Mathematics

Are mathematical statements existential statements? I ask because we're taught that set theory is, in a sense, foundational to all mathematics, and most of the propositions considered in set theory essentially assert the existence of particular sets.

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I'd separate the question whether mathematical statements are (often) existential from the question of the status of set theory. (Sure, we can construct faithful proxies inside set theory for most of the structures that mathematicians are interested in. But it is a moot question whether this makes set theory foundational in any good sense at all.)

Now, many mathematical statements are pretty uncontroversially not existential, but have the form "if anything is A it is B". So the theorem that anything which is a finite division ring is commutative doesn't tell us that there are such things as finite division rings, but only what they must be like if they do exist.

But of course many other common or garden mathematical theorems certainly do look existential. "There are an infinite number of prime numbers" looks existential -- and it is naturally read as implying that there are prime numbers (lots of them!). "There are four regular star polyhedra" looks existential -- and it is naturally read as implying that there are regular star polyhedra. And so it goes.

Perhaps appearances are deceptive, however. Perhaps these superficially existential statements are really non-committal "if ... then ..." statements in disguise. So really what we are saying, for example, is something along the following lines: if any collection of things has the natural-number structure, then it contains an infinite number of things filling the prime-number role -- leaving it open whether there is anything that satisfies the antecedent of the conditional. Alternatively, maybe "there are an infinite number of prime numbers" is a statement made inside an essentially fictional mode of discourse (the arithmetical fiction), and no more really implies that there are numbers than "Sherlock Holmes lived at 222B Baker Street" implies that there really was such-and-such a man living along Baker Street.

Now, "if ... then ..."-ism and fictionalism are problematic as stories about the status of mathematics. I'm not recommending either position! But they are enough to illustrate the point that it isn't, perhaps, so obvious after all that prima facie existential mathematical statements have to be construed as really being kosher existential statements.

Stewart Shapiro's Thinking about Mathematics gives a terrific introduction to some of the issues hereabouts.

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