Putnam's argument against the possibility of nominalising 'distance-statements' in "Philosophy of Logic" (1972)

Question

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?

Answer 1

As I read this:

Even if r1 and r2 are integers, this idea creates an infinite number of points in space that have names, the (1-D) grid of candidates for object b that puts object b N meters plus-or-minus 1 centimeter away from object a.

These points are named things in space, but any physical realization of this naming (labeling the points) would fail, because there can be only finitely many physical objects to assign as names.

Numbers have to be something other than an abstract representation of a possible physical labeling scheme. Because they represent a labeling scheme that is impossible to make physical. You need at least a notion of naming that allows for a construction or checking procedure that may need to be executed an arbitrarily large number of times. And that means there has to be more going on than naming as we encounter it normally.

Therefore: Physics itself requires concepts that cannot be captured by physical realization and simple naming of objects.

As I see it, the argument seems far too obvious to require this level of detail, but overkill forestalls future wasted negotiations...