Parsons’s Mathematical Thought: Sec. 35, Intuition of finite sets

Suppose we accept that “it is not necessary to attribute to the agent perception or intuition of a set as a single object” in order to ground arithmetical beliefs. Still, we might wonder whether some such intuition of sets-as-objects might serve to “give an intuitive foundation to theories of finite sets“.

But Parsons finds problems with this suggestion too. One difficulty can be introduced like this. Suppose I perceive the following array:

$$$$$$

Then do I ‘intuit’ six dollar signs, a single set of six dollar signs, a set of three elements each a set of two signs, or even a set containing the empty set together with a set of six signs? Which way do I ‘bracket things up’?

$$$$$$
{$$$$$$}
{{$$}{$$}{$$}}
{{}{$$$$$$}}

The possibilities are many — indeed literally endless, if we are indeed allowed the empty set (and what is our intuition of that?). So it seems that the “intuition” here has to involve some representational ingredient to play the role of the brackets in the various possible bracketings. But then we are losing our grip on any putative analogy between intuition and perception (as Parsons puts it, “in a perceptual situation involving the application of certain concepts, we not expect that a linguistic of other embodiment of the concepts should be perceptually present in that very situation”).

Secondly, note that we can in fact give a theory of those “bracket terms” — putatively for hereditarily finite sets constructed from a given domain D of individuals — which uses a relative substitutional semantics. That is to say, we can start with a first-order language for which D is the domain, add terms for hereditarily finite sets of elements from D, and variables and quantifiers for them, which we then interpret substitutionally relative to D. Parsons spells this out in an Appendix, but the general idea will be familiar to readers of his old paper on ‘Sets and Classes’. And the upshot of this, Parsons says, “is that if we take the relative substitutional semantics as capturing a speaker’s understanding of the language of hereditarily finite sets … then we largely remove the motives for characterizing awareness of such sets as initution”. That’s a significant “if” of course: but we might indeed wonder why we should take elementary talk about finite sets (and sets of those, and so on) to be more committing than the substitutional interpretation allows.

Note that this isn’t to say that we have entirely eliminated a role for intuition. For on the relative substitutional interpretation we still need the idea of sequences of individuals from D. And we might suppose that that notion is grounded in intuition. But even if true, that still falls well short of the original thought that we could need intuitions of sets-as-objects to give a foundation to theories of finite sets.