Can I demonstrate a proposition without a rigorous proof?

Question

Writing a paper, I want to declare a certain proposition, but is only supported by other research's empirical result. In this sort of case, what can I name to my argument in my paper? CONJECTURE is so much informal and seems to be so lazy. Otherwise, is it possible just to write down PROPOSITION in my paper without rigorous proof, but with some procedures attaining to that proposition including previous empirical results?

Thank in advance!

Answer 1

As I understand it:

• you have an interesting mathematical sentence $\varphi$
• you think $\varphi$ is true
• you have a novel argument $\alpha$ as to why $\varphi$ is true, yet $\alpha$ falls short of being a genuine proof of $\varphi$

My advice is therefore as follows.

1. You should only call $\varphi$ a "proposition" if:

   • You prove $\varphi$ in the paper, or
   • You tell the reader where a proof of $\varphi$ be found, or
   • The proof of $\varphi$ is so straightforward that you don't bother proving it in the paper, although you have personally proved it in your own notebook and can reconstruct the proof given not-too-much-time.

2. Otherwise, "conjecture" is the right label for $\varphi$. If it is someone else's conjecture, you may wish to call it "so-and-so's conjecture."

3. Tell your reader that you will present an informal argument for $\varphi$, which nevertheless falls short of a genuine proof. Then make the argument $\alpha$ as clearly as possible. Best case scenario, it will be clear where the gaps are. Worst case scenario, at least you may have gestured in the right direction; this may be useful to someone who can see how to make your intuition rigorous.

4. There is another possible issue, which is that you may feel like you should be able to prove $\varphi$, but nonetheless you haven't been able to. So, you don't want to embarrass yourself by calling it a conjecture in a situation where some of your readers can easily see a proof. This is both reasonable, and quite normal; I suggest talking to some friends or acquaintances who know mathematics, and see if they can prove it. This will be hard if you don't have the right connections; if you're at university, talking to some of your lecturers would be a good idea. Otherwise, consider emailing some researchers in the field and very politely explaining your situation, and that it would help you immensely if they could spend a little bit of time trying to prove $\varphi$. Explain that if the problem is indeed difficult, then simply knowing that it is difficult would be helpful to you; and that if the problem is easy, then you knowing how to prove $\varphi$ would help you immensely with the task of publishing your paper.

Answer 2

Technically, it is a conditional proof. In practice, it suffices to convert your result into a conditional statement. Two examples

Under the Riemann hypothesis, the following inequality holds...

If P = NP, then the polynomial hierarchy collapses.

If your result relies on a someone else's result, you may write.

S. Else proposed in [1] the following conjecture

Conjecture [S. Else, 2014] ...

Proposition If Else's conjecture is true, then my result is correct.