March 15, 2008

Cantor Rulez!!! (A Philospher Writes…)

Why Cantor matters (and Wittgenstein's just an interesting historical oddity): "Writing decades after Cantor's death, Wittgenstein lamented that mathematics is 'ridden through and through with the pernicious idioms of [Cantor's] set theory,' which he dismissed as 'utter nonsense' that is 'laughable' and 'wrong'." (from Wikipedia's entry on Cantor).

From the Olympian heights of philosophy, mathematics must seem so grubby, and, well, useful, but to this reader, it's hard to get past Cantor's transfinite numbers for examples of abstract beauty and the stringent clarity of the purely counter-intuitive. I think the first time I read about — and understood — the various comparative transfinite cardinalities of the integers, the rationals, the reals, etc., (and, crucially, the associated proofs) was when I realised I could do mathematics in ways I can't do arithmetic (don't ask me to add things up). To a young Woy Woy boy struggling with high school in the foreign reaches of Canberra, this was a revelation.

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