What Numbers Could not Be

by Paul Benacerraf

Genres: Philosophy, Mathematics

Unknown Binding

Published January 1, 1965

About the author

Paul Joseph Salomon Benacerraf was a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. Benacerraf was appointed Stuart Professor of Philosophy in 1974, and retired as the James S. McDonnell Distinguished University Professor of Philosophy.

Reviews

[Hobart Mariner · Rating 4 out of 5]

Interesting essay that describes the difficulty of "reducing" numbers to sets. Arithmetic, as described by Peano's axioms, is characterized by the simple relation of "successor" (which has to abide by certain obvious rules). It is quite simple to list out a family of sets that plays by these simple rules: {}, { {} }, {{{}}}, ... is just one. Therefore it is common to model the system of numbers as a system of sets. The problem is that the sets in question have properties (like set-membership) which are not intrinsic to arithmetic; different set models will disagree over the answer to questions like "Does 7 belong to 11?"

For Benacerraf this is fatal, or almost fatal. For how can we decide among the infinitely many set-theoretic models of arithmetic? If one of them is correct, there has to be an argument in its favor. He dispatches the notion that numbers can be replaced with their predicates (the number m is replaced with the predicate "Set x has m elements") on more or less solid set-theoretic grounds. Thus we are left with the question "What numbers are sets, really?" In his words, "There seems to be little to choose among the accounts. Relative to our purposes in giving an account of these matters, one will do as well as another, stylistic preferences aside." Here is where I felt myself starting to disagree with him. I was reminded of a basic mathematical idea, namely the Set-Vector space adjoint pairing. If you have a finite-dimensional real vector space V (consisting of vectors v as well as the operations of addition and scaling), you can "forget" the structure and just treat V as a set, GV (on which there are no operations defined). After doing this, you can form the free vector space on the set GV, which is denoted FGV: this is a vector space with as many coordinates/dimensions as there were *vectors* in the original space: typical elements of FGV are expressions like -2[0_V] + 2[v ]+ 3[w] + 5[z] which are never reduced down to what their sum is in V (0_V is the zero vector in V -- but it becomes a nonzero vector [0_V] in FGV). Now, the question, in purely vector space language, is "How many different ways can we map FGV linearly and onto V?" The answer is "infinitely many," because defining an map only requires that we choose any basis {e_1,...,e_n} of V and then choose a linearly independent set {f_1,...,f_n} of FGV to send f_k -> e_k. The rest of the linear transformation can be defined in any way we choose (extend a basis for FGV, etc.), and there are infinitely many ways to choose {f_1,...,f_n}. In Benacerraf's view, perhaps, we would say we have an infinite number of candidates for a surjective linear map FGV -> V. None of them would enjoy any feature *as surjective linear maps* that the others do not. Yet there is one special linear map from FGV to V which is more important than the others: the map that sends [v] to v. This one enjoys a special feature that any set map h from a free vector space FX to V will factor through a set map X -> GV, free-ified, and then composed with epsilon. (A mouthful, I know.) The important thing is that you can have infinitely many choices for a possible "Model" of something in a larger structure (like arithmetic in set theory) and yet there can be a truly canonical and "natural" choice that stands out from the others.

The last chunk of the essay when he talks about "ways out" of the tight net of refutations he has placed around set-theoretic foundations doesn't land for me. First off, he dodges Frege's thing about "Is Julius Caesar a number?" by saying that we only look at thins that are of the same basic "category." You can only ask if two lamp-posts are the same, etc. This feels like a cheat to me. To someone whose vision and sense of the world has been sufficiently blurred, it is perhaps possible that any such category division could seem artificial. Maybe when you're super drunk, it seems possible that Julius Caesar is a number...

The other thing that comes at the end is his patch that enables people to more or less go on identifying numbers with sets as before. You can make the identification 3 = {{{}}} as long as you don't *assert* it. The former is something you do to get a meta-mathematical model going so that some calculations can be done; the latter is closer to a metaphysical statement of truth. The identifier does not claim that they have discovered the true nature of 3 by stating 3 = {{{}}}, they only find it a useful means of representing some characteristics of the number '3' in set-theoretic language. So we replace the whole question of "What are numbers?" with the more pragmatic question of "What abstract structures model number-ness?" Benacerraf thinks that going beyond this is silly, because all of arithmetic is "on the surface" so to speak -- there is no deeper truth than the formal relations that exist between these abstract goblins. He revisits some insight of Quine that *any* infinite progression (totally ordered set without limit points) will suffice as a model of the integers, hence none of them can actually be said to be the numbers. Instead of one "true" set of integers there are infinitely many possible versions of the integers; arithmetic becomes that science which you can apply to any "infinite progression." On one face this seems obvious: isomorphism of structure is the replacement for actual metaphysical existence. On the other, there is something a little special about 1,2,3,... I was reminded again in his section assailing the question of "What is a number?" with a question I heard a lot when I was teaching math: "What is a vector?" To which the stupid but tempting answer was "It's a member of a vector space."

Anyway, I like the essay, even if it seems to move around in a bit wider orbit than you initially think. The bits about transitive and intransitive counting, as well as the central thought experiment about Ernest and Johnny, are both really accessible frameworks for thinking about fundamental questions regarding the integers. And I really like the way he wraps it up, referring to E+J: "They think that numbers are really sets of sets while, if the truth be known, there are no such tings as numbers; which is not to say that there are not at least two prime numbers between 15 and 20."

[Victoria · Rating 5 out of 5]

The Johnny and Ernie example was kinda superfluous - could have been more straight to the point with its argument. Otherwise, the challenge is quite clever