Mar. 29th, 2011

So I've always been deeply disturbed by Gödel's incompleteness theorems. I've tried to explain to people why they freak me out so much, but I haven't had much success. I've had people tell me that I don't understand what they mean.

I think I'm kind of in a place to explain why I find this so disturbing, and as a corollary why I find mathematical logic so amazing.

I was raised on a very odd kind of fundamentalism. I think it was mostly a by-product of how intelligent, yet how constrained by her husband, my mother was. Theology was the only thing she could really turn her mind towards back then. She taught me to regard the Bible as containing the axioms and inference rules for life, that all actions should be derivable from scripture. Exegesis was the proof theory for morality, essentially.

Of course, the Bible itself was to be used to justify its own validity and I got very used to the idea of there being an absolute arbiter of truth, a system for deciding what was right and wrong that was beyond question because it had proved its own soundness and completeness.

When I was in my early teens I fell in love with mathematics, and looked for the same kind of certainty there that I had in my theology. You can see where I'm going with this, I'm sure, and that it was a sock to the gut to find out that the idea of a perfect, self-justifying, formal mathematics was a pipe-dream. It's impossible! There is always doubt about the ground level, because you have to use informal mathematics to justify your formal system. On some level, the idea still makes me feel a little sick to my stomach.

Yet as I've let go of that theology my mother and I developed, I've come to love the study of logic itself. Particularly, I'm enamored with constructive mathematics - the reduction of formal mathematics to programs that construct, in some sense, real artifacts that have the properties we want. It may not be the perfect, self-justifying, system I thought I needed but I find it beautiful regardless.

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