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"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."

Felix Klein

What notions are used but not clearly defined in modern mathematics?


To clarify further what is the purpose of the question following is another quote by M. Emerton:

"It is worth drawing out the idea that even in contemporary mathematics there are notions which (so far) escape rigorous definition, but which nevertheless have substantial mathematical content, and allow people to make computations and draw conclusions that are otherwise out of reach."

The question is about examples for such notions.

The question was asked by Kakaz

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In mathematics, by mathematicians. Everything is clear? I suppose mathematics is still live nowadays... – kakaz Feb 25 '11 at 21:08
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"Everything is well defined in modern mathematics" - We really don´t know that for sure yet (i.e. consistency of ZFC)... "Mathematics is more about correctness than about truth." -I would argue that it is more about the relative truth. Being about correctness is contains too much of a self-purpose... – efq Feb 26 '11 at 1:55
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This question has a meta thread: tea.mathoverflow.net/discussion/968/… – JBL Feb 26 '11 at 1:57
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@Qiaochu - I understand Your position, but I think nobody have to fear vagueness in such situation. It is just another soft-question on math-overflow. It is some fun. I know that You are professionals but I am an amateur. I would like to play with mathematics. Usually it is worth of mention what do we use without proper definition, fighting between intuition and complicated formalism, and possibly why. I do not understand why question which is obviously interesting and have potential to broaden horizons for many people is so controversial. – kakaz Feb 27 '11 at 19:05
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Closing this thread seems more like punishing someone for being an amateur rather than enhancing the quality of the site. Now and throughout history, I believe, a large percentage of the most interesting mathematics revolves precisely around those notions that are used but not (yet) clearly defined. A big list of such subjects seems extremely valuable to me. Vote to reopen. – Louigi Addario-Berry Mar 1 '11 at 16:01

34 Answers 34

Maybe situation in Matroid theory where there is several strict axiomatization schemes but its equivalence is not easy to prove, is interesting here. Probably there should be some generalization which would tie this different approaches into one, more or less obvious notion. There is even terminology connected with that phenomenon by G.C.Rota Cryptomorphism http://en.wikipedia.org/wiki/Cryptomorphism

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What's not clearly defined here? – Qiaochu Yuan Feb 26 '11 at 13:52
    
matriod itself - which has several strict and not easy-equivalent definitions. It is like a finger pointing on the moon, and we still look on a finger... But OK - I am probably terribly wrong. – kakaz Feb 26 '11 at 16:41
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This is a bad example. The notion is precise, it does not matter whether some equivalence is trivial or not. – Andrés E. Caicedo Feb 26 '11 at 17:35
    
@Andreas - yes, probably You are right. – kakaz Feb 26 '11 at 17:51
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This is not precisely what the question about imprecise notions was about, but it is a somewhat related issue. There is no excuse for the large number of down votes. – Gil Kalai Mar 1 '11 at 15:35

In analysis, the concept of a limit at infinity vs. a limit at a real number $r$.

Typically, there is a whole list of definitions of various limits $\lim_{x \rightarrow a} f(x) = b$, depending on whether $a$ and $b$ are ordinary reals or $\pm \infty$. You may have 9 separate definitions of the limit, one for each case. This situation repeats itself any time a limit is used implicitly, for example if an integral converges to a real or to $\pm \infty$, a series converges, and so on.

Everyone knows that these definitions are really the same, but it seems more cumbersome to have a single unified definition than to have separate definitions that are, informally, the same concept. It is this covert "intuitive sense" in which all the definitions are the same that is not clearly defined.

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I don't think this is in the spirit of the question. If one really insists, it is possible to provide a compact definition of those "nine different limits" (consider extended reals, and so on...), and there is absolutely nothing unclear about it. – Mariano Suárez-Alvarez Sep 30 '11 at 2:18

The definition of a number is kinda fuzzy. Is the alph null a number?

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Perhaps this answer is being voted down because $\aleph_0$ is a cardinal number, which is a well-defined notion. But the issue of which algebraic objects constitute "number systems" is kinda fuzzy, yes. – Mark Grant Feb 26 '11 at 7:40
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Nobody really uses the notion "number" in modern mathematics without specifying what kind of number is meant. – darij grinberg Feb 26 '11 at 10:08
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@darij: doesn't that support what Frank is saying? – Carl Mummert Feb 27 '11 at 1:32
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The question asks for things that are both poorly-defined and used; darij is pointing out that it is not used. – JBL Feb 28 '11 at 3:36

How about the natural numbers $\mathbf N$ and the real numbers $\mathbf R$? If they were both clearly defined, then (for example) mathematicians would agree that the continuum hypothesis has a truth value, even if that value is not known. But there's not such agreement, and some will dispute that CH even has a meaning, much less a truth value. Even keeping it just to $\mathbf N$, all "definitions" that I know of are circular (e.g. the Peano axioms in second order logic just kick the unclarity up to the level of the predicates that the induction axiom quantifies over). The Hilbert $\omega$-rule is similarly self-referential. Yet we (mostly) agree that all arithmetic formulas (even, say, $\Pi^0_{100}$ formulas) do have truth values, that there's a (not effectively describable) first-order theory of "true arithmetic", etc. It just comes back to "the naturals, I mean the ordinary naturals, you know, 0, 1, 2, 3..." which comes across as a little bit faith-based ;-).

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There are Peano axioms in first order logic, where the induction axioms becomes a (countable) infinite axiom scheme. If these axioms have a model at all, then everything we can deduce from them holds in all models. On the other hand, we know from Gödel that there will always be undecidable statements concerning $\mathbb N$, which we could decide by introducing new axioms. I understand that this may be unsatisfactory, but somehow we have to live with this state of affairs. Once you have settled with $\mathbb N$, there is no problem with $\mathbb R$ at all. – Sebastian Goette Dec 5 '15 at 18:36

protected by Andy Putman Dec 6 '15 at 2:45

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