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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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:
Here is the very first paragraph of T. Y. Lam's book entitled Serre's conjecture (taken from Springer LNM 635):
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
and a bit further down:
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:
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 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.
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.
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.
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.
Adding to previous answers, if one is determined to procure a conjecture, here's my advice.
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:
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.