tea.mathoverflow.net - Discussion Feed (Notions used but not rigorously defined)

Todd Trimble comments on "Notions used but not rigorously defined" (13539)

2011-03-01T10:58:20-08:00

redacted

thierryzell comments on "Notions used but not rigorously defined" (13536)

2011-03-01T09:53:55-08:00

For the record (and if I remember correctly), the book "Stratified Morse Theory" by Goresky and MacPherson begins with a historical account of the development of algebraic topology. The authors wax quite wistful, towards the end of that section, about how easy it was back in those days to write papers and have results even though the precise definitions had yet to be nailed down properly. They assert that such a sustained level of vagueness was what allowed the subject to take off, and that it would be much harder to do so today.

I believe this is the spirit of the question being discussed here. It's easier to think about the historical example than to jump directly into the notions that are currently non-rigorous. The lack of rigor is probably more obvious after the fact.

It's a legitimate question, though I'm not sure how enlightening the answers will be.

Qiaochu Yuan comments on "Notions used but not rigorously defined" (13535)

2011-03-01T09:27:18-08:00

Great! Then I have no objections to the question.

gilkalai comments on "Notions used but not rigorously defined" (13534)

2011-03-01T07:29:56-08:00

Just to make it clearer I added the above quote to the question. I thought that this was precisely what the OP was asking, but since this is a CW we can certainly edit it (even if we modify the precise original intention).

Qiaochu Yuan comments on "Notions used but not rigorously defined" (13533)

2011-02-27T14:58:04-08:00

I think 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.

I agree that this is a fascinating question, but I disagree that this is the question the OP is asking. (In fact, I don't know what the OP is asking! That's what I mean when I say that the OP is being imprecise.)

Mariano comments on "Notions used but not rigorously defined" (13532)

2011-02-27T14:11:25-08:00

Emerton, but was any of the proposed definitions for F_1 somehow unclear or not precise? Correct, as it applies to definitions (as opposed to statements) can only mean, I think, satisfactory or, to be more verbose, that it captures the idea.

I think asking about definitions which are seen to be only unsatisfactory, approximative implementations of some idea is quite sensible. The fact that this occurs (and it surely occurs!) is an extremely interesting aspect of mathematical work; in particular, this is an aspect of maths that is completely missed by the popular view on the subject and on our work. But I think that the question in question should be made precise, if that is indeed what it is being asked.

A completely different question, «what are definitions that were unclear or unprecise?» in the sense of not completely defining a concept (I cannot think of any example of this right now) could also be asked... and right now I don't think it is at all clear which of the two is being asked!

Emerton comments on "Notions used but not rigorously defined" (13530)

2011-02-27T12:58:38-08:00

I don't agree with the objections to this question. I think 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. (Much as calculus did, well before it was put on a rigorous footing in the 1800s.)

E.g. despite Mariano's comment above, as far as I know there was no candidate definition of mixed motive prior to Voevodsky's work, and the definition of the abelian category of mixed motives (as opposed to its associated derived category) has still not been made. Nevertheless, there are many examples in the literature using the (not-yet-established) theory of mixed motives to draw significant conclusions.

In another direction, I remember when I was a student that there were many enumerative computations in algebraic geometry that could be made by computation in an appropriate quantum field theory, but which were not rigorously justified. Again, the situation has improved somewhat since then, but as far as I know there are still quantum field theoretic computations that cannot be made rigorously.

I also think that the hypothetical field F_1 is an example; despite the competing definitions that may have been offered, I don't think it is clear yet which (if any) of them is the correct.

There are other examples that I know of (e.g. the hypothetical Langlands group attached to a global field), and I'm sure there are examples that I don't know about as well.

In general, I think that it's an interesting state of affairs in mathematics when we have specific notions with strong predicative power which are not yet made rigorous, and it seems worthwhile to document some examples. If nothing else, it may implicitly offer suggestions to some readers as to where to direct their efforts.

Todd Trimble comments on "Notions used but not rigorously defined" (13528)

2011-02-27T06:29:23-08:00

One possibility for a more precise question is: what concepts in mathematics have been given multiple candidate definitions that have not been proven to be equivalent? (The candidate definitions may themselves be -- in fact in modern mathematics typically are -- impeccable, completely rigorous.) For example, there are various candidate definitions for F_1, for the notion of (weak) n-category, etc. The notion of matroid does not qualify (as far as I understand) because the different formulations have been proven equivalent.

I don't think we have to get into discussions about "what is a concept?", etc. The criterion is simply that there is a noun or phrase like F_1 that people talk about and have proposed rigorous definitions of, but there is not yet a unique definition (up to "cryptomorphism", to use a coinage of Rota).

Qiaochu Yuan comments on "Notions used but not rigorously defined" (13527)

2011-02-27T06:17:07-08:00

@markvs: because my opinion is that the OP's question, as it stands, is too imprecise to be appropriate for this site. I can imagine more precise versions which are, but those aren't the versions being asked. I disagree that closing it again is silly (all this means is that I am disagreeing with five people) and agree with Yemon's objections.

markvs comments on "Notions used but not rigorously defined" (13526)

2011-02-27T05:05:21-08:00

@Qiaochu: Why do you think that the OP should change the question just because you want him/her to do that? (Personally I think that your objections do not make any sense, but it is beyond the point and I do not want to discuss it.) Also the question has been closed and reopened. Closing it again is silly.

HJRW comments on "Notions used but not rigorously defined" (13525)

2011-02-27T04:59:35-08:00

Yemon, I was (probably mis)remembering a bit of Lucky Jim. Maybe I'll dig it out in the morning. No offence intended.

Qiaochu Yuan comments on "Notions used but not rigorously defined" (13524)

2011-02-27T04:38:20-08:00

The OP has been unwilling to make the question more precise. I am voting to close.

Mariano comments on "Notions used but not rigorously defined" (13522)

2011-02-26T18:15:35-08:00

I agree with Yemon here.

The answers are mainly about ideas ("$q$-analogue", say), not with actual mathematical objects. The search for the correct category of motives does not mean that each of the candidates considered was not clearly defined, and so on.

The problem is not with the answers but with the question, which does not ask anything clearly defined...

Yemon Choi comments on "Notions used but not rigorously defined" (13521)

2011-02-26T16:27:32-08:00

BTW, Henry, I am not quite sure why you think a dictionary would help over and above a lifetime of speaking the language...

Yemon Choi comments on "Notions used but not rigorously defined" (13519)

2011-02-26T14:42:34-08:00

Henry, this is somehow my entire point: if a curve is a notion, is Quadratic Reciprocity a notion? Is (conjectured) Mirror Symmetry a notion? Is weak almost periodicity a notion and if so what are the things which are weakly periodic in which sense?

I am reminded of Russell's tongue-in-cheek, quasi-Socratic remarks about it not being so obvious that one plus one equals two.

The original question blurs natural language with mathematics in a way that I think could lead to poor answers. It does not give examples (such as the one Kevin alludes to above) of what (s)he is after.

kakaz comments on "Notions used but not rigorously defined" (13516)

2011-02-26T01:59:55-08:00

*If You have problems with the meaning of the words "used" and "clearly defined" You may always contribute by answering: "clear definition in use", do not You? Sorry I cannot resist;-)*

Kevin Ventullo comments on "Notions used but not rigorously defined" (13515)

2011-02-25T22:09:18-08:00

From a historical point of view, I think this question is relevant:

http://mathoverflow.net/questions/54932/how-were-moduli-spaces-defined-before-functors

Steve Huntsman comments on "Notions used but not rigorously defined" (13514)

2011-02-25T21:03:28-08:00

@Ben--In the same vein, consider most physically interesting quantum field theories in 3+1 dimensions, especially pure glue Yang-Mills.

Ben Webster comments on "Notions used but not rigorously defined" (13513)

2011-02-25T19:24:35-08:00

Alex- I disagree. I think conformal field theory is a great example of something which is mathematically interesting, lots of people study (and do good mathematics inspired by), but which everyone knows no one can rigorously define (yet).

Alexander Woo comments on "Notions used but not rigorously defined" (13512)

2011-02-25T19:17:58-08:00

Not a suggestion to reopen, but a sort of answer:

We don't know.

If we knew some notion was used but not rigorously defined, we would either figure out how to define it or stop using it.

If you look at the history of mathematics, you will see that very many terms were thought to be sufficiently well defined until some mathematician(s) poked a hole in the common understanding(s) of the term.

I must also say that I don't like the suggested interpretation of the Klein quotation. It makes it look as if 19th century mathematics was some deficient variant of modern mathematics, rather than a slightly different cultural practice altogether. (cf. Wittgenstein's remarks on Frazier's _The Golden Bough_)

Zev Chonoles comments on "Notions used but not rigorously defined" (13511)

2011-02-25T19:15:10-08:00

After reading David's comments I've reconsidered and cast the final vote to reopen.

HJRW comments on "Notions used but not rigorously defined" (13510)

2011-02-25T19:00:17-08:00

What is a "notion"? When is it being "used"?

Yemon, at the risk of being facetious, if you honestly don't know the answer to these questions then I refer you to any English dictionary. (This is not to say that I disagree with your other objections.)

The serious point I'm trying to make is that words don't have to be defined to mathematical levels of precision if they're being used outside the technical context of mathematics.

markvs comments on "Notions used but not rigorously defined" (13509)

2011-02-25T18:41:24-08:00

Voted to reopen.

Georges Elencwajg comments on "Notions used but not rigorously defined" (13508)

2011-02-25T18:30:13-08:00

It is used in the sense that there are conferences on the subject and that great mathematicians like Soulé write articles about that field.

Qiaochu Yuan comments on "Notions used but not rigorously defined" (13507)

2011-02-25T18:23:25-08:00

It is debatable the extent to which that notion is "used." Again, I think the OP is misinterpreting the quotation he uses.

Georges Elencwajg comments on "Notions used but not rigorously defined" (13506)

2011-02-25T18:20:39-08:00

An addition to David's list: the field with one element, F_1.

Yemon Choi comments on "Notions used but not rigorously defined" (13505)

2011-02-25T18:00:20-08:00

Speaking more bluntly than I would normally allow myself to: my problem with the question is fourfold:

1) In my view the answer to the question is arguably "everything" - the issue is what you mean by clearly defined. Structurally? Interpreted within a particular model?

2) What is a "notion"? When is it being "used"?

3) I think that it will be easy to give many (and frequent) poor answers to the question, with a somewhat subjective component. (If something isn't "clearly defined", could it be because you just haven't read the right literature?)

4) I do not think that the level at which David interprets the question is one which the questioner will greatly appreciate (but my opinion is coloured by previous questions (s)he has raised on MO).

Qiaochu Yuan comments on "Notions used but not rigorously defined" (13504)

2011-02-25T17:21:58-08:00

Personally I don't like the question because the OP seems to be misinterpreting the quotation. But I guess that's not a good reason to close the question.

David Speyer comments on "Notions used but not rigorously defined" (13503)

2011-02-25T17:06:51-08:00

I have some sympathy with the question http://mathoverflow.net/questions/56677/what-notions-are-used-but-not-clearly-defined-in-modern-mathematics-closed , and I strongly disagree with the notion that every concept used in current mathematics has a rigorous definition.

A few examples: People at the n-category cafe seem to spend a lot of time discussing what the definition of an "infinity-category" should be. My understanding is that one of the reasons Lurie and Nadler-Ben Zvi's work is so exciting is that we seem to be getting close to agreeing on a definition. (Not my area, though, so I could be wrong.)

As I understand it, we don't have a rigorous definition of a "conformal field theory". (Again, not my area.)

Getting into combinatorics/combinatorial representation theory, I don't think there is a generally accepted definition of a "bijective proof" or of a "manifestly positive formula", which are things people in those fields spend a lot of time looking for.

Of course, there are also words like "trivial" and "elegant", but I think it is obvious that we don't want to define those!

I'm not sure this question will draw any great answers, but it might be interesting to see what other concepts people know of which are missing definitions.