What's In a Name?

What’s in a name? Would not a turdorch* by any other name smell as sweet?

Today's question is inspired (or perhaps provoked) by this Steve Landsburg claim: "well formed statements about arithmetic are either true or false, regardless of whether they have proofs or disproofs." For example, the following: "Every even number is the sum of two primes."

Steve Landsburg has a series of posts on the foundations of arithmetic. I won’t try to summarize them in any detail, so it’s not likely that you will be able to follow my argument below without reading them at least in part. There is a central point I want to dispute, that “every well-formed arithmetical statement is either true or false,” whether provable in some axiomatic description or other.

This claim assumes that numbers, and their arithmetic, have an existence independent of that axiomatic description. As a philosophical point, I tend to agree, but the point is hardly self-evident. The strange thing to me is that Landsburg, despite his great knowledge of math, acts as if he were oblivious to the great controversies that shook the foundations of arithmetic a century ago. Frege, Russell, and others attempted to put arithmetic on a sounder foundation by reinterpreting numbers in terms of what they regarded as the more primitive notion of sets. This project collapsed precisely when Russell pointed out that there were in fact apparently well-formed statements about set theory that could neither be true nor false. (“Is the set of all sets not members of themselves a member of itself?) Gödel and others subsequently showed that there was no possible set of axioms that could prove all possible statements in arithmetic.

The fundamental question is whether existence is independent of description, and I think our author picks a poor example:

SL:

“A purported map of Nebraska can be inconsistent; Nebraska itself can’t be.”

I think this claim illustrates exactly where SL goes wrong. What is Nebraska anyway? I say that it’s a geographical and political entity defined by geographical criteria and political rules. Some lines on a map and some laws constitute its very existence. It didn’t exist before those lines were drawn and those laws were adopted. It is entirely possible that those laws and lines may be mutually inconsistent, and in fact a significant business of courts is adjudicating real or apparent contradictions of just that sort. I can’t think of a meaning for the word Nebraska that isn’t defined in just some such way. We know of objects of seemingly more fundamental nature: this computer, the electron, or the blogger/economist Steve Landsburg, for example. If we look in enough detail, though, even these can start to look a bit fuzzy at the edges. Ultimately we depend on some rules of thumb or definition to distinguish one thing from another, and these rules and definitions can at least potentially contain contradictions.

Are mathematical entities like sets and numbers like Nebraska in that respect or not? I don’t think that the answer is obvious.

*or rose.