Популярно о конечной математике и ее интересных применениях в квантовой теории

Dear Felix, I have finally read your letter; sorry for the delay, it's a very busy time for me.

I am not a physicist, and I am not familiar with the culture of the physics community, so I cannot comment on the phenomenon that you lament about: lack of acceptance, or even a meaningful criticism, of your theory. What I can try to comment upon is the mathematical side of the discussion.

In my opinion, everything in mathematics can be used to create models of natural phenomena, be this the classical differential calculus or calculus of finite differences, be this standard or nonstandard analysis (in which infinity is not a limit), be this based on the right of integers or the modular arithmetic (the rings Z/pZ), be this classical or constructivist logic. From the mathematical point of view, one needs to obtain a consistent and, preferably, elegant theory capable of explaining the relevant phenomena in the framework of the model at hand. From this point of view, both the Galilean and the Lorentz transformations are parts of mathematics on equal footing, although RT provides a more accurate description of the nature than NT.

My criticism of your article – and in it I agree with the referee – is that it essentially just a declaration that one can build quantum theory based on the rings Z/pZ, but it doesn't provide examples or any details. Perhaps one needs to read your book to appreciate your approach, but one cannot expect the reader to be familiar with the book. It well could be that your subject is too technical to be explained in an expository article in a magazine for general mathematical audience.

You also seem to claim that mathematics could be rebuilt starting with the rings Z/pZ, instead of Z (Peano arithmetic). This may be the case, but such an undertaking would take an enormous amount of work and, in my opinion, even if successful it will have little bearing on modeling nature. As I said, no mathematics is off limit if it's relevant in description of nature, and there is no need to rebuild the foundations for this purpose.

It may not be directly relevant, but let me mention something that is close to my research interests. Recently, the field of discrete differential geometry has emerged, and it continues to be an active research area (the name itself is an oxymoron). The situation is somewhat similar to what you described: instead of smooth objects, such as curves and surfaces, one studies discrete ones (polygons, polyhedra), and the former can be obtained from the latter as the limiting objects. Btw, this discrete differential geometry is intimately related with completely integrable systems, which are so common in mathematical physics.

These are my thoughts.

Best regards, yours Sergei