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Tuesday, August 2, 2011

Mathematical Intuitions

I recently had a brief discussion about the non-existence of numbers with a friend and student of physics. I'm a skeptic when it comes to the existence of numbers. I think math is something we do, and that numerals and their corresponding words are tools without referent. There is no number 2, but only various roles filled by the numeral "2" and the word "two." I wouldn't even say the numeral "2" denotes a rule (or rules) for its use.

My friend agreed there was something obviously correct about my approach, but said that we still have to wonder about the correspondences. I didn't have time to respond, but the remark seemed to betray a common intuition about mathematics: that mathematical equations and theorems in some way correspond to facts. I don't think there's any reason to suppose this is true. Math does not correspond to anything, just like hammers don't correspond to anything. I don't think there's anything corresponding to the number two, for example. Nor must there be anything corresponding to the equation 2 +2 = 4, or to any of Peano's axioms, and so on.

What we might want to account for isn't correspondence so much as utility. Why does math work? The obvious answer is: because organisms have evolved to do certain things. Roughly speaking, math is the formalization and utilization of systems of quantification in the identification and deployment of patterns. Why does that work? Well, why does the heart work? This is something that we are able to do, for some evolutionary reason. There is a good answer, or set of answers, but not a philosophically puzzling one, even if we don't know all the details yet.

There might be another aspect of math that seems philosophically puzzling: that is, why does it seem like mathematical theorems are discovered, and not invented? In some sense, the theorems of mathematics seem to already be "out there." But where is "out there?"

One possible answer: As we develop our mathematical system (or systems), we constrain the possibilities for their development. A mathematical theorem is not simply invented out of nothing. So what is "out there" are the parameters of possible mathematical theorems determined by the mathematical system we are already using--or, perhaps even better: determined by our innate capacity for mathematical invention, which is often integrated with the constraints of our current mathematical system.