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 …
5 thoughts on “What’s so great about sets?”
Or maybe, less convolutedly: when any mathematical theory is formalised, ie put into a form that makes the proofs of its theorems absolutely perspicuous qua proofs, what it is for the theory’s sentences to be true, or true on an interpretation, is essentially given in the language of set theory.
So, can anyone tell me (our host, if no-one else is up for it?): (i) Is this true? (ii) Does this amount to a special role for sets of the sort that our host was talking about?
Don’t worry about patronising me; I’m a complete ignoramus.
Look, I’m a complete amateur about all this stuff, but is there anything at all to the following line of thought?
The original point of predicate logic was to formalise mathematical inference and proof in such a way that there was no need to appeal to ‘intuition’, or anything like it, at all. The possibility of such formalisation, and the means by which it is effected, are pretty fundamental to our understanding of what mathematical proof is. But if the logical system underlying the formalised version of any mathematical theory is going to shed this sort of light on mathematical thinking, it has to have, or impart to the theory it underlies, certain good-making properties like soundness and, where available, completeness, decidability etc. But this sort of evaluation of formalised theories and their underlying logics is just model theory (isn’t it?), whose subject-matter is interpretations, ie essentially set-theoretical structures. I’m probably not putting this in the right way, but it seems to me that set theory is much more central to understanding interpretations of formalised theories than it is to understanding a lot of other mathematical contexts in which the language of set theory crops up, often in seemingly dispensable or per accidens kinds of way.
So the thought is that this might be a special place set theory has in maths: it comprehensively underpins our methods of establishing the adequacy of the calculi that make possible the translation of any mathematical theory into a form that makes the proofs of its theorems absolutely perspicuous qua proofs.
I wonder what will happen if the sort of view expressed by Edward Frenkel in his NY Times review of Max Tegmark’s Our Mathematical Universe catches on. He wrote:
However, there is a lot more to math than such mathematical structures. Objects other than sets are necessary, and they have now become widespread. Moreover, there is an effort underway to overhaul the foundations of math in which set theory is no longer central. So mathematical structures constitute but a small island of modern mathematics.
The formulation of ZF set theory, and other related varieties, always seemed very ad-hoc to me. It’s almost as if the axioms were reverse engineered to produce desired theorems and solve the outstanding paradoxes of the time, but their multifarious other consequences were ignores. Perhaps exemplifying this attitude is the Axiom of Choice – the Banach-Tarski theorem was intended to show how absurd Chocie really was, and that it was far too ‘powerful’ to represent reality, in fact, but for some odd reason most mathematicians did not question Choice because of it, but rather said “this is a strange theorem that we have to accept” despite not only complete counter-intuitiveness, but a general disagreement with any observations of reality. Perhaps this is my philosophy (which tends to confirm with an observational/empirical basis for mathematics) railing against the school of formalism, but I still haven’t read any convincing argument for the validity of the ZF system; certainly it captures “enough” mathematical reasoning to do virtually all (if not all) mathematics, but the evidence seems to suggest it goes far beyond this.
“People often like to cite the paradoxical decomposition of the unit sphere given by Banach-Tarski. “Yes, it doesn’t make any sense, therefore the axiom of choice needs to be omitted”.
To those people I say that they know too little. The axiom of choice is not at fault here. The axiom of infinity is. Infinite objects are weird. Period. End of discussion.” (Asaf Karagila)