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What are the basic things (about when and how) to be kept in mind while proposing a conjecture in Mathematics. Should it accompany solid efforts at proof. When should any one think of proposing a conjecture.I'd like to know some of these details just out of curiosity.

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That depends what you're asking. You can conjecture anything you want based on anything you want - but the better you are able to justify that conjecture (e.g. by providing data or heuristic reasoning), the more likely other mathematicians are to listen to you or believe you. –  Billy Aug 4 '11 at 8:34
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I once attended a talk where the speaker proposed two conjectures that one could quickly see contradicted each other. I thought that was terrible manners. –  Andres Caicedo Aug 4 '11 at 9:15
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There are three close votes on this question, but I don't see any arguments for closing, only interesting responses. I vote against closing. (That means the next close vote should be registered here to cancel mine, instead of casting an actual close vote.) –  joriki Aug 4 '11 at 12:01
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@Mike Jones: Dear Mike, Ramanujan was not an amateur: he was a Fellow at Trinity, a Fellow of the Royal Society, and was publishing in (at the time) leading journals, writing joint papers with other leading number theorists of his day. Regards, –  Matt E Aug 5 '11 at 0:55
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@Matt E: And Robert Burns was NOT a farmer. And Elvis Presely was NOT a truck driver. Whenever anyone makes the big time, their origins are forgotten. –  Mike Jones Aug 5 '11 at 22:08

4 Answers 4

up vote 27 down vote accepted

First of all, there are various attitudes towards conjectures. Let me describe two rather famous instances before making some remarks:


I'd like to start with "Serre's conjecture".

In his famous paper Faisceaux Algébriques Cohérents (often called simply FAC) the Fields medalist J.-P. Serre mentioned a specific technical problem:

Serre's "conjecture"

Loose translation:

Let us point out that, if $V = K^r$ (in which case $A = K[X_1,\cdots,X_r]$), we don't know whether there exist finitely generated projective $A$-modules that are not free, or, what amounts to the same, whether there exist non-trivial algebraic vector bundles with base $K^r$.

Here is the very first paragraph of T. Y. Lam's book entitled Serre's conjecture (taken from Springer LNM 635):

On P.243 of his famous article Faisceaux algébriques cohérents (FAC), Serre wrote: 'On ignore s'il existe des $A$-modules projectifs de type fini qui ne soient pas libres' ( $A = k[t_1,\ldots,t_n]$, $k$ a field); shortly thereafter, the freeness of finitely generated projective modules over $k[t_1,\ldots,t_n]$ became known to the mathematical world as "Serre's Conjecture". Serre might have objected to the fact that what he raised essentially as an open problem became his "conjecture" by world acclamation, but the fine distinction between "Serre's Problem" and "Serre's Conjecture" may now be safely left to the deliberation of the mathematical historian. Culminating almost twenty years of effort by algebraists, D. Quillen and A. Suslin proved independently, in January, 1976, that finitely generated projective modules over $k[t_1,\ldots,t_n]$ are, indeed, free.


The second instance I'd like to mention appears in Wolfgang Lück's work frequently. He is an internationally acclaimed topologist whose recent work centers around the Baum-Connes Conjecture and related questions. A quick search on the page containing his list of publications reveals that the word "conjecture" appears no less than 42 times. To be fair, I should note that a good many of them are titles of talks and conference names, but nevertheless the word conjecture is used quite often in his surroundings. The talks I saw by him always mentioned a few of these conjectures and also proved a few instances.

Here's a random sample from his Inventiones paper The relation between the Baum-Connes Conjecture and the Trace Conjecture

Modified Trace Conjecture

and a bit further down:

Farkas Conjecture


This should illustrate two rather diametrically opposed attitudes to "conjectures" by two very famous mathematicians. Serre rather relucantly pointed out an open problem which was considered so interesting that it became his conjecture, more to his dismay than to his joy, I might add. Its resolution by Quillen certainly contributed to his earning the Fields Medal while Suslin didn't receive it for reasons somewhat unclear to me, but that's a different story...

On the other hand, Lück uses the word conjecture on a daily basis and the conjectures he talks about and his results concerning them are considered very interesting and ground-breaking by many people (including me of course).

Bear in mind however that both Lück and Serre are mathematicians known world-wide and more than deservedly so.

For more modestly talented people like you and me I think it is far more appropriate to follow Serre's example and content ourselves just hinting at open problems. It is probably a safer bet to assume that neither you nor me will ever phrase a question considered important, interesting and difficult enough by many enough to get elevated to the status of conjectures. Running ahead and throwing around conjectures is simply not an appropriate attitude I think. But that's just a piece of unsolicited advice and my two cents.


So far, I haven't really addressed your question "how to propose a conjecture". My response is simply: "Don't."

(But see Matt E's answer for reasons why formulating conjectures may be justified, appropriate and useful).

Here's a brief explanation of my advice:

In order for a conjecture to be taken seriously a good many sociological conditions must be fulfilled:

  • You must be taken seriously by your audience.
  • You must know what you're talking about.
  • You must be convinced that the conjecture is an important problem (not only you must consider it important but others should be likely to agree).
  • You must be sure that the conjecture is unlikely to produce silly counter-examples or be confirmed quickly and easily.

If you formulate a conjecture, this might be interpreted to say that you're thinking that all these conditions are fulfilled. Out of reasons of modesty I think it is therefore best to refrain from making them, at least as long as you haven't reached a certain status of recognition in your research area.

After having made some contributions to your field you will probably be in position to judge whether formulating a conjecture rather than a question or a problem is appropriate or not, for instance for reasons as listed by Matt E in his answer. I am not in position to give any advice on that.


Added: joriki made a good point that deserves to be mentioned more prominently:

[...] I wholly agree with your thoughts on modesty. However, it seems you're treating the word "conjecture" as synonymous with "(potentially) famous conjecture", "conjecture (potentially) named after someone". I think it's not immodest to use it in another sense, as in "There is an obvious pattern here that suggests the conjecture that ... for all n." That doesn't have to imply that you think this is a problem that will raise the interest of generations of mathematicians and advance mathematics immeasurably.

For instance, Popper uses the word in this more mundane sense in his "Conjectures and Refutations", e.g.: "The actual procedure of science is to operate with conjectures: to jump to conclusions -- often after a single observation [...]."

I do see these points and objections. I don't have anything much to add or object to that since I don't know Popper's writings very well. Thank you for raising this point joriki.


Rajesh, you are of course free to follow this interpretation but take my answer as a word of warning that some people tend to think of conjectures the way I tried to express it above. The word "conjecture" is loaded and it is better to be safe than sorry.

Let me display an excerpt from the text Some Hints on Mathematical Style by David Goss:

Some Hints on Mathematical Style

I'll finish by recommending reading these hints (based on remarks by Serre) in full. Read them attentively and think about the points raised there. You may or may not agree with all the points but I think it should be profitable in any case.

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"...you must be convinced that all these conditions (and probably a good many more) are fulfilled. This assumption is already an indication of immodesty (I believe) and should be avoided for these reasons alone." - I gave a +1 for this. Probably OP should call what he has a "guess" for the time being, and then he can call it a "conjecture" when he's prominent already... :) –  J. M. Aug 4 '11 at 11:16
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Very nice post, Theo, thanks -- entertaining, informative, and to the point. I wholly agree with your thoughts on modesty. However, it seems you're treating the word "conjecture" as synonymous with "(potentially) famous conjecture", "conjecture (potentially) named after someone". I think it's not immodest to use it in another sense, as in "There is an obvious pattern here that suggests the conjecture that ... for all $n$." That doesn't have to imply that you think this is a problem that will raise the interest of generations of mathematicians and advance mathematics immeasurably. –  joriki Aug 4 '11 at 11:56
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@joriki: I still think "guess" is a much simpler and less loaded word than "conjecture". At the very least, it won't trigger the ire of professional mathematicians, and if you're wrong... "well, it was only a guess..." :) –  J. M. Aug 4 '11 at 12:06
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"Research problem" is better than "guess". If you're wrong you still get credit for identifying an interesting research problem. –  Douglas S. Stones Aug 4 '11 at 12:56
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Maybe someday, when I grow up, grow cojones, and become rockstar-famous, I will rename my "list of plausible hunches" to "Glorious List of Conjectures™"... :D (too bad I can't upvote any more today) –  J. M. Aug 4 '11 at 16:31

I think you should prove a few theorems before you propose any conjectures. Better yet, publish a few papers. Anyone can make conjectures; yours won't stand out from the others until people think something coming from you is worth looking at.

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Not only can anyone make conjectures, but if you don't have some kind of mathematical taste, you're probably not going to make interesting conjectures. Most conjectures made by most people will either be too easy or too hard, and in any case they will probably not lead to interesting research programs. –  Qiaochu Yuan Aug 4 '11 at 12:51
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Combining two responses: You can conjecture anything you want whenever you want. But probably the only people who will take notice are your peers: those with similar experience (or lack of experience) in mathematical research. ... The big important conjectures were mostly promulgated by important mathematicians who had already published much work in the area of the conjecture. They may then add their conjecture to a paper they are publishing. Or add it to a lecture they are giving to other mathematicians. –  GEdgar Aug 4 '11 at 12:53

Conjectures play different roles in different areas of mathematics. My impression is that they seem to be more common, and perhaps more accepted as a mode of mathematical expression, in highly structured areas such as algebraic number theory, algebraic geometry, and algebraic topology. My own expertise is in the former two areas, and here there are many very famous open conjectures: the Riemann hypothesis, the Birch---Swinnerton--Dyer conjecture, the Tate conjecture, the Hodge conjecture, Langlands' conjectures, resolution of singularities in characteristic $p$, and so on. Many of them are named after the particular individuals who first posited them. Not everyone believes in their validity, but many do, and they gain their legitimacy not only (or not even primarily) from the stature of the name they are attached to, but also from the sense that they fit well with other known and expected results, and with the general directions and themes of research in the field. Because of the highly structured nature of these fields, it is not so easy to find new statements that fit well with what is already known or suspected to be true!

Of course, the credibility of a conjecture can be enhanced by obtaining evidence in its favour, but some things are difficult to compute, and the credibility of a conjecture often rests on other considerations besides just verification of particular cases. As I've indicated, a sense of fitting well within the known framework of the subject, while simultaneously extending this framework in a new and interesting way, is often an important criterion for a plausible and interesting conjecture.


As for making conjectures oneself:

Different people view conjectures with differing levels of sacrosanctness, and this (as well as other considerations) affects their view on the wisdom or otherwise of making conjectures.

For example, in his answer, Theo suggests that it is probably best not to make conjectures at all, primarily for reasons of modesty. Gerry Myerson suggests that one should at least be proving good theorems in an area before making conjectures in it, which is certainly sensible advice.

My own view is the following: I think that it is good to make conjectures if you sincerely believe them; they help to focus (one's, and other people's) research, and draw attention to what is important in a given line of investigation. However, just in the interests of not damaging one's own reputation too much, it is probably not sensible to make a conjecture unless one has a very good feel for the subject in question (meaning that one should be proving theorems in the subject that are near to the cutting edge). Even when one is at the cutting edge of research, there is always a possibility that one's conjecture will be wrong, due (typically) to a misunderstanding of or lack of knowledge of certain phenomena and (counter-)examples. (This is an occupational hazard of making conjectures!)

If one suspects that a certain phenomenon might be true, but is unconfident in one's intuition, or unsure whether it is always true or what the precise hypotheses might be, one can always raise a question instead, rather than stating a definitive conjecture.

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Dear Matt, I am a bit unsure how to proceed. My own position is best expressed in the extract of David Goss's Hints on mathematical style at the very end of my answer. I do agree with everything you say and concede that I have overstated my case. I would be willing to remove the part you seem to disagree with (is my understanding correct that your disagreement concerns mainly the part containing the boldface "Don't" up to and without the "added" part?). However, I can't do that without affecting the context of your answer and I don't want to do that. Regards, –  t.b. Aug 4 '11 at 17:38
    
@Theo: Dear Theo, In general terms, I think we can agree to disagree; in fact, we probably don't disagree that much, but are simply approaching the question from different starting-points. Regarding particulars, I will edit my answer a little, so that it reads less negatively. After I do this, it will probably free you up to edit your answer (if you want to --- but please don't feel obliged to) without affecting mine. Best wishes, –  Matt E Aug 4 '11 at 17:55
    
@Theo: Dear Theo, Just to elaborate a little more: yes, my disagreement is with the section of your answer that you indicated. I do agree with the sociological conditions you list, but I just draw a different conclusion: namely, it makes sense to be cautious in formulating conjectures, but it is not necessarily immodest. Our difference is probably in part a matter of how often we expect that these sociological conditions might actually be fulfilled, but also probably reflects a (completely reasonable) difference in temperament. Best wishes, –  Matt E Aug 4 '11 at 18:07
    
Dear Matt, Thank you for your explanations and also for your thoughtful answer. I have slightly edited my answer and I hope that I removed the part that said it was necessarily immodest to formulate conjectures. That was not really my intention and I apologize if I have inadvertently offended you (I lack a better word, it seems a bit too strong). Best wishes, –  t.b. Aug 4 '11 at 18:59
    
Dear Theo, Thanks for your reply (and I wasn't offended at all!), Best wishes, –  Matt E Aug 4 '11 at 19:27

Adding to previous answers, if one is determined to procure a conjecture, here's my advice.

  • The most important point: You lose very little by phrasing a conjecture as a research problem. There should be some tangible gain before making a research problem a conjecture (if anyone was thinking it, it's almost surely not going to be named after you).

You not only need to be convinced that your conjecture is important, you need to demonstrate its importance. Typically this is done by proving "An affirmative answer to Problem X implies Conjecture Y is true", where Y is e.g. a published conjecture, some result of some importance, etc. If you cannot find any significant implications, then you need to explain why it is important in its own right.

If your conjecture an unimportant conjecture, then you risk getting egg on your face if it is wrong, and having your conjecture ignored if it is right.

You also need to demonstrate that you have made a serious attempt to (a) verify its correctness and (b) prove it:

  • Attempt to prove it. If your attempts fail, then put into words why it fails. For example, "If one attempts to resolve Problem X by using Techniques A, B and C, then we encounter [blah blah blah], thereby thwarting the attempt."
  • Attempt to prove the easiest cases of the conjecture. Again, if you fail, it's best to put into words why your attempts failed. Does the conjecture have any generalisations of which you could prove a special case?
  • If relevant, use a computer (or random data, etc.) to verify it for as many cases as possible.

After this, my advice would be to discuss it to someone (via private communication) who is knowledgeable in the field. Do not yet describe it as a conjecture. Also, I wouldn't immediately disturb a world-class expert (unless they happen to be your supervisor, friend, etc.). If they think it's worthwhile, consider telling others.

If you continue to get good feedback, then the next stage would be to present it at e.g. a research seminar, or a local conference. Again, make sure you phrase it as a question or a research problem "Q: Is it true that [blah blah blah]? I have observed [X, Y and Z] which seems to suggest it is true." Expect questions like "Have you tried A, B and C?", which hopefully you know how to answer from your attempt at proving it.

Afterwards, you might consider talking about it at a major international conference (assuming you don't have anything else better to present).

...and finally, after withstanding the test of time, it's probably still best to just call it a research problem.

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Dear Douglas, I disagree with your suggestion of the terminology "research problem" (hence my downvote). The word "conjecture" is well-established in mathematics. I agree that is probably not good to throw it around willy-nilly, but this is not a reason for never using it. (See my answer for more on my own view-point.) Regards, –  Matt E Aug 4 '11 at 16:42

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