it's possible to define precisely, and I think it's ultimately dependent on our intuition about these things being in the right tune. This is one of the reasons believing in the Trinity is essential for doing philosophy, as that is one concrete case we know of where the one and the many are equally ultimate, yet conflicting. It serves as the archetype for all the similar cases we come across in all areas of philosophy and human experience.

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20:00 (utc-5)
edited: 21:09 (utc-5)