|Abstract||In recent times there have been a number of proposals for a nominalistic philosophy of mathematics. These proposals divide into two quite distinct camps: those who take mathematical propositions to be true, and those who take them to be untrue.2 Both options face substantial difficulties, but let’s focus on the first option. The problem here is in asserting that mathematical propositions such as ‘there exist infinitely many complex roots of the Riemann zeta function’ are true (as this one surely is) and then to go on to deny that there are any complex numbers. To do this just seems inconsistent, or at least “intellectually dishonest” (Putnam, 1971, p. 347). One way to approach this problem is to reinterpret the mathematical claims in question so that they come out true, but do not refer to mathematical objects. So for example, Geoffrey Hellman  interprets mathematical claims to be about possible structures. Such options, since they do not take mathematical claims at face value, must employ a non-uniform semantics and this is thought, by almost everyone, to be a significant price to pay for one’s nominalism. The problem is particularly acute when one considers mixed mathematical and empirical statements such as ‘there exists a planet with mass m and location (x, y, z) and a function G that describes the gravitational potential of the planet at time t’. Here different parts of a single sentence must be treated differently—the talk of planets (and perhaps fields) is treated literally but the mathematical parts are treated non-literally. Apparently the only alternative to reinterpreting mathematical discourse is to follow Hartry Field  and deny the truth of mathematical propositions. But this option is very counterintuitive.|
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