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Set theory enables us to turn assumptions into theorems. But it seems like few mathematicians are concerned with foundations and view the rest of mathematics, for example analysis, as having a life of its own. But the use of sets is widespread. The question is: how can 1) we not be concerned with foundations and 2) use sets in every part of math? Aren't (1) and (2) conflicting? I would understand if sets were just a convenient tool to make arguments. But they're not: they're a critical component. We can't "abstract" away from them (or: nobody seems to try).

If we're not using a formal notion of "set", then what is the meaning of statements of the form "there exists a set such that ..."? Does it mean instead "these objects can be thought of as being together"? And if so, why use the word "set" to describe something totally informal?

isthisreallife
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  • I think (1) and (2) are conflicting. As to the first question, I think this is because we do not take it too seriously and do math just for fun. – heptagon Sep 23 '16 at 18:02
  • Why would we need to be concerned with foundations in order to use the ones already laid down by others? – Tobias Kildetoft Sep 23 '16 at 18:31
  • Informal does not mean non-rigorous. All (correct) mathematical arguments could be in principle formalized in $ZFC$, but that doesn't mean they should be – Exit path Sep 23 '16 at 18:31
  • leibnewtz: I'm not sure what you mean. How do you judge correctness? Isn't the modern standard of "correct" taken to be "provable from axioms (usually ZFC)"? – isthisreallife Sep 23 '16 at 20:42

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The historical truth of the matter is, mathematicians have rarely been very concerned with foundations - until their informal reasoning runs into trouble; then they scramble to introduce axioms, create models of this or that, and otherwise introduce rigor into their fields. Once they have done that, and have figured out where the pitfalls of earlier work lie and how to avoid them, nearly all of them go right back to informality. As long as one mathematician can understand what another is saying - and informality helps a lot there - they generally don't pick nits (except where those nits are actual lapses of reasoning, of course).

[if modern formal methods have led you to assume mathematicians would always argue precisely and rigorously, then the history of the development of the Calculus would shock the heck out of you...]

PMar
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  • But "truth" depends on context. Isn't that the lesson from rigorization? In one geometry two lines might never meet, in another geometry, two lines will always meet. Is it really nitpicking to be critical towards claims of existence? – isthisreallife Sep 25 '16 at 16:33