Популярно о конечной математике и ее интересных применениях в квантовой теории

Следующая попытка: Taiwanese Journal of Mathematics. Их ответ был такой:

We do not have a full referee report, but quick opinions gathered suggest that it would be difficult to convince the editorial board to accept the article. Therefore, rather than begin a refereeing process that could take months, I am returning your manuscript to you now so that you have the chance to submit it elsewhere without delay.

Не буду комментировать этот ответ, но, во всяком случае, ответили через два дня и на том спасибо.

Еще одна попытка: Notre Dame Journal of Formal Logic. Ответа не было больше месяца и спросил о статусе:

Dear Professor Pillay,

The status of my paper is “With Editor” from November 6th, i.e. the paper is with the Editor for more than a month. I understand that the Editor is very busy. On the other hand, the paper is short (10 pages) and in my understanding rather simple. My understanding is that the paper is under review, right? Could you, please tell me when (even very approximately) the referee reports are expected.

    Thank you. Sincerely, Felix Lev.

Т.е., по наивности я думал, что раз назначен редактор, который держит статью больше месяца, то статья на рецензии. Но в течение менее часа получил такие ответы от главного редактора:

Dear Felix,

The Editor-in-Chief on the philosophy side (Mic Detlefsen) died in October. We have appointed a replacement who will take over his papers. There are papers submitted in July which have not been dealt with yet.

    Anand Pillay (Editor-in-Chief)

Dear Felix,

Actually I took the opportunity to look at your paper myself, and I can say quickly that it is not suitable for the Notre Dame Journal. The statement about $Z$ and the $Z/pZ$ (i.e. $F_p$) is obvious. (Also if you are interested there is a big literature about «pseudofinite» structures in logic. Easily found on google.)

So I will reject the paper.

Regards,

    Anand Pillay

Dear Felix,

As I said in the email to you I am rejecting the paper. Sorry.

    Anand Pillay
    Editor-in-Chief
    Notre Dame Journal of Formal Logic

Т.е. в первом ответе он как бы оправдывается, что из-за того, что бывший главный редактор, отвечающий за философию, умер, многие статьи задерживаются. Во-первых непонятно, почему он решил, что моя статья относится к философии. И непонятно, если статус был “With Editor”, то какому редактору послали. Ведь явно не тому, который умер, а тогда непонятно зачем он оправдывается. Но второй ответ пришел через 40 минут. Т.е., за эти 40 минут он посмотрел статью и решил отвергнуть. Из его ответа ясно, что до этого никто статью не смотрел, несмотря на статус “With Editor”. А третий ответ пришел через 5 минут.

Из его ответа ясно, что он статью не понял и не пытался понять, и, скорее всего, он не в состоянии понять. Я послал ему такое письмо:

Thank you for the info about your decision on my paper. I will not appeal the decision. However, let me note that when I send a paper to a journal, I am interested not only whether the paper will be accepted or not but also in knowing the opinion of qualified referees.

In fact, you were my referee and my understanding is that, although the formal status was “With Editor” for more than a month, nobody looked at the paper till Dec 10th, when it took you less than 40 minutes to come to the conclusion. From the formal point of view the reason of rejection was “The statement about Z and the Z/pZ (i.e. F

) is obvious.” And also you advise me to look at the literature on «pseudofinite» structures. I would be very grateful if you answer the following questions.

I understand that the statement is simple, have no doubt that for you the statement is indeed obvious and several mathematicians said the same. However, in my understanding, in mathematics the statement that something is obvious needs to be explained. Could you, please give me a direct reference where this statement is proved and how the limit is understood. You and several mathematicians told me that this is obvious from ultraproducts, «pseudofinite» structures etc. and I agree. However, those notions are rather sophisticated. My paper is titled “A new look at potential vs. actual infinity”. Those notions are discussed in the framework of actual infinity. The mentality of many mathematicians is that problems with characteristic 0 are fundamental while finite rings or fields can be used as something auxiliary for tackling those problems. My observation is that the majority of mathematicians do not care that standard mathematics has foundational problems (as follows e.g. from G?del's incompleteness theorems and from other considerations). My hope was that NDJFL does care about this.

My math professor was a famous mathematician M.A. Naimark, and I was very impressed by his lectures on calculus and group representations. As I note in the abstract, the technique of standard math involves only potential infinity while the basis does involve actual infinity: the theory starts with Z, then rational, real, complex numbers and sets with different cardinalities are introduced etc. As a rule, in mathematics legitimacy of every limit is thoroughly investigated but in standard math textbooks it is not even mentioned that Z is the limit of Z/p (by the way, Z/p=F

only if p is prime) and nothing is said on whether the limit is legitimate. The matter is that when Z/p is replaced by Z we arrive at standard math which has foundational problems.

I came to my ideas from physics where I proved that quantum theory based on finite math is more fundamental than quantum theory based on standard math: the latter is a special degenerated case of the former in the formal limit p??, and in my paper I argue that analogously, standard math is a special degenerate case of the former in the formal limit p??.

So I believe that the fact that Z/p?Z when p?? should be proved without reference to ultraproducts, «pseudofinite» structures etc. but directly by analogy with the proof that some sequence (a

)?? when n??. Unfortunately, this is not easily found in google and the majority of mathematicians prefer to work with Z from the beginning without caring whether or not Z is a limit of a finite set.

Основной смысл письма такой: в математике не должно быть аргумента, что просто что-то очевидно. Любое утверждение, что что-то очевидно должно доказываться или объясняться. И получил в ответ два письма написанные с интервалом 22 минуты:

I am not saying that your Statement 1 follows from some machinery such as ultraproducts, I am just saying that Statement 1 is obvious. Given integers a, b for all stuff. large primes p, a+b, a·b in Z coincides with a+b, a·b in F

.

Let me clarify. It is not just obvious it is a matter of definition. For a prime p, the field F

consists of elements 0,1….,p-1, with addition and multiplication modulo p. Namely for a, b in F

, a+b (in F

) is the remainder when a+b is divided by p. Likewise for a·b. So by definition, if a,b are in F

, then for p bigger than max of a+b, a·b, a+b and a·b coincide in Z and in F

.

Т.е., вначале он пишет, что это очевидно и пытается объяснить. А через 22 минуты уже пишет, что это не только очевидно, но и является определением.

Я написал большой ответ:

I am disappointed with the treatment of my paper at NDJFL. For more than a month the status was “With Editor”, and my na?ve 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

) 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

” (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

consists of elements 0,1….,p-1, with addition and multiplication modulo p. Namely for a,b in F

a+b (in F

) is the remainder when a+b is divided by p. Likewise for a·b. So by definition, if a,b are in F