Fictionalist Nominalism and Applied Mathematics

For at least a century, mathematics has been a fertile source of arguments against nominalism. Among such arguments, arguments that are based on applied mathematics have at least one important advantage over arguments based on pure mathematics. Nominalists, upon being convinced that the Banach-Tarski Theorem-which has no conceivable application to the physical world-was inconsistent with nominalism, might well be willing to say, "So much the worse for the Banach-Tarski Theorem." But few nominalists would be willing simply to toss NonLinear Partial Differential Equations for Scientists and Engineers (by Professor Lokenath Debnath; "a very hard act to follow," according to one reviewer) into the fire. It would be a heroic nominalist indeed who was willing to dismiss that book-or any of thousands of other books of a similar nature-with the words, "Does it contain any abstract reasoning concerning quantity or number? Yes. Commit it then to the flames: for it can contain nothing but sophistry and illusion."In my view, arguments from applied mathematics are particularly difficult for fictionalist nominalists to deal with.1 In section 1, I will present an argument based on applied mathematics against nominalism simpliciter. (The argument is coextensive with the section. Although I in fact accept the argument, Section 1 is no more than a presentation or statement of the argument, and none of the declarative sentences it contains should be taken to be an assertion of the author's.) In section 2, I will present a modification of the argument of section 1 that is directed specifically against fictionalist nominalism. In section 3,1 will consider a reply to the argument of section 2 that is based on the common fictionalist contention that "the mathematical fiction" can be shown to be a "conservative extension" of nominalistically acceptable discourse.1. An Argument for Rejecting NominalismLet us consider a simple problem whose solution requires an application of mathematics to the physical world.2 We suppose that Alice wishes to find the number of cans (cylindrical cans, each with the dimensions of a "sample" can she has in her possession) of Sherwin-Williams Porch and Floor Enamel (her favorite floor paint) needed to paint the floor of the Mary Lou Beasley Reading Room-a rectangular room, the length of whose sides she is in a position to measure-given that one needs 0.02 cubic units of this paint to cover a surface whose area is 100 square units. ("Units" are units of linear measure: meters, centimeters, nanometers, inches, leagues, parsecs.) This simple problem, as befits a simple problem, is easily solved. To solve it, Alice need only apply the following general principle to the floor of the Beasley Room:1.1 The rectangular-surface/paint-can numerical requirement principleNRP For any rectangular surface R and any numbers3, x, y, z, and w: if x and y measure the lengths of two adjacent sides of R in Ls, and if z measures the height of a can of paint4 in Ls and w measures its diameter in Ls, then the number of cans of paint that are needed to paint R is5 0.00025 (xy/w2z).Note that NRP has been presented in the form of a schema. Instances of this schema are obtained by replacing each occurrence therein of the dummy symbol 'Ls' with (the same) plural "expression signifying a unit of linear measure" ("ULM"). Examples of ULMs are: 'meter', 'inch', 'league', 'nanometer', 'terrestrial polar diameter', 'cubit', and 'astronomical unit'. Thus, one instance of NRP will contain the phrase '. . . ifx and y measure the lengths of two adjacent sides of R in meters, and if z measures the height of a can of paint in meters . . ,'.6To accept or assent to NRP (or any schema) is to accept or assent to ('to commit oneself to accepting'? 'to be prepared to assent to'?) all its instances.Alice need only apply NRP to the case at hand-yes. But in order to apply NRP to the case at hand, she will first have to know of two numbers that they measure the lengths of two adjacent sides of the floor of the Beasley Room in meters (or in nanometers or in parsecs . …