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... .)
Thanks 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
John Carlos Baez@tao - One "safe space" for airing less precisely formulated speculations seems to be blog articles. But blogging seems to have declined in popularity with young mathematicians.
Another is Twitter, though I've mainly seen category theorists speculating there, not so much other kinds of mathematicians.
I hope Mathstodon will become another place for informed speculation!
DANIEL POLLACK@johncarlosbaez @tao It is interesting to contrast the tone of what counts as informed speculation among mathematicians compared to scientists in other fields. Mathematicians seem to be less comfortable getting out of the comfort zone of what they can (at least conceivably) prove. Chemists, Astronomers, Physicists, Biologists and Statisticians all seem able to engage with others in their discipline more broadly outside of their immediate areas of expertise. The culture of rigor/proof seems to stifle many mathematicians from voicing their insights in a broader context. IMHO this is a loss for the entire scientific community,
Martin Escardo@johncarlosbaez @tao I very much hope everybody here, including students, feel safe to speculate, with the senior people reacting nicely, even if they have grounds to disbelieve the speculations. It is an important part of a mathematical career to learn how to speculate, and the only way is trying. Sometimes we will be successful, sometimes we will fail badly, and sometimes we will instead inspire variations of our thoughts.
fresh@tao Yes, but as soon as someone has left the circus, the ideas are gone, as well. I have a representation for Lie algebras that is trivial for semisimple ones, and non-trivial for solvable ones. It is not really complicated
A(L):={f | [f(X),Y]+[X,f(Y)]=0} and X.f:=[ad X,f].
This is again a Lie algebra and therefore opens the doors to really many cute toys. That little thing looks like a nice key to treat the solvable world better than it is now, but I cannot find the damn door lock.
A(L):={f | [f(X),Y]+[X,f(Y)]=0} and X.f:=[ad X,f].
This is again a Lie algebra and therefore opens the doors to really many cute toys. That little thing looks like a nice key to treat the solvable world better than it is now, but I cannot find the damn door lock.
Avani@tao
This pattern seems to recur throughout the sciences. I am thinking of Nobel winners like Crick who have gone on to ponder consciousness, or, more recently, the shap direction George Church's work took once he was established. Countless others must be thinking exciting, blue sky thoughts, but living in fear of the opinions of their next NSF panel. The only exceptions I've seen are some people who have stable private funding, though I don't see that as a scalable solution.
This pattern seems to recur throughout the sciences. I am thinking of Nobel winners like Crick who have gone on to ponder consciousness, or, more recently, the shap direction George Church's work took once he was established. Countless others must be thinking exciting, blue sky thoughts, but living in fear of the opinions of their next NSF panel. The only exceptions I've seen are some people who have stable private funding, though I don't see that as a scalable solution.
Arthur Détaille@tao personnaly as a student, I find it very helpful to face my contradictions or learn by making huge mistakes. My professors push us to speculate, and I find it very very rewarding to be wrong (or right) in this safe environment
Florian Felix@tao I really like that my advisor basically constantly encourages to just draw a completely wild picture and see where it leads.
Volker K. 🏳️🌈@tao This reminds me on this talk by Atiyah 2018 in Heidelberg, where "he claimed to have solved the Riemann hypothesis, Hilbert's eighth problem, by contradiction using the fine-structure constant." [Wikipedia]. Of course nobody objected given the age and the contributions Atiyah made during his lifetime (some non-technical journalists fell for his numerous claims, though).
Minimaliste13@tao If a young and/or unkonwn mathematician was to propose an amazing conjecture accounting for several mysterious things and predicting heretofore unnoticed phenomena, would we really frown at him?
Based on the experience of Beilinson when he stated his conjectures in the late 70's, I would think that on the contrary the community would react very positively.
@minimaliste13 Conjectures are indeed one "socially acceptable" mechanism for mathematical speculation, but perhaps there should be others as well. One modest change in culture, for instance, would be to encourage "Conclusions" sections in papers or talks, which we usually don't do in pure mathematics for some reason.
@tao It's true that very few math articles do that simple thing which is common in several other fields: outline possible future research and/or ask questions (Barry Mazur does that very regularly, but he stands out in that respect among people I have read).
Dan Goodman@tao I remember at a department dinner during my maths PhD I suggested to a table of eminent mathematicians that it might be interesting to have a journal focused on speculative but informed mathematical ideas. Total silence around the table until one of them said "you mean like a biology journal?". Guffaws all round.
I'm now a neuroscientist. 😉
Jon Awbrey@tao From the same thread ...
https://mathoverflow.net/questions/38639/thinking-and-explaining#comment90487_38639
@tao In a related story ...
