## Are Natural Mathematical Problems Bad Problems?

One unique aspect of the conference “Visions in Mathematics Towards 2000” (see the previous post) was that there were several discussion sessions where speakers and other participants presented some thoughts about mathematics (or some specific areas), discussed and argued.  In the lectures themselves you could also see a large amount of audience participation and discussions which was very nice.

Let me draw your attention to  one question raised and discussed in one of the discussion sessions.

### 3.4 Discussion on Geometry with introduction by M. Gromov

Now, lets skip a lot of interesting staff and move to minute 23:20 where Noga Alon asked Misha Gromov to elaborate a statement from his opening lecture of the conference that  the densest packing problem in $R^3$ is not interesting.  In what follows Misha Gromov passionately argued that natural problems are bad problems (or are even stupid questions), and a lovely discussion emerged (in 25:00 Yuval Neeman commented about cosmology in response to Connes’s earlier remarks but then around 27:00 Vitali asked Misha to name some bad problems in geometry and the discussion resumed.) Misha made several lovely provocative further comments: he rejected the claim that this is a matter of taste, and argued that people make conjectures when they absolutely have no right to do so.

Misha argues passionately that natural problems are stupid problems

Actually one problem that Misha mentioned in his lecture as interesting (see also Gromov’s proceedings paper Spaces and questions), and that was raised both by him and by me is to prove an exponential upper bound for the number of simplicial 3-spheres with n facets. I remember that we talked about it in the conference and Misha was certain that the problem could be solved for shellable spheres while I was confident that the case of shellable spheres would be as hard as the general case.  He was right! This goes back to works of physicists Durhuus and Jonsson see this paper On locally constructible spheres and balls by Bruno Benedetti and  Günter M. Ziegler.

##### (Disclaimer: I asked quite a few questions that were both unnatural and stupid and made several conjectures when I had no right to do so.)

encore

Vitali Milman attacked the solution of the 4CT as “bad”and Segei Novikov disagreed and referred to the proof as “great”.

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### 4 Responses to Are Natural Mathematical Problems Bad Problems?

1. Gil Kalai says:

In my view the sphere packing problem both in dimension 3, in other specific dimensions, and asymptotically is a great problem. I do agree that sometimes deeper understanding of matters lead to less natural questions that are better. I do think that we can discuss the importance of questions while understanding the limitation of such discussions.

2. Jon Bannon says:

It is interesting that in the discussion nobody made the distinction between a natural question and a naive question. It is possible for a question to be natural (in the sense of being un-forced) while sitting on a substantial edifice of prior work. One only needs to look at 20th century algebraic geometry and the work of Grothendieck to see that this distinction is meaningful. I’m thinking particularly of the sense that Grothendieck accumulated a vast collection “trivial” observations from which something highly nontrivial surfaced. Of course Gromov was being provocative, but it is pretty clear that his notion of natural actually means naive…as being accessible to the average mathematician on the street…

• Gil Kalai says:

Dear Jon, Maybe Gromov referred to “natural” in response to the way people in the audience referred to some specific questions like the spacial sphere packing problem and the four color problem. How do you regard, Jon, the problem of densest sphere packing in R^3? as natural? as naive? both? neither? (likewise 4CT?)

• Jon Bannon says:

Well, I personally think that a problem is interesting if working on it generates a lot of new and interesting mathematics in the community. Problems that are encountered from many directions that are holding up progress in understanding many areas are good problems, provided new and surprising things come from investigating these problems. A problem is a seed from which new mathematics grows, I guess, playing the role of focusing the attention of the community on the space of facts surrounding the problem. This is dependent on the time and place of the problem, and if there is no approach or fruitful ideas that come from the problem I am not sure such a problem is good for mathematics, because students and researchers will waste too much time and effort being stuck rather than developing things in the landscape.

The late Ken Appel gave a very nice talk about computer proofs before he died, and I wish I had followed up with him to get his slides. He clearly had thought quite a lot about the nature of proof (in light of his and Haken’s computer-aided proof of 4CT), and he made the claim that is is very likely that many statements, even simple ones, may be far to complicated to prove by humans. If this is the case, then one of the central concerns of pure mathematics may be to find “naturality” in mathematics. Perhaps mathematicians should not even pay attention to those results which are likely to require computer assisted proofs. This said, it is easy to imagine a situation where we develop very intuitive automated theorem provers and proof assistants which allow us to function much like Grothendieck…maybe there will be some kind of way to aid our intuition about making conjectures if we set the provers to verify/prove many very natural things in order to allow us to look at the resulting landscape in order to more easily make conjectures. (Totally science fiction, I know, but the idea is to imagine what computers can meaningfully do for the day to day practice of imaginitive mathematics…)

My own view of sphere packing and 4CT is colored by the above opinion that naturality is the mark of mathematics as a human practice, even if this is “local naturality” in that we work out what is evident while standing on prior facts…this is not synonymous with naive questioning by the mathematician on the street…but it may be synonymous with relentless naive questioning by the relevant mathematical specialist on the street! I think that these questions initially were asked in a time where the belief that simple questions should admit simple (or at least understandable to humans) solutions, and in light of developments in contemporary mathematics this may not work out (P vs NP and all…) I’d surely like to see a human proof of these things, provided there is a viable approach that doesn’t reduce obviously to tilting at windmills.