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.

I agree with the position that there are objective "correct answers" to questions about (first-order) arithmetic in a way that there probably are not for questions like CH. However, I think that the "mental picturing" argument you outline here is misguided, and the objectiveness of arithmetic has to do with other reasons.

ReplyDeleteFor one thing, it would seem strange to assert that someone who was blind from birth, but with otherwise normal mental capacities, could never justifiably know facts of arithmetic like the rest of us just because they cannot visualize a number line... surely they could conceive of arbitrarily large finite natural numbers in their own ways. By the same token, I would argue that we can deduce objectively true facts about mathematical objects that we cannot hope to visualize, such as 4-dimensional speheres -- we can compute their volumes, figure out how to pack them with optimal efficiency, etc. (Now I'm not sure if I'm being overly literal-minded in my interpretation of "mental picture" here, so I apologize if I am.)

I would say that our right to make objective claims about arithmetic has nothing to do with "mental pictures" as I think you mean them, but rather with the fact that there is a clear, coherent idea behind what "the totality of integers" is which is lacking in the idea of "the totality of all sets." Maybe that sounds a little mystical but I don't know how better to say it. (And a nitpick: I wouldn't say that it's our conception of individual sets, per se, which is vague, but rather our conception of what the *totality of all sets* should be.)

Hi John,

ReplyDeleteI am sorry for not noticing this comment earlier!

Re: mental picturing arguments, I am actually quite skeptical of appeals to mental pictures myself. I wrote the post to say that *even if you like mental picture arguments* they don't to distinguish CH from the numbers in any very strong way.

re: clear concepts, I can understand feeling like something is unclear about the height of the heirarchy of sets, but do you also think that the notion of e.g. "all possible subsets" of the integers is unclear? I say this because questions like CH would be settled by a determinate fact about the width of the hierarchy of sets, even if it is ultimately vague how the hierarchy of sets is supposed to go.

p.s. Does that make sense? My day started fairly early and sometimes my writing skills get exhausted :( .