I've got to go reread my Shapiro. But before his smooth writing bewitches me, let me note down the very simple objection that I am currently unable to see how he would answer.

Structuralism is traditionally motivated by the desires to address a problem from Benacerraf: that there are multiple equally good ways of interpreting talk of numbers as referring to sets, so that either answer to "what set is the number 3" seems unprincipled. But now:

If you are not OK with plentiful abstract objects, you can't believe there are abstracta called structures.

If you are OK with plentiful abstract objects, then you can address this worry by just saying that the numbers and sets are different items. Certain mathematics textbooks find it useful to speak as though 3 were literally identical to some set, but this is just a kind of "abuse of notation" motivated by the fact that we can see in advance that any facts about the numbers will carry over in a suitable way to facts about the relevant collection of sets named in honor of those numbers. One might argue that analogous abuse of notation happens all the time in math e.g. writing a function that applies to Fs where you really mean the corresponding function that applies to equivalence classes of the Fs. This route seems like a much less radical move than claiming that basic laws about identity fail to apply to positions in a structure e.g. there is no fact of the matter about whether positions in two distinct structures (like the numbers and the sets) are identical.

I think Shapiro takes the OK with plenitudinism horn. Resnik has something more like the no fact of the matter view. I think. I too would need to dig into the relevant texts to be sure that gets their views right.

ReplyDeleteYes i think so too. But then my question is: if you are OK with many different kinds of objects why not say that the numbers and sets are *distinct mathematical objects*, so three isn't literally identical to the set paired with three by either reduction. this would seem to address the benacerraf point that equating 3 with either set seems unprincipled. Any idea what he would say?

ReplyDeleteThe book is on my desk back in the library. I think he does specifically say that numbers aren't sets. I'm not sure he says it specifically in the context of addressing the Benacerraf point, but it's clear that ar structuralism is put forward in opposition to set theory reductionism with that among the motivations.

ReplyDeletehey jrshiply, maybe I am not being clear.

ReplyDeleteThe point is this: Once you have numbers != sets, (which plain non-reductive platonism will give you) why do you need structuralism?