Monday, March 14, 2011

On Aboutness

We sometimes meet dismissive statements like this:
“Mathematics appears to be not about anything at all.  On such a view, it has no subject-matter.”

When confronted with such a pronouncement, we resort to the Weierstrass Penguin Test:  namely, does the same objection apply to penguins?  What are they “about”?  Well, in the colloquial sense, they are all about :  playing, swimming, sliding on their tummies, and all the rest;  but that is not a meta-level aboutness.  They are not the less funny and fat, for all that.

When someone speaks of mathematics, dismissively, as not being “about” anything, we suspect he never made it past simple arithmetic, and finds the integers pale and spectral  next to nice red apples.   And it is true, you cannot eat the number 2 [**]; but then, neither can you really take the square root of a couple of apples.

There is plenty for mathematics to be about. Geometry is about spatial structures.  Number Theory is about Our Friends the Integers.  Topology is about sets together with a neighborhood system. Algebra is about relations that generalize those familiar from simple arithmetic.  And so on.  The reason it is hard to say what mathematics overall is “about”, other than the tautological-sounding “mathematical objects”, is that it is so rich and deep and broad.  What, for that matter is literature ‘about’, or politics, or scholarship?

[**]  Actually, according to the frantically nominalist view of Bertrand Russell,  you can -- or a part thereof at least.  Namely, those two apples, which, along with the Everley Brothers and much else, go towards making up the unimaginable vastness of duple things, which his Lordship once fancied were simpler than das, was der liebe Gott gemacht. To such extremities is the atheist obliged to travel.

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