Terence TaoOne of the quirks of professional mathematics is that researchers are discouraged from speculating too far beyond the range of what they can actually prove. To quote Minhyong Kim (from https://mathoverflow.net/a/38694/766): "it's almost as though definite mathematical results are money in the bank. After you've built up some savings, you can afford to spend a bit by philosophizing. But then, you can't let the balance get too low because people will start looking at you in funny, suspicious ways."
It can take some conscious effort for junior mathematicians, once they actually have earned enough "theorem credits" to afford to speculate, to actually venture opinions and make broader conclusions. Easier to play it safe and only stick to what their theorems and results can objectively verify. While this does cut down on a lot of nonsense, I sometimes wonder if we should have more spaces to encourage mathematical speculation.
We do have the mechanisms of posing open problems and formulating conjectures, which work well enough, but we don't really have a culture of throwing out less precise, but still informed, speculation on where a subject might be headed. (Except perhaps during conference teas and dinners. Which is one good reason to attend such conferences... .)
Yangfeng JiThanks for all the responses! It may help to distinguish two types of speculation (cf. https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/): "pre-rigorous speculation", in which one asks "dumb" questions (cf. https://terrytao.wordpress.com/career-advice/ask-yourself-dumb-questions-and-answer-them/) before fully knowing the field, and "post-rigorous speculation", in which one shares informed insights and opinions from one's rigorous understanding of the field.
I think we should encourage both types, in appropriate venues of course (and separated from traditional "rigorous" work).
Enrique GME@tao Might you please suggest a space for post-rigorous speculation - maybe even for definitively provable or disprovable observations that might lead to interesting work and which do not appear to have been made in the literature? Not the stack platform please.
@Egarcia I think there have been several good suggestions already in this thread. I myself am partial to including Conclusions and Discussions sections in papers (and talks). In February at a conference http://www.ipam.ucla.edu/programs/workshops/machine-assisted-proofs/ that I am co-organizing, we will have a panel to discuss the future of the field. Occasionally we do solicit "perspectives" articles from experts, e.g., https://www.amazon.com/Mathematics-Frontiers-Perspectives-V-Arnold/dp/0821826972
