In chapter V, The Inadequacy of Nominalistic Language, of Philosophy of Logic (1972), Putnam argues that there can be no nominalistic "translation scheme" of sentences of the form "the distance between d is r1 plus/minus r2" unless one is willing to postulate an actual infinity of physical objects. His argument, on page 38-39, is the following:
If there are only finitely many individuals, then there are only finitely many pairwise nonequivalent statements in the formalized nominalistic language. In other words, there are finitely many statements S1, S2, ..., Sn such that for an arbitrary statement S, either S is equivalent to S1 or S is equivalent to S2 or ... or S is equivalent to Sn, and moreover (for the appropriate i) S is equivalent to Si follows logically from the statement "the number of individuals is N" [a proof of this is given as a footnote]. But if we have names for two different individuals in our "language of physics", say, a and b, and we can express the statements "the distance from a to b is one meter plus/minus one centimeter", "the distance from a to b is two meters plus/minus one centimeter", etc., then it is clear that we must have an infinite series of pairwise nonequivalent statements... Thus any "translation" of "the language of physics" into "nominalistic language" must disrupt logical relations: for any N, there will be two different integers n, m such that the false "theorem": If the number of individuals is N, then 'the distance from a to b is n meters plus/minus one cm.' is equivalent to 'the distance from a to b is m meters plus/minus one cm.' will turn out a true theorem of logic if we accept the translation scheme. Thus a nominalistic language is in principle inadequate for physics.
To start with, I don't understand what pairwise nonequivalent statements are in this context and why there should only be finitely many of them if there are only finitely many individuals. I understand pairwise nonequivalent in the sense that in a set of pairwise nonequivalent statements, any two statements will be nonequivalent to each other. But what am I to make of these statements? Are they of the form "the distance between d is r1 plus/minus r2"? Further than that I'm quite lost. Can someone clarify the argument?