Philosophy of Mathematics by Linnebo Øystein;

Author:Linnebo, Øystein; [Linnebo, Øystein;]
Language: eng
Format: epub, pdf
ISBN: 9780691161402
Publisher: Princeton University Press
Published: 2017-01-15T07:00:00+00:00

7.2 HARTRY FIELD’S STRATEGY FOR NOMINALIZING SCIENCE

According to Field, “the only serious argument for platonism depends on the fact that mathematics is applied outside of mathematics” (1989, p. 8). In order to undermine this single serious argument, he sets out to show how nominalists too can explain applications of mathematics to the empirical sciences.6 If successful, this explanation will show that mathematics is not, after all, indispensable to empirical science.

Field’s argumentative strategy is inspired by a toy example due to Putnam, discussed in connection with game formalism (cf. §3.2). We observed that first-order logic provides an ontologically innocent way to make finite number ascriptions. For example, the claim #xFx = 1 (which is “platonistic” because of its reference to the number 1) can be “nominalized” as ∃x∀y(Fy ↔ x = y).7 We also observed, however, that it is exceedingly impractical to work with such nominalistic number ascriptions, as the formulas and derivations quickly become too long to be surveyable. It is hugely advantageous to allow the claims of ordinary platonistic arithmetic too, as well as bridge principles that link these claims with the nominalistically acceptable number ascriptions. For example, if there is a single F, a single G, and nothing that is both F and G, we can use the equation 1 + 1 = 2 to infer that there are precisely two things that are F-or-G. This means using a detour through discourse about abstract objects such as numbers to simplify our reasoning about the concrete. Are such detours safe? Do we know that they won’t take us from true nominalistic premises to a false nominalistic conclusion? In the present example, we do. We can prove that everything that can be established via a platonistic detour can also be established directly, remaining strictly within the limits of what is nominalistically acceptable.8 In technical parlance, we can prove that platonistic arithmetic is conservative over the mentioned part of nominalistic arithmetic.

In light of this toy example, it is easy to explain the gist of Field’s strategy for nominalizing science. The idea is simply to extend the strategy from the toy example to science in general. This dauntingly ambitious aim involves two separate tasks. First, we need to do to every scientific theory what we did to finite number ascriptions, namely to “nominalize” the theory by reformulating it in a way that avoids all commitment to abstract objects. Second, we need to show that the detours via platonistic mathematics are benign and serve only to simplify reasoning that could in principle be conducted while remaining at the nominalistic level. That is, we need to show that the platonistic theory is conservative over the nominalistic one. Suppose Field is right that the two tasks can be carried out. Then science can in principle be done in a nominalistic way (by the first task). It might nevertheless be expedient to apply mathematics to science as a device that is useful for practical purposes, although in principle eliminable (by the second task). In short, a

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