“..a strong conviction that they always result from a fairly systematic process — and that the opposite impression, that some ideas are incredible bolts from the blue that require “genius” or “sudden inspiration” to find, is an illusion that results from the way mathematicians present their proofs after they have discovered them.”

Hmm, it will be quite exciting to discuss it. I do not think the issue is how proofs are presented. Sudden developments are not in contradiction with systematic processes (just think about graph processes.) And also there are, from time to time, inspirational surprising sharp turns.

“The third thing I’d like to do reflects another interest of mine, which is avoiding duplication of effort. I’ve spent a little time thinking about whether there is a cheap way of getting a Behrend-type bound for Roth’s theorem out of these ideas (and I’m not the only one). Although I wasn’t expecting the answer to be yes, I think there is some value in publicizing some of the dead ends I’ve come across.”

This is also a very interesting topic. Duplicating efforts and certainly competition can be beneficial, but presenting or hearing (sometimes) paths of attacks that failed can be interesting and of some value.

]]>Written from the perspectie of an AS Maths/Further Maths student.

]]>The classification of ideals depends on the structure of the ring. For functions on F_3^n over a finite field of characteristic not 3, the ring is a product of fields corresponding to characters, and any ideal just is a sum of fields corresponding to some subset of the characters – this is just Fourier analysis.

In characteristic 3, the ring is generated by the translation operators x_i =T_{e_i}-1 in the directions of a basis, each of which satisfies (T_{e_i}-1)^3 = 0, so the ring is F_3[x_1,…,x_n]/(x_1^3, …,x_n^3) – a nilpotent ring. This ring has many ideals which don’t really have a nice classification. The space of low degree polynomials corresponds, dually, to high degree polynomials in the x_i. You could restrict attention to special ideals, like ideals generated by polynomials – these correspond to the spaces of polynomials with all monomials appearing having multidegree in a certain downward-closed set of tuples of natural numbers.

However I think you can show that all spaces of functions that have the key property needed for this proof are quite close to the space of low degree polynomials.

]]>To me though this seems like just the start. I typed ‘dissemination of mathematics’ into Google about six months ago and I’ve just typed it in again. Much the same has happened this time, although there are several new sites of interest. Basically, you get resources and initiatives concerned with mathematics education, some of which like http://iccams-maths.org/ seems like great stuff at a glance. Another site I found the other day in a similar vein was https://undergroundmathematics.org/, although I came to this a different way.

This kind of thing isn’t about the dissemination of mathematics amongst professional mathematicians themselves, however. Dissemination sideways if you like. Rather it’s about dissemination from the top down, if that’s the right way of putting it. It’s worth mentioning that this is a very laudable goal and anyone who’s ever taught mathematics at any level should appreciate these initiatives.

But what about dissemination sideways? To me to be able to fully disseminate mathematics sideways means that it must be formalised, however unfashionable amongst the majority of mathematicians this continues to be. The reason is that with formalised mathematics you have a corpus that is discoverable and usable, although I’m searching for the words here. I don’t think that formalisation is an end in itself and nor do I think that a formalised proof is necessarily any more worthy just because it has been verified by a computer. It just seems that it’s the only true place to start.

I think that mathematics that has been formalised has a structure and nature that somehow makes it much more amenable to being discoverable, reusable, searchable, etc. I got very excited when I watched a talk https://www.youtube.com/watch?v=Is_lycvOkTA by Thomas Hales again recently, since he seemed to be striking similar chords in places.

The arxiv, open-source web-based journals and an increasing number of on-line mathematics databases are fantastic but to me they still seem, because of their interfaces, to some degree like mathematics at a distance, again I’m searching for the words. They’re like Wikipedia on steroids. The content is more refined and detailed, so much so that it’s hardly a comparison, but the means for interactivity and collaboration are still limited. These sites might serve as resources for a mathematician taking part in a Polymath project, for example, but they do not serve as its context or arena. They afford the learning of mathematics amongst mathematicians, which is great obviously, but I think they are limited in affording the *doing* of mathematics. Both of these are needed for discovery of new mathematics or improvement of existing mathematics, I think.

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