Популярно о конечной математике и ее интересных применениях в квантовой теории - Феликс Лев

статью, где эти проблемы обсуждаются на популярном уровне. Есть несколько известных журналов, которые публикуют популярные статьи по математике, и мне казалось, что такая статья будет этим журналам интересна.

Моей первой попыткой был журнал "The Mathematical Intelligencer". Editorial policy журнала говорит, что они не принимают обычный математический стиль теорема-доказательство, т.е., все должно быть на популярном уровне для широкой аудитории. Один из главных редакторов – Сергей Табачников, который закончил мехмат МГУ. Когда я учился в МФТИ, то некоторые думали, что мехмат МГУ – чуть ли не высшая каста. В связи с той проблемой, которую сейчас обсуждаю, мне было интересно знать мнение математиков, т.к. мне казалось, что уж им очевидно что такое конечное кольцо или конечное поле.

Рецензия на мою статью была такая:

Reviewer 1: I have read the article, and do not recommend publication. I am in principle very interested in things like ultrafinitism or questioning the role of the real numbers in physics, but this article struck me as having very little to say about such matters that wasn't too obvious to count as a genuine contribution. For instance, everybody understands (or at least all serious mathematicians and physicists understand) that infinite precision is not possible. So we use the real numbers not because we think that they map directly on to reality, but because it turns out to be convenient to do exact calculations within the real number system, obtain exact answers, and then use those exact answers to make predictions that can be verified, not exactly of course, but often to a high degree of precision. An argument against the real numbers has to offer some advantage of using a different system.

Ясно, что я написал appeal:

Author’s appeal on Editorial Decision

The decision to reject my paper was based on the advice of Reviewer 1, and there were no other referee reports. The motivation of Reviewer is as follows.

Reviewer says that “everybody understands that infinite precision is not possible” and that “So we use the real numbers not because we think that they map directly on to reality, but because it turns out to be convenient to do exact calculations within the real number system, obtain exact answers, and then use those exact answers to make predictions that can be verified, not exactly of course, but often to a high degree of precision.”

At this point, my approach is completely the same as the approach of Reviewer. However, Reviewer concludes the report with the sentence: “An argument against the real numbers has to offer some advantage of using a different system.”, and only this sentence is the reason for the advice to recommend rejection.

This sentence shows that Reviewer even did not carefully read the paper. From the very beginning of the paper, I explain that mathematics with infinitesimals cannot be universal. For example, as I note, many physicisists “… say that, for example, dx/dt should be understood as Δx/Δt where Δx and Δt are small but not infinitesimal. I ask them: but you work with dx/dt, not Δx/Δt. They reply that since mathematics with derivatives works well then there is no need to philosophize and develop something else.” Thus, the mentality of these physicists on the application of real numbers is the same as the mentality of Reviewer.

I fully agree that mathematics with infinitesimals is very powerful in many applications. However, I note that “The development of quantum theory has shown that the theory contains anomalies and divergences.”

The idea of the paper is to explain on popular level the results of my monograph “Finite mathematics as the foundation of classical mathematics and quantum theory…” recently published by Springer. Even the title of the monograph shows that “advantage of using a different system” is discussed in detail, and in the manuscript, I explain on popular level why finite mathematics is more general (fundamental) than classical one. I note that my results are fully in the spirit of the history of science. For example, nonrelativistic theory works in many cases with a very high accuracy, but it cannot explain phenomena where it is important that the speed of light c is finite and not infinitely large. I note that, analogously, in nature there are phenomena (e.g., gravity) which can be explained only in the framework of finite mathematics where it is important that the characteristics p of the ring is finite and not infinitely large.

The Reviewer's remarks show that he/she is completely unfamiliar with the fact that the problem of infinities is one of the main problems of quantum theory and many famous scientists wrote that fundamental quantum theory should be based on finite mathematics.

In summary, the Reviewer’s advice to recommend rejection is completely unfounded. My paper satisfies all the requirements specified in the editorial policy of “The Mathematical Intelligencer”. I would be grateful if the Editorial decision is reconsidered.

Так как на этот appeal долго не было ответа, я написал Сергею Табачникову по-русски:

Уважаемый Сергей,

Решил написать Вам по-русски о моей статье, которая только что отвергнута в “The Mathematical Intelligencer”. Проблема не в том, что она отвергнута, а в том на каком уровне она отвергнута. Вначале очень кратко о себе. Я закончил МФТИ, в России защитил кандидатскую и докторскую и работал в Дубне. У меня много статей в известных журналах, а недавно Springer опубликовал мою монографию “Finite mathematics as the foundation of classical mathematics and quantum theory. With applications to gravity and particle theory”. Более подробно данные обо мне есть в моем ORCID: https://orcid.org/0000–0002–4476–3080.

Одна из основных проблем квантовой теории в том, что теория построенная на классической математике (с бесконечно малыми, непрерывностью и т.д.) приводит к расходящимся выражениям (проблема бесконечностей). Поэтому многие известные ученые предлагали, что самая общая (фундаментальная) квантовая теория должна быть построена на конечной математике. В книге строго доказано, что классическая математика является частным вырожденным случаем конечной математики в формальном пределе p→∞, где p – характеристика поля или кольца в конечной математике. Смысл