Benacerraf WNCNB

Today's article is the beginning of the contemporary philosophy of mathematics, at least along with Benacerraf's other article we'll read. It is referred to so much that it is generally referred to as WNCNB.

Probably the majority of the philosophy of mathematics today is concerned with problems related to "structuralism." "Structuralism" is a bad choice of name, since it has nothing to do with structuralism. The reference is to structures in the mathematical sense of the term. Structuralism is the view that what mathematicians (or at least number theorists) study is what Benacerraf calls "abstract structure," that is, what all the isomorphic models of, for example, the natural numbers, have in common.

Benacerraf's article is the beginning of modern mathematical structuralism. He didn't take himself to be proposing structuralism, and there are antecedents dating back, it is usually said, to Dedekind. (Cantor deserves, but doesn't get, much credit here, and both are probably actually following Riemann in this regard.) But it is anachronistic to regard modern structuralism as a tradition starting with them. It starts, I believe, with Benacerraf.

He starts with a fairy tale—actually, he starts with remarks about intransitive and transitive counting and recursive progressions. I'll get to those later. (A later article, commenting about what he now believes about these issues begins with a yiddish joke.)

Two children, Ernie (Zermelo) and Johnnie (von Neumann) learn set theory before they learn arithmetic. It is then easy to tell them what the numbers are. They are just the members of a certain set of sets with a relation , <, on the set. Once they are told the names of the members of the set and how to count with them, and the definitions of 0, successor, +, , everything about the numbers follows from their properties as sets.

Everything appears to be great. They learned what the numbers are; we see that numbers are not sui generis but just more sets, and we see how what are usually taken to be basic axioms about numbers can in fact be derived from principles about sets, reducing the number of basic kinds and of basic principles. This fairy tale is almost Quine's position, known as set-theoretic monism. I usually call it set-theoretic imperialism.

Unfortunately, this fairy tale does not have a happy ending:
Ernie's numbers are

that is, Johnnie's numbers are

Thus, they both "know" which sets are the numbers, but they have different sets. At most one of them can be right. Remember, this is supposed to be a story about what the numbers are, not about what can serve as numbers. Quine is under no illusions, he just doesn't think there is any antecedent fact of the matter about such things, and so he can stipulate.

Who is right? Benacerraf argues, and this is what is new and true in this article, that both versions, and indeed all other proposals for what numbers are (including ones using things other than sets), fare equally well: they get the central points right, and differ on details about which we have no intuitions or prior knowledge. The Russell (on behalf of Frege) case needs special treatment because Frege, and many Fregeans, think his analysis has some claim to be fundamental.

Since all the candidates differ only on things about which we know nothing (or, possibly, as Benacerraf grants, that are all false), there can be no preferred candidate for what the numbers really are, that is, no candidate more specific than just the numbers. They do seem to be sui generis and not reducible to sets, classes, categories, or anything else.

One might claim that there is a fact of the matter about which things the numbers are, but we don't know, and have no way to determine, which. Benacerraf views that "possibility," without argument, as absurd. In desperation, however, many years later, we find philosophers adopting related positions, and also trying to rehabilitate some Fregean story.

Adopt Benacerraf's reasonable view about that for now. Now what?

The first-level conclusion is that numbers are not objects (or, perhaps, that they are sui generis, in which case they are weird and incomplete).

In the second-to-last section of the article, Benacerraf gestures at a positive theory. That section is called "Way Out." Nearly everyone has taken him to have meant "The Way Out." He didn't. He meant, here I am departing from anything any reasonable person would take seriously, since I seem to have painted myself into a corner.

The problem of saying what the numbers are, at least in the form in which Benacerraf approaches it, originates with Frege. For Frege, the identity relation is fundamental to our notion of object (Quine: "no entity without identity") and it is a relation with domain all entities. Thus, for example, either Julius Caesar is equal to the number 43, or else he isn't. Thus, Frege needs to find out exactly which objects the numbers are.

Why can't one answer that question simply by saying, they are the numbers? That isn't good enough for a variety of reasons. The simplest is that it leaves unanswered such questions as, Are the natural number 1, the integer 1, the rational number 1, the real number 1, and the complex number 1, 5 distinct things, or all the same?

The reason that has been emphasized is that numbers are "incomplete." The usual axioms for numbers say arithmetical things about them, but say nothing about whether they are abstract or concrete, sets or objects in categories, Roman dictators or not, and so on. Thus, even on this minimal view, we have to supplement the axioms we agree on somehow, and a simple "and they don't have any other properties" won't do for at least two reasons:

1. On that view they are neither abstract nor not abstract, and so we lose the law of the excluded middle.
2. In fact, they do have additional properties: they are abstract, not located in space, never ruled Rome, ... .

Benacerraf's first, tentative proposal, is that the notion of entity is no good. That is, not only are there no such objects as numbers, there are no such objects as entities. There is no general notion of identity between arbitrary objects, only objects of the same "category" or "kind." Geach has argued that that is true anyway on grounds having nothing to do with mathematics. That view goes against fundamental ideas of Frege and Quine, and so it makes even its adherents nervous. The biggest problem is the one Quinean holism immediately suggests: No one has a reasonable definition of "category."

Benacerraf's other proposal is that there are no numbers, what there are is progressions, and hence structures of the type of the natural numbers and that talk of numbers is talk of what is true in all of those structures or in the corresponding abstract structure.

Thus, to say that '2+2=4' is true is not to say something about particular objects, 2 and 4, but rather to say that '2+2=4' is true in the Johnny structure and the Ernie structure, and indeed in all such structures.

Benacerraf's strategy, it seems, is to eliminate numbers while preserving number talk as something more abstract than our usual talk about objects. As always, whenever a philosopher proposes a way to eliminate something, another employs the same method to introduce it. There are "eliminative structuralists," like my characterization of Benacerraf, and structuralists who try to say what "abstract structures" are, and take, for example, numbers to be "roles" in such structures.

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Benacerraf distinguishes intransitive from transitive counting: intransitive counting is just reciting the names of the numbers in order. Transitive counting is using the ability to count intransitively by pairing the utterances of the numbers with things by, for example, successively touching them, in order to find out how many there are. Benacerraf claims that to serve as numbers, a structure must not only be of the right isomorphism type, but must also have with it a specification of how to count intransitively. Quine argued that all one needs is the structure. Why? Given the structure, one can use it for counting, nothing further is required. Benacerraf says two further things are required: you need to be able to find the successive elements of the structure ("recursively") and you need to know whether you started with 0 or 1, and that is established by the counting procedure.

One doesn't need the numbers to count, only there names, which is related to his remarks about the roles of the words.

-- ProfessorShaughanLavine? - 07 Feb 2005 - 07 Feb 2007