Популярно о конечной математике и ее интересных применениях в квантовой теории

Никто не спорит, что аппарат классической математики очень мощный и во многих случаях (но не всех) работает с очень большой точностью. И в таких случаях совершенно незачем применять конечную математику. Ситуация полностью аналогична тому, что есть в физике. Например, нерелятивистская теория является частным вырожденным случаем нерелятивистской в формальном пределе c?? (где c – скорость света), но в повседневной жизни нерелятивистская теория работает с очень большой точностью и тогда незачем применять релятивистскую теорию. Аналогично, классическая физика является частным вырожденным случаем квантовой в формальном пределе h?0, где h – постоянная Планка. Но в тех случаях когда классическая физика работает с большой точностью, незачем применять квантовую теорию; например, незачем описывать движение Луны уравнением Шредингера; в принципе это возможно, но приводит к неоправданным усложнениям. В своей статье я пытался на популярном уровне объяснить результаты своей книги. Ясно, что книга могла выйти только после одобрения высококвалифицированными рецензентами. И даже из названия книги ясно, что ней подробно приведены аргументы в пользу конечной математики. Статья полностью соответствует всем критериям editorial policy Вашего журнала. Однако, статья отвергнута с такой рецензией:

И я привожу эту рецензию.

Т.е., менталитет рецензента такой, что раз классическая математика во многих случаях работает, то незачем философствовать и применять что-то другое. Как я пишу в статье, такая философия у многих. Рецензент пишет, что должны быть аргументы в пользу применения другой математики. Но весь смысл статьи как раз в том, чтобы привести такие аргументы и ясно, что рецензент даже не читал внимательно статью. От статуса "Under review" до "Reviews completed" прошел всего один день так что рецензия была написана очень быстро. Статья была в редакции больше месяца и я надеялся, что за это время будет подготовлена серьезная рецензия. Однако, из рецензии ясно, что рецензент даже не читал статью внимательно.

Я только что отправил Prof. Fernando Gouvea мой appeal который приложен. В editorial policy ничего не говорится про appeals, но это обычная практика, что автор имеет право на appeal. Надеюсь, что мой appeal будет рассмотрен.

Заранее благодарен.

    С уважением, Феликс Лев (Felix Lev).

Его ответы были такими:

Dear Dr. Lev:

Let me start by asking you to communicate with the magazine in English: I am the only member of the editorial board who speaks Russian, and we conduct all the magazine-related business in English. Thank you for your understanding.

Concerning your article and the editors" decision, let me assure you that the referee is a highly qualified mathematician who had studied your article in detail. Obviously, he will remain anonymous to you, but let me say that, as an author, I'd be very happy to have a reviewer of this caliber and quality. Let me be clear: we firmly stand by his recommendation to reject the article.

Let us look at your arguments.

You claim that "любое явление, которое объясняет классическая математика, в принципе может быть объяснено с любой точностью в конечной математике, если p очень большое. Но есть и явления, которые могут быть объяснены только если p конечное, а не бесконечное." (every phenomenon explained by classical mathematics, in principle, can be explained with arbitrary precision in finite mathematics, if p is very large. But there also exist phenomena that can be explained only if p is finite, and not infinite).

What does this statement mean? Perhaps your book provides enough detail, but it is not at all clear from your article. You illustrate it by examples, claiming that arithmetical identities such as 10+20=30 are ambiguous, whereas 10+20=30 (mod 40) or 10+20=5 (mod 25) are not. These examples are unconvincing, and they do not clarify the meaning of the general statement above. More generally, you describe the simple relations between the rings Z/Zp and Z as an argument toward this general statement; we find this unconvincing as well.

You say that the purpose of your article was to popularize your book. Unfortunately, this goal was not achieved: it is not clear from your article whether the approach that you promote is capable of obtaining new results or of consistently explaining known results in a new way. Perhaps one needs to read your book to make sense of your theory and to appreciate it, but your article comes short of being compelling.

The final paragraph of your article (in bold) is its main message. In our opinion, to convince the reader of the truth of this credo, and even to make precise sense of it, would need much more elaboration than presented in your article.

Sincerely yours,

    Sergei Tabachnikov
    The Mathematical Intelligencer
    Professor of Mathematics, Penn State

И затем он послал второй ответ:

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

P. S. Thank you for the note about D. B. Fuchs. He is 81 now, and we continue our collaboration, working on a joint paper now.

Мой ответ на его письма был такой:

Dear Sergei,

Thank you for your response to my detailed letter. However, you probably will not be surprised if I say that I am disappointed with your response. You asked whether my approach “is capable of obtaining new results or of consistently explaining known results in a new way”. I was very glad that you asked this question and hoped that you will read my response. But now I am not sure that you were interested in my response at all and probably you decided from the beginning that the answer is negative.

I tried to answer your question in such a way that (in my understanding) the answer should be appreciated and understood by any mathematician, even by students of mathematical departments. For example, I give a popular explanation why in modular mathematics I have one irreducible representation (IR) which splits into two IRs in the formal limit p??. This (mathematically beautiful!) example immediately shows that, even from a pure mathematical point of view, modular theory is more general than standard one. Since you said nothing about this example then either you even did not read it at all or were unable to understand it.

I also give other simple MATHEMATICAL examples which show that there are cases when modular theory can solve problems which standard theory cannot. However, again, no specific comments on those examples are given, and so you either did not read those examples or were unable to understand them.

I understand that everybody has his/her own problems, and nobody can insist on what other people should or should not read. But it is beyond any logic that you asked a question and said nothing explicit about my response. You say: “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.”

In my paper and the last letter, I give many simple MATHEMATICAL arguments but neither you nor the referee give any comments on these arguments. And so again, you either did not read the arguments or were unable to understand them. In the literature, criticism is defined as “the practice of judging the merits and faults of something”. But since there is no sign that you and the referee tried to understand my arguments, the word “criticism” in your letter is fully inappropriate. If it were only a discussion between two people, then everybody has a full right to read or not to read what he/she wants. But, in the given content, your opinion is understood not only as your personal opinion but as the opinion of the readers of your journal. I am not sure that your understanding of this opinion is realistic. For example, several physicists and mathematicians told me that they would be interested in reading a popular discussion of my approach. You say, “It well could be that your subject is too technical to be explained in an expository article in a magazine for general mathematical audience.” But I just tried to explain my results in an extremely popular (expository) level, and, in my understanding, this is fully what the editorial policy requires.

You explain to me that “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”. According to the present knowledge, quantum theory is the most general model of nature which mankind has developed. So, in the content of your letter, your words can be understood only such that you do not think that my theory is elegant and capable of explaining the relevant phenomena. But in your letter, as I already noted, I do not see any sign that you tried and/or were able to understand what my theory is capable of.

Now let me comment on the following extract of your letter: “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.”

In my paper and the last letter, I note that (during the last 80 years) there is a great problem that, by using the existing mathematics, physicists cannot construct a quantum theory which is mathematically consistent and can explain many existing experimental data. This is acknowledged by famous scientists and even Nobel Prize laureates (as I noted, even one of them wrote a paper titled “Living with Infinities”). Also, many authors and even some Nobel Prize laureates wrote papers conjecting that the ultimate quantum theory will be based on finite mathematics. Of course, constructing such a theory would take an enormous amount of work. However, in your opinion “, even if successful it will have little bearing on modeling nature” and “there is no need to rebuild the foundations”.

So, you do not know about existing fundamental problems, do not have ideas how to solve them, the opinion of famous scientists is not important to you, but your opinion is that “there is no need to rebuild the foundations”. So, your remarks are like those from the known Chekhov’s story “Letter to a learned neighbor” when a man writes to his neighbor: “You say that there are spots on the Sun; this cannot be because this can never be”.

You note that there are even similarities between our approaches because both start from a discrete approach. You will probably be indignant if someone, without any attempt to figure it out, says that your results are only declarations. You will probably say that you already have recognized works on this topic and this topic is related to generally recognized problems. But I can also say that I have papers in so-called prestigious journals, there are many other papers on this topic, and even Nobel Prize laureates wrote about this.

In summary, you asked me a question, I tried to answer this question in detail, there are no sign that you and the referee read my arguments and/or were able to understand them, but you say that my paper “is essentially just a declaration”. Giving negative statements about my arguments without any explicit mentioning them, contradicts scientific ethics and is simply indecent. I understand your last letter such that you do not want to spend any time on our discussion, and I also think that your attitude is such that any further correspondence is meaningless.

    I wish all the best to you and your journal. Felix.

P. S. Your statement that “both the Galilean and the Lorentz transformations are parts of mathematics on equal footing” is not correct. As explained in the famous Dyson’s paper “Missed opportunities” published in 1972 in Bull. Amer. Math. Soc., the Lorentz group is more general than the Galilei one because the latter can be obtained from the former by contraction c??. In turn, being semisimple, the Lorentz group (it is more correct to talk about its covering group SL(2,C)) has a maximal possible symmetry and cannot be obtained from a more symmetric group by contraction. And, as I tried to explain in the paper, finite mathematics is more general than classical one because the latter can be obtained from the former by contraction p.

Как обычно, я привожу эту длинную переписку, понимая, что вряд ли кто-то захочет всю ее читать. Но я должен привести эту переписку, чтобы не сказали, что я тенденциозен, утверждая, что рассмотрение моей статьи не соответствовало ни editorial policy ни научной этики. Действительно, Сергей Табачников пишет, что мои утверждения unconvincing и что мои аргументы – только declaration.

Не знаю, понимает ли он, что такие утверждения без всякой попытки обосновать их, противоречат научной этике. И еще, как я пишу в своем ответе, его слово "criticism" не соответствует своему значению. Определение "criticism" такое: "the expression of disapproval of someone or something based on perceived faults or mistakes". То есть, предполагается, что даны какие-то аргументы. А его письма не содержат никакого намека, что он хоть в чем-то пытался разобраться. И рецензия тоже показывает, что, как я отмечаю, рецензент даже не прочитал внимательно статью. Вначале он пишет, что ему интересен ультрафинитизм, потом пишет, что реально можно что-то посчитать только с действительными числами, а если не так, то должны быть аргументы. Но весь смысл статьи, чтобы привести такие аргументы и, похоже, он этого даже не понял.

Моя следующая попытка – Archiv der Mathematik, и ответ Editor-in-Chief Ralph Chill такой:

"…We are aware of your having discussed this paper with Clemens Heine, and we have valuated the paper by ourselves. It is true that the paper is of general nature, and could be of interest for a broader audience, but we feel that the paper does not fit into this particular journal. We want to encourage you to submit your paper to a another journal, where it certainly will find its place…

То есть, он признает, что "It is true that the paper is of general nature, and could be of interest for a broader audience… ". То есть, он фактически признает, что статья полностью соответствует editorial policy. Но, почему-то, "but we feel that the paper does not fit into this particular journal". И он ведь математик, а не поэт, поэтому когда он апеллирует к своим чувствам, то это странно. Казалось бы, вопрос очень простой: соответствует статья editorial policy? Да или Нет? Ясно, что я написал appeal, но он на него не ответил. То есть, опять-таки, он, наверное, не понимает, что такой ответ противоречит научной этике.

Моя следующая попытка – Expositiones Mathematicae. Их editorial policy такая:

Our aim is to publish papers of interest to a wide mathematical audience including graduate level students. This is a peer-reviewed journal that publishes papers in all branches of Mathematics under the headings "Main Articles" and "Mathematical Notes":