Here’s something I wrote a while back to answer a question on math.stackexchange about why sets and set theory should (or shouldn’t) be thought to have a special place in maths. Following a link on a related matter I found myself directed back to this piece of my own: I think I still quite like it. So here it is again …
There is a long, fascinating, and often-told story about the nineteenth century project for the rigorization of analysis, and about the re-construction of classical mathematics in terms of natural numbers and sets of natural numbers and sets-of-sets of natural numbers, etc. etc. And if we are feeling particularly austere we can even re-construct the naturals in a pure set theory which lacks urelements, so everything gets implemented in pure set theory. There are lots of good recountings of the story — here’s a short one with lots of pointers to more: http://plato.stanford.edu/entries/settheory-early/
I mention the history because it explains why set theory has long been thought to have a special “foundational” place in the architecture of mathematics. But does it really? Can category theory (for example) provide an alternative foundation? And anyway, now we’ve got over our wobbles from about a hundred-and-twenty years ago, when some thought classical mathematics was threatened by paradoxes of the infinite, does mathematics in any sense need universal “foundations”?
Big questions indeed, and the general question about some supposed need for “foundations” is not wanted I wanted to comment on here. But here’s one line of thought that I’ve encountered from mathematicians, not so often mentioned by philosophers, which perhaps underlies some of the continuing nods to the special place of set theory.
Suppose working on Banach spaces, or algebraic topology, or whatever, I conjecture all widgets are wombats. And then the bright young grad students try to prove or disprove Smith’s Conjecture.
Next week, Jane turns up to class claiming to have refuted the conjecture by finding a structure in which there is a widget which isn’t a wombat.
Well, what are the rules of the game here? What kit is Jane allowed to use in her structure building? To give her a best shot at refuting the conjecture, she perhaps ideally wants some kind of all-purpose kit that only minimally constrains what she can build. She wants the mathematical equivalent of a Lego kit where you can pretty much attach anything onto anything, rather than the equivalent of a building kit you can only make toy houses from, or one you can only make toy cars from. (Perhaps Smith’s Conjecture still works fine for, so to speak, houses and cars.)
What the standard sets of the iterative hierarchy seems to provide is just such an all-purpose mathematical Lego kit. We start with some things (or if you like, with nothing at all), and then we are allowed to put them together however you like into new things, and then we are allowed to put what we’ve got together however we like ad libitum, and to keep on going as long as we like. Precisely because the rules for building new sets allow maximising at every step (the idea is at each level we are allowed every possible new combo, and there is no limit to the levels), we really do get an all-purpose structure-building kit. And having such a mathematical Lego kit is just what Jane ideally needs if she is to have untrammelled free rein in coming up with her widget which isn’t a wombat.
Or so the story goes, in outline …