Parsons’s Mathematical Thought: Secs 33, 34, Finite sets and intuitions of them

So where have we got to in talking about Parsons’s book? Chapter 6, you’ll recall, is titled “Numbers as objects”. So our questions are: what are the natural numbers, how are they “given” to us, are they objects available to intuition in any good sense? I’ve already discussed Secs 31 and 32, the first two long sections of this chapter.

There then seems to be something of a grinding of the gears between those opening sections and the next one. As we saw, Sec. 32 outlines rather incompletely the (illuminating) project of describing a sequence of increasingly sophisticated but purely arithmetical language games, and considering just what we are committed to at each stage. But Sec. 33 turns to consider the theory of hereditarily finite sets, and considers how a theory of numbers could naturally be implemented as an adjunct to such a theory. I’m not sure just what the relation between these projects is (we get “another perspective on arithmetic”, but what exactly does that mean? — but, looking ahead, I think things will be brought together a bit more in Sec. 36).

Anyway, in Sec. 33 (and an Appendix to the Chapter) Parsons outlines a neat little theory of hereditary finite sets, taking a dyadic operation x + y (intuitively, x U {y}) as primitive alongside the membership relation. The theory proves the axioms of ZF without infinity and foundation. I won’t reproduce it here. In such a theory, we can define a relation x ~ y that holds between the finite sets x and y when they are equinumerous. We can also define a “successor” relation between sets along the following lines: Syx iff (Ez)(z is not in x and y ~ x + z).

Now, as it stands, S is not a functional relation. But we can conservatively add (finite) “cardinal numbers” to our theory by introducing a functor C, using an abstraction axiom Cx = Cy iff x ~ y — so here “numbers are types, where the tokens are sets and the relation ~ is that of being of the same type”. And then we can define a successor function on cardinals in terms of S in the obvious way (and go on to define addition and multiplication too).

So far so good. But quite how far does this take us? We’d expect the next step to be a discussion of just how much arithmetic can be constructed like this. For example, can we cheerfully quantify over these defined cardinals? We don’t get the answer here, however. Which is disappointing. Rather Parsons first considers a variant construction in which we start not with the hierarchical structure of hereditarily finite sets but with a “flatter” structure of finite sequences (I’m not too sure anything much is gained here). And then — in Sec. 34 — he turns to consider whether such a story about grounding an amount of arithmetic in the theory of finite sets/sequences might give us an account of an intuitive grounding for arithmetic, via a story about intuitions of sets.

Well, we can indeed wonder whether we “might reasonably speak of intuition of finite sets under somewhat restricted circumstances” (i.e. where we have the right kinds of objects, the objects are not too separated in space or time, etc.). And Penelope Maddy, for one, has at one stage argued that we can not only intuit but perceive some such sets — see e.g. the set of three eggs left in the box.

But Parsons resists at least Maddy’s one-time line, on familiar — and surely correct — kinds of grounds. For while it may be the case that we, so to speak, take in the eggs in the box as a threesome (as it might be) that fact in itself gives us no reason to suppose that this cognitive achievement involves “seeing” something other than the eggs (plural). As Parsons remarks, “it seems to me that the primary elements of a story [a rival to Maddy’s] would be the capacity to classify what one sees … and to recognize identities and differences” — capacities that could underpin an ability to judge small numerical quantifications at a glance, and “it is not necessary to attribute to attribute to the agent perception or intuition of a set as a single object”. I agree.

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