Популярно о конечной математике и ее интересных применениях в квантовой теории

I am disappointed with the treatment of my paper at NDJFL. For more than a month the status was “With Editor”, and my naïve expectation was that somebody is reading the paper. However, only after my query you spent 40 minutes or less and the only reason of rejection was “The statement about Z and the Z/pZ (i.e., F_p) is obvious”. In mathematics the statements that something is obvious should always be explained but I thought that since Editor-in-Chief of such a prestige journal makes such a statement then maybe indeed it is extremely obvious, and I don’t understand something trivial. That’s why I wrote that I would be very grateful if you explain your words. However, I was amazed by your response.

In the first email you continue to state that Statement 1 is obvious: “Given integers a, b for all stuff. large primes p a+b, a·b in Z coincide with a+b, a·b in F_p” (probably “stuff.” is a misprint of “sufficiently”) but after 22 minutes you wrote another email:

“Let me clarify. It is not just obvious it is a matter of definition. For a prime p, the field F_p consists of elements 0,1….,p-1, with addition and multiplication modulo p. Namely for a,b in F_p, a+b (in F_p) is the remainder when a+b is divided by p. Likewise for a·b. So by definition, if a,b are in F_p, then for p bigger than max of a+b,a·b, a+b and a·b coincide in Z and in F_p.", so now you are saying that this is the matter of definition.

My first remark is technical. The problem deals only with rings and has nothing to do with division. So, it is not necessary to consider the field F_p and only primes. The problem is whether Z is the limit of rings R_p=(0,1,…p-1) (with operations modulo p) when p→∞.

Again, in mathematics, mathematical statements should be formulated unambiguously such that different interpretations should be excluded. For this reason the words “for any” and “there exist” are often used in mathematical statements. However, saying about a and b you are not using those words, and one can only guess what you mean. Consider you “definition” “So by definition, if a,b are in F_p, then for p bigger than max of a+b,a·b, a+b and a·b coincide in Z and in F_p.” literally. For example, if a=0 and b=0 then for p > 0, a+b and a·b coincide in Z and in F_p. Or if a<10 and b<10 then for p > 100, a+b and a·b coincide in Z and in F_p.”. So your “definition” is indeed obvious.

I guess that probably you meant something like this: for any p₀ there exists a set S such that for any a+bϵS,a·b ϵS, a+b and a·b coincide in Z and in F_p for any p >= p₀, and card(S)→∞ when p₀→∞.

However, even if my guess is correct, this still cannot be a correct definition that R_p→Z when p→∞. The definition should be such that not only for two elements from S their sum and product coincide in Z and in R_p but that it is possible to find a number n such that for any m<=n the result of any m operations of multiplication, summation or subtraction of elements from S should be the same in Z and in R_p, and that n→∞ when p→∞.

The exact formulation of the definition is given in my paper, and I prove that with this definition indeed R_p→Z when p→∞. As I said, the definition should be to some extent analogous to the definition that the sequence (a_n) →∞ when n→∞: for any M>0 there exists n₀ such that a_n>=M for any n>= n₀.

I asked several mathematicians to give me a reference where this is proved but nobody gave such a reference. The response of some of them was analogous to yours: this is obvious. Then I asked that if this the case then why in mathematical textbooks this is not even mentioned and standard math starts from Z from the beginning, but again no response. As I wrote, they don’t care that standard math has foundational problems (as follows e.g. from Gödel’s incompleteness theorems and other considerations). But when I asked Prof. Zelmanov (who is the Fields Medal laureate) he did not say that this is obvious and advised me to look at Terence Tao’s blog where ultraproducts are considered. In my paper I thank Prof. Zelmanov for his advice and refer to the blog.

Technically indeed it is possible to prove that R_p→Z follows from the results on ultraproducts although in ultraproducts they consider only fields and their goal is to use finite fields for proving some features of fields of characteristic zero. Nevertheless, this is not a direct proof, and the construction is rather sophisticated.

In summary, I think that, with the probability 99.99 %, in the literature there is no direct proof that R_p→Z when p→∞ and so my proof is new. Let me note that my paper contains not only this result: I explain that this result is the first step in proving that finite math is more fundamental than standard one: the latter is a special degenerated case of the former in the formal limit p→∞.

However, it seems obvious that you even did not try to carefully read my proof and other results of the paper. You noticed that I prove that R_p→Z, immediately (within minutes) decided that this is obvious (as you say, even without the machinery of ultraproducts) and immediately wrote a rejection. I am amazed that the attitude to my paper at such a prestigious journal was on such a level.

For me it is not a great tragedy that my paper will not be published in NDJFL. I have no doubt that the results are fundamental, they will be acknowledged sooner or later and published elsewhere. However, I treat such an attitude to me as disgraceful from the professional point of view. Such an attitude in fact means that you treat me as unprofessional who submitted to NDJFL a junk which does not deserve consideration.

Of course you have a right to have such an opinion. However, if you think that your attitude was a mistake I would be grateful if you tell me this and will be fully satisfied. I understand that we are only people, everybody makes mistakes, you are very busy handling such a journal, you have to look at many papers and probably some of them are indeed junk, so probably mistakes in your work are inevitable. However, decent people acknowledge that they make mistakes when this becomes obvious.