Do Peano’s axioms define the natural numbers? Would it be possible for any system of axioms to do so? Some musings after Mathieu Marion’s article ‘Wittgenstein on Surveyability of Proofs’. (I don’t think there’s anything original in here; it’s just me thinking through the topic.)

What are the natural numbers? Of course, they’re 0, 1, 2, 3, and so forth. But what is, say, 0? And what do the words “and so forth” actually do? One customary way of thinking about this is that the natural numbers are defined by Peano’s axioms. 0 is a natural number; every natural number n has a successor S(n); m=n if and only if S(m) = S(n); 0 is the successor of no natural number; and a few more axioms having to do with equality and induction. Nice! But can these axioms really define the natural numbers?

Or, to ask what may be the same question in a slightly different way: can these axioms pick out the natural numbers among all other things? Are they true only of the natural numbers? Not at all. As Russell pointed out, they are just as true about the “natural numbers bigger than 99” (with the succession relation being the customary ‘+1’). Or about the even numbers (with the succession relation being the customary ‘+2’). Or about all numbers of the form 1/(2^n) (with our customary 1 being the zero element and the succession relation being the customary division by 2). So the Peano axioms do not, in fact, define the naturals.

There are several way to try and get around this. First, one could claim that “natural numbers bigger than 99” doesn’t fit the axioms, since it’s first element is 100 and not, as the axioms require, 0. But this is to overlook the fact that it is an open question whether the ‘0’ of the axioms is the same as the ‘0’ or our everyday language. Or rather, if that is not an open question, then the axioms presuppose a way to identify 0 and hence cannot be said to define the natural numbers. Also, the first element of our second construction (even numbers) actually does start with the customary 0.

Second, one could suggest that the Peano axioms by themselves may be powerless to define the natural numbers, but they become capable of doing so once we add the axioms for addition. After all, the axiom a + 0 = a is true for the naturals, but not true for the naturals bigger than 99; since 100 + 100 = 100 is false (and 100 is the zero element in this construction).

But how do we know that 100 + 100 is false? Of course, this requires us to identify the ‘+’ sign with our usual addition on the natural numbers. But this just throws us back on the original problem. If we presuppose the ability to perform this identification, then we presuppose an ability to simply state that the axioms are to be understood ‘in the usual way’, and if it is necessary for us to do say, then the axioms clearly fail to define their intended domain. On the other hand, if we take the axioms as defining the ‘+’ operation, then we clearly have to say that this ‘+’ is actually our usual (+ x – 100). Or, in the particularly nice case of the (1/n^2) progression, that the ‘+’ sign is our usual multiplication.

Third, one could suggest that the natural numbers just are what the Peano axioms define; and that the very question of whether they succeed at this definition must therefore be mistaken. According to this way of thinking, it make no sense to ask whether the axioms are really about the natural with our customary addition, or whether they are really about the sequence (1/n^2) with our customary multiplication. For this precisely presupposes a grasp of these different domains that is independent of the axioms; and that is what any kind of ‘formalist’ conception of mathematics sets out to deny.

Well and good. But there is a price to pay for this formalist move. For we surely do have a pre-axiomatic grasp of the natural numbers: we count things. The formalist move forces us to say that whatever the mathematician is talking about, she is not talking about that. If there are three apples on the table, and we ask the mathematician how many apples there are on the table, then she can either speak in the language of everyday and say “three”; or she can speak in the language of mathematics, but then the answer has to be “could be anything, or possibly nothing, I first need to see the entire formal structure of your counting procedures.” But this answer is a disaster. For we can never show enough of our counting procedures simply by performing them: no amount of performances could ever exclude non-standard interpretations or even failures to obey the rules of arithmetic at all. (It may always turn out that, say, the successor of 167842646703 is 0!) So instead we will have to give the general rules of our counting procedures. But this will either be a formalist mathematical system — the Peano axioms again — which do not themselves describe counting; or we will have to find some way of linking the rules to our practice, which brings us back to stage zero of this entire argument.

One could completely cut the link between mathematics and the practice of counting, I suppose. But this would make mathematics devoid of any use; and, perhaps even more tellingly, it would leave us in desperate need for a science of counting — forcing us to reinvent mathematics under a different name.

So — so much for formalism?