Here is a post I had written a while ago, but never put up on this blog. It touches on some odd things that Russell says, primarily as they relate to Frege and Benacerraf’s subsequent discussion. 

In the philosophy of math seminar, one of the things we discussed was Benacerraf’s classic “What Numbers Could not Be.” Benacerraf motivates his view, structuralism, by presenting an example of two kids who learned about numbers through set theory. Their set theoretic representation of the numbers are different, but they are isomorphic. They agree on all arithmetic truths but disagree about what sets the numbers are. Benacerraf uses this to argue for some conclusions about mathematical ontology and identity. Skipping over some stuff, one of the things that Benacerraf concludes is that numbers are the roles in an arithmetic structure, roughly. He wants to say that there is no point in talking about numbers, or mathematical structures generally, in terms more specific than  structure preserved up to isomorphism.

Did anyone ever think that numbers were something more specific than that? Did anyone think that they were things that could be characterized further? Surprisingly, Russell. early in his Introduction to Mathematical Philosophy, cites Frege as making an advance over Peano in characterizing arithmetic. There are two parts to Russell’s claim. First is that Frege defined Peano’s three primitives, number, successor, zero, in Fregean logic. Second Peano’s characterization of arithmetic admits of many different interpretations. Presumably, Russell is implying that Frege somehow characterized the natural numbers not just to isomorphism but to identity.

Russell fleshes out this second point with some examples. One could take 0 to be 100 and start the successor function there. Or, one could take any countable, linearly ordered series and find a copy of the natural numbers there. This is an odd complaint, especially since Peano’s characterization of arithmetic is done with the second-order induction scheme, so it specifies the natural numbers uniquely up to isomorphism.

Why would one think that Frege’s characterization of arithmetic does not succumb to this “problem”? Here’s a stab at that. Frege’s construction of the numbers starts with set containing just the empty set as 0. This is obtained by taking the extension of the concept “not self-identical.” This is the set of things x such that x is not identical to x. From there, he proceeds by defining the successor of a number n to be the number belonging to a concept F for which there is an object x such that the number n belongs to the concept “falls under F but is not x”. This is done by taking forming a series of concepts, “is a member of the sequence ending in number m”. The extension of this concept will have m+1 members. Two numbers will be equal when concepts they belong to can be put in one-one correspondence. Finally, the number of a concept F is the extension of the concept “is equal to the concept F”, which is to say the set of all sets in one-one correspondence with F. So 2 will be the set of all pairs, three the set of all triples, and so on.

This sort of construction can’t specify the numbers more than up to isomorphism, for reasons that Russell pointed out. Once we do this construction, we could interpret the successor function as taking us two steps up the ladder, so to speak. This would still make all of Peano arithmetic come out true, or would if consistent.

At this point, Frege didn’t have his sense/reference distinction, so that can’t be appealed to to sort out the difference between the two. One might be able to use it to argue that number words refer to one particular set or other, but this runs into the waiting arms of Benacerraf’s article. It doesn’t look like Frege’s version specifies the numbers past isomorphism any more than Peano’s does.

In the  sections of the Grundlagen where Frege is giving his construction, sections 55-83, he doesn’t indicate any inclination to specify the natural numbers in the way Russell claimed. He does reduce Peano’s three primitives to things definable in his logic. In fact, later in the Grundlagen, section 101, Frege does not seem to care that different interpretations of the symbols are possible in the case of complex numbers. He says,

We are tempted to conclude that it is quite immaterial whether i means a second or a millimeter or anything else, provided only that our laws of addition and multiplication hold good; everything depends on that, and the rest we need not bother about. Well, perhaps it is indeed possible to assign a whole variety of different meanings to a+bi, and to sum and product, all of them such that those laws continue to hold good; but it is not immaterial whether we can or cannot find some such a sense for those expressions.

This is similar to what Benacerraf says. Unlike Benacerraf, Frege doesn’t think that some identity statements are nonsensical. It seems to me that Russell is mischaracterizing Frege and Russell’s views are nuts. Why then is Frege even in the Benacerraf article? There are two reasons, I think.

The first reason is that Frege identifies numbers with certain sets, which on Benacerraf’s view is entirely superfluous. They aren’t any particular set, as shown by Benacerraf’s article, so Frege must be wrong. I’m not sure why Frege couldn’t get on board with Benacerraf here. He seems fine to say that a number could be interpreted in any number of ways as long as the numbers and operations on them are interpreted appropriately. He didn’t do this explicitly in his construction, but it looks like something he would be fine with.

The other reason is that Benacerraf wants to draw some conclusions about the bivalence of identity. He wants to deny that identities with sortal mismatches have a truth-value at all. Frege, famously, held that for any expression you had to be able to settle the truth of an identity claim involving it. This, I think, is a separate issue from the previous reason. I’m not sure what a Fregean response to this would look like, but I’m also having a bit of trouble with Benacerraf’s argument on this point.