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Is most of mathematics independent of set theory? Reading this quote by Noah Schweber:

most of the time in the mathematical literature, we're not even dealing with sets!

it seems that the answer to my question is "yes". But why? When I read in the mathematical literature, sets appear everywhere – we need them in the definitions of groups, rings, vector spaces, ... However, is there some truth in the quote, and in what sense?

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EDIT: for a variety of reasons, I think I should give an explicit caveat here. Obviously I believe the things in my answer below - otherwise I wouldn't have written it. But I am sure there are many, many mathematicians smarter than me who would strongly disagree with some point, or even all of it. I think this is an area where it is easy to generate heated discussion, so I want to be clear that everything below reflects my own views, and that I am aware that I am subject to a number of biases, both mathematical and philosophical.

OK, let me clarify what I meant with that remark. Certainly its naive interpretation is blatantly false, and I could have written much better, but I did mean something by it, and I stand by that.

Incidentally, I think it may be possible to read this as anti-set-theory. That's definitely not my position - I consider myself a set theorist (and a computability theorist, but that's less relevant at the moment! I even think that mathematics could benefit from more of the community more seriously engaging set-theoretic issues; what I say below is descriptive, not prescriptive.

It is very well-known that most of mathematics does not care about foundational matters; I think, for example, that the vast majority of mathematicians would be unable to state the axioms of ZFC. And that's fine! Math long predated the emergence of a generally accepted foundation, and there are and will continue to be challenges to that (or any) foundation (I'm speaking in particular of homotopy type theory, about which I know nothing but have heard is very cool and fundamentally not about sets; maybe someone who knows better can step in, here?).

But I claimed something stronger, that most math is not about sets. So in what sense to mathematicians not talk about sets? Well, obviously we do in a sense - e.g. a group is universally defined as "a set such that . . .". My point is that this definition is vague - the notion of what a "set" is here is the naive one, and naive set theory is inconsistent. So while mathematicians use the word set, the use of any precise notion of set is generally not part of mathematical practice - it's accepted that any natural use of sets won't run afoul of the paradoxes of naive set theory, and will be formalizable routinely (if tediously) in, say, ZFC.

So my point is that while mathematicians use the informal concept of "set" all over the place, most of the time we do not use any precise notion of "set". But it gets worse: there are widely-used parts of math which are fundamentally non-set-theoretic, in the sense that casting them in set-theoretic terms is extremely unnatural: e.g. do you really think of a real number as an equivalence class of sequences of rationals? Demanding a set-theoretic background means abandoning structuralism; as a set theorist, I approve of this, but I also recognize that in doing so I'm breaking with a large number of mathematicians, if not the majority. See also here and here.

What I claim, in short, is the following: although the informal concept of "set" is of course universal in math, to very large extent math is opposed to working in any specific formalization of the concept, instead accepting as a matter of practice that the portion of naive set theory that is invoked will be consistent. In particular, lots of mathematical practice is fundamentally non-set-theoretic.

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  • Why does approving that demanding a set-theoretic background means abandoning structuralism imply that you are breaking with a large number of mathematicians? – Please Help Feb 21 '17 at 20:05
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    @PleaseHelp I think you've slightly misinterpreted what I wrote (or I wrote unclearly) - I mean that I approve of demanding a set-theoretic background, even though it forces a break from structuralism. I think most mathematicians (after reading about both perspectives, if they're unfamiliar with them) would disagree with me on this - they (I suspect) would argue that a strong form of structuralism, incompatible with what I'd consider truly set-theoretic foundations, is preferable. – Noah Schweber Feb 21 '17 at 20:07
  • @NoahSchweber: Thanks for your clarification. I wonder now: Aren't structural set theories also truly set-theoretic foundations? – Please Help Feb 21 '17 at 20:10
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    @PleaseHelp Honestly, it depends who you ask. I would (strongly) say "no," and (I think) most mathematicians interested in foundational issues would at least partly agree with me here. But I also imagine there are many on the structuralist side who view their approach as the "true" understanding of the set concept, and ZFC-ists (like me) as misunderstanding what is really going on. – Noah Schweber Feb 21 '17 at 20:11
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    In summary, there seem to be two arguments here. One is that it's possible to operate in math mostly without knowing the axioms underlying set theory, using only a relatively naive notion of sets. The other is that there are areas of mathematics which don't fit nicely into set theory. The first argument seems irrelevant, because these mathematicians don't know any precise system of foundations. It's a little bit like saying that if a native English-speaker can't spell out the rules of grammar explicitly, then those rules don't underlie his speech. The second argument, even if true... – user49640 Feb 21 '17 at 20:31
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    doesn't in any way support the contention that "most of the time in the mathematical literature, we're not even dealing with sets!" In fact, the very example given of the real numbers contradicts the argument. While we seldom consider real numbers themselves as sets, we constantly use the fact that the real numbers form a set, and this fact is constantly present in the minds of mathematicians in a variety of situations. Just because we can almost always replace $\mathbf{R}$ with any isomorphic field doesn't mean that the property that $\mathbf{R}$ is a set isn't used. – user49640 Feb 21 '17 at 20:36
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    @user49640 I think you're confused. The claiming that we don't deal with sets is on an ontological level, not a formal one. – Stella Biderman Feb 21 '17 at 20:39
  • By the way, I certainly wouldn't down vote this answer, as it is the best possible explanation of what is meant in the quote. – user49640 Feb 21 '17 at 20:39
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    @user49640 I wasn't saying that $\mathbb{R}$ isn't thought of as a set - I was saying that individual reals aren't thought of as sets! E.g. the question "Is $2\in\pi$?" would strike most mathematicians as meaningless, even though in set theory it's perfectly meaningful (and true if we view reals as left Dedekind cuts!). – Noah Schweber Feb 21 '17 at 20:40
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    I understand that that was what you meant, but it hardly supports the quote. And if the argument is that, psychologically speaking, mathematicians are operating in a version of set theory with atoms, that doesn't mean they're not dealing with sets. It seems completely normal that mathematicians should disregard, at times, the precise definitions of certain objects they deal with. That seems unavoidable irrespective of the foundations chosen. So saying that mathematicians don't think of real numbers as sets of rational numbers or of Cauchy sequences doesn't mean they're not using set theory. – user49640 Feb 21 '17 at 20:47
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    @user49640 Set theory with urelements still doesn't work, though. Think of the torus vs. the Klein bottle; pick a point on the torus (that is, in the torus when viewed as a set), and ask whether that point is on the Klein bottle as well (that is, in the Klein bottle when viewed as a set). In set theory, distinct urelements can always be compared; but again, this question is meaningless in mathematical practice, the point being that urelements in different contexts are incomparable. And this is fundamentally non-set-theoretic; type theory is a better model here. – Noah Schweber Feb 21 '17 at 20:51
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    I still think you're jumping from the fact that there are some situations where the nature of a certain object being a set can be ignored to an unfounded conclusion that sets aren't often needed. Obviously when you talk about "a point on the Klein bottle," you're talking about something that is much more complicated than a point in $\mathbf{R}^4$. One wouldn't expect the two points you mentioned to be able to be compared in that way, regardless of the system used. The fact that you'd need a particular embedding of the two spaces in, say, $\mathbf{R}^4$ to make that comparison is unsurprising. – user49640 Feb 21 '17 at 21:01
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    Or more accurately, an embedding of each space together with a distinguished point. – user49640 Feb 21 '17 at 21:03
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    @user49640: in set theory any two objects can be compared! Detractors of Noah's opinions may like to look into Benecerraf's paper What numbers could not be. Or perhaps ask themselves questions like what is $\Bbb{N} \cap \Bbb{P}(\Bbb{N})$ or $\Bbb{R} \cap \Bbb{C}$? – Rob Arthan Feb 21 '17 at 21:37
  • "Set theory with urelements still doesn't work, though. Think of the torus vs. the Klein bottle" — The torus? The Klein bottle? There can be many toruses and many Klein bottles, and either you talk about properties of all of them (which is what you do most of the time), then it does not make sense to ask about the identity of a single point (Say I'm standing on the north pole of a planet. John is standing on the north pole of a planet. No two persons can stand on the same place. Does that mean I'm John? Not necessarily; I might stand on Earth's' north pole, and John on Mars'), or you … – celtschk Feb 22 '17 at 08:38
  • … talk about a specific Klein bottle and a specific torus, then the question "pick a point on the torus (that is, in the torus when viewed as a set), and ask whether that point is on the Klein bottle as well" is indeed answerable: The chosen torus may intersect the chosen Klein bottle, and if they do, you'll find the same point in both. – celtschk Feb 22 '17 at 08:41
  • I'm very confused. You start your answer with "OK, let me clarify what I meant with that remark". But I have no idea what remark you are talking about. You aren't the original questioner so it can't be referring to anything in the question. And I looked at the edit history and your first version starts with that as well so its not something added in later referring to something you said in a first edit. This question thus feels like it is missing whatever remark you are referring to. I assume it was a summary of your overall position but it currently just reads really confusingly... – Chris Feb 22 '17 at 09:50
  • Actually, just realised that you were the commenter linked to in the original question. Perhaps edit in some acknowledgement of that. The quote was in the original question and there wasn't much need to go read the source (since the question made perfect sense to me without going to source). – Chris Feb 22 '17 at 09:52
  • Note: this sparked a related question: Whats does "most of mathematics mean? – Bill Dubuque Feb 22 '17 at 14:05
  • @PleaseHelp I notice you've unaccepted this answer - is there a point you'd like me to further address? – Noah Schweber Jun 22 '19 at 19:55
  • @NoahSchweber: He/she also downvoted all three answers of ours nearly simultaneously, so I don't think there is any proper reason. – user21820 Jun 23 '19 at 14:53
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You can think of set theory as a low-level programming language, like Assembly; it works directly with sets in a way analogous to how Assembly works directly with bits and bytes. Some people work with low-level programming languages, like the people who need to write compilers or operating systems or whatever (I am totally making this up, not being a programmer myself), but most people are content to work with high-level programming languages, like Python. These are in some sense ultimately built on low-level programming languages (something somewhere has to actually execute the Python interpreter), but

  1. the low level has been abstracted away so you don't have to think about it, the same way you don't need to learn how your operating system works in order to program in Python, and

  2. the low level could be replaced by something else, and as long as it's still good enough to support the high level, it doesn't really matter, the same way you can program in Python on both a Windows machine and a Mac.

So yes, in some sense it's true that when you're doing mathematics you're (usually) secretly dealing with sets, in the same way that in some sense it's true that when you're programming you're secretly manipulating bits in your physical computer. But most people neither want nor need to think all the way down the layers of abstraction like that, because / and it doesn't matter what the bottom layer of abstraction looks like as long as it can run the layers above it.

Qiaochu Yuan
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    Hal Abelson once said "computer science is not about computers," and it's in roughly that sense that we might say "mathematics is not about sets." – Qiaochu Yuan Feb 22 '17 at 04:35
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Most mathematics can be translated into some suitable set theory such as ZFC. That is certainly true! However, it is totally different from the claim that most mathematics deals with sets. From the point of view within ZFC, of course it is true because in ZFC there are nothing else except sets! But in fact, most ordinary mathematics can be translated into a very very weak system called ACA, which has no internal notion of sets of sets. Furthermore, there are alternative systems such as type theories that some logicians even argue to be more natural than ZFC as a foundation for mathematics. That is naturally a subjective opinion, but the objective fact is that there are indeed different formal systems that can 'do the same thing', so to speak, even though one might 'think' of everything as sets while another 'thinks' that there are urelements.

Ultimately I agree with the main point in Qiaochu Yuan's answer. Namely that mathematics is nearly always based on existing structures that obey some properties. For elementary number theory, we do not care what are the internals of each natural number as long as the collection of natural numbers together with the arithmetic operations on them, as a whole, satisfy the Peano axioms. This abstraction would be a valid reason to say that most of them when we use natural numbers we do not actually deal with sets, since it does not matter even if natural numbers are urelements!

Of course, this mirrors the idea in programming that we write high-level programs and do not deal with CPU instructions in the sense that we do not actually care how our programs are translated to CPU instructions. We only care that our program behaves in a way that is solely determined by the high-level programming language. Take Java for example. On different machines it is necessarily translated into different CPU instructions, but that is done by the Java environment; in our terms the Java language is the abstraction that frees us from 'bare-metal' concerns. In the same way our mathematical axiomatizations (like PA for natural numbers) frees us from foundational concerns.

Finally, there is a huge benefit to working with abstractions than with underlying implementations. In case we ever wish to use a different foundational system for whatever reason, abstraction makes it far easier to identify which parts transfer over without change to the new system. We can in fact precisely reason about such general transfer via interpretability.

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