Tuesday, November 1, 2011

Pictureability and Definiteness in Mathematics

A fairly popular position in philosophy of mathematics is the following: there are definite right answers to questions about arithmetic, but not to questions about set theory (especially not to questions like the Continnum Hypothesis whose answers can be changed by forcing), because we have a clear notion of what the numbers are supposed to be like but not what the sets are supposed to be like.

This claim about clarity is sometimes supported by appeal to notions of mental picturing. The idea would be that we can picture all the numbers, but that we can't picture things like `all the subsets' of the numbers, or the hierarchy of sets as an item that contains, at each level, all subsets of the levels below. But I want to ask: In what sense can we picture all the numbers? In what sense CANT we picture the sets?

All the ways I can think of of picturing all the numbers involve some kind of "..." e.g. you might picture all the numbers by having a mental picture like the physical picture I have drawn below:

| || ||| |||| ....

When we use this picture to conceive of the structure of the numbers, we are employing a highly non-trivial method of picturing, which lets us represent the state of there being a countable infinity of numbers by considering some finite collection of drawings like the above.

And people certainly do *in some sense* draw pictures of the hierarchy of sets. At the beginning of a set theory class (and at the beginning of my college analysis class) the professor will frequently draw up a V on the board. In a way that is a picture of the heirarchy of sets. And, presumably, one could mentally picture the heirarchy of sets in the same.

For this reason I am (a little) skeptical about justifying the claim that we have a definite idea about the sets but not the numbers by appeal to considerations about what we can mentally picture. We can picture the numbers if we are willing to let "..." represent the idea of all finite successors to a stroke symbol. We can picture the hierarchy of sets if we are willing to let the increasing girth of the V represent the idea that each level of the hierarchy of sets contains all possible subsets of the level below. What (mathematical) objects one can picture depends on what methods of picturing we will accept, as allowing one to genuinely conceive of a situation by entertaining a mental picture.

At the very least, I think that if someone wants to argue that claims about numbers are definite and claims about sets are not on the basis of appeals to mental pictures, they owe us some kind of story or explanation for why one method of picturing is OK and the other is not.