A million plus one equals a million. Theory of Relativity

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To the reader


Despite a certain number of smart words and surnames, the article is quite accessible to the perception of nemathematics. Despite the provocative title, the article is not frictional. Read on health.

Prologue


The beginning of the twentieth century was rich in revolution - both political and scientific. For example, then the axiomatization of mathematics was in full swing. It happened violently, dramatically. Cantor's “naive theory of sets” was buried by the Russell paradox, the limited axiomatics of Zermelo-Frenkel showed, already in the thirties, Gödel’s incompleteness theorem.

In physics, a special theory of relativity made a revolution. The discovery of Einstein, based on the work of Maxwell, Lorentz and other scientists, postulated some counterintuitive properties of physical reality, in particular - the Lorentz velocity addition. It turned out that if runner B moves in the same direction as pedestrian P, speed P relative to the ground is 2 m / s, and speed B relative to P is 5 m / s, this does not mean that speed B relative to the ground is 7 m / from. According to relativistic physics, speed B will be approximately 6.9999998 m / s. A small difference for practical tasks, but a huge one from the point of view of world perception. It turned out that quantities that used to be folded like apples cannot be added in this way.

This is not to say that in the middle or end of the twentieth century there were fewer revolutions. Rather, they became commonplace. The situation, when the foundations of science are collapsing, has transformed from an extraordinary into an expected one. Since then, several more “micro-revolutions” have taken place in the foundations of mathematics, for example, Robinson's “non-standard analysis” (which I hope to write a separate article about). And some revolutions were brewing, but never happened. My story will go about one of them.

I almost became a freak


One of my first thoughts after a head-on acquaintance with SRT was this: what if speed addition is just normal, “correct” addition, and our “ordinary” arithmetic addition is a construct that has nothing to do with reality? At small values, Lorentz addition is practically indistinguishable from ordinary. What if it works in other areas? For example, if we pour two liters of water in a bucket first, and then another five, then suddenly we get not seven liters, but six with six nines and one eight after the decimal point? Or another value, depending on what in this case is considered the "speed of light."

However, the volume of water is a complicated thing. It depends on temperature, subject to (purely theoretically, with vanishingly small probabilities, but still) random fluctuations, and if you go to the microscale, it becomes completely unclear how to measure it. But what if relativism sneaked into the holy of holies - into the natural series itself? Suppose, suddenly there is such a big pile of apples that adding another one will not change the number of apples in it?

When I got acquainted (again, in a cap) with quantum physics, this gave new ground for my thoughts. Under certain conditions, an electron can be in two places at the same time. Perhaps it is not a matter of some incomprehensible wave-particle duality, but the fact that a unit, under certain conditions, is equal to two?

I was very proud of the breadth and originality of my thoughts. 1

Notebook entries


This is the name of the story of Julio Cortazar, written in 1980, before my birth and long before I gained the ability to reason about the properties of natural numbers. It begins with the fact that in the Buenos Aires metro there was a discrepancy in the number of incoming and outgoing passengers: those who left the metro were 4 less than those who entered. A thorough search was conducted, but no passengers, nor any indication of how or where they disappeared, was found. The main character of this circumstance seemed frightening.
… меня удерживала на поверхности одна достойная внимания теория Луиса М. Бодиссона. Как-то полушутя я упомянул при нем о том, что рассказал мне Гарсиа Боуса, и как возможное объяснение этого явления он выдвинул теорию некой разновидности атомного распада, могущего произойти в местах большого скопления народа. Никто никогда не считал, сколько людей выходит со стадиона «Ривер-Плейт» в воскресенье после матча, никто не сравнивал эту цифру с количеством купленных билетов. Стадо в пять тысяч буйволов, которое несется по узкому коридору, — кто знает, их выбежало столько же, сколько вбежало? Постоянные касания людей друг о друга на улице Флорида незаметно стирают рукава пальто, тыльную сторону перчаток. А когда 113 987 пассажиров набиваются в переполненные поезда и их трясет и трет друг о друга на каждом повороте или при торможении, это может привести (благодаря процессу исчезновения индивидуального и растворению его во множественном) к потере четырех единиц каждые двадцать часов.
I will not retell the further plot of the story, it is beyond the scope of this article. One way or another, I received another confirmation that if an interesting thought occurred to me, then I was definitely not the first to whom it came to mind.

On the dogma of the natural series


Just last night I finished reading The Apology of Mathematics by V. A. Uspensky. A very interesting collection of articles intended for the humanities, but also related to the issues of the philosophy of mathematics, which are interesting to the person who distinguishes the derivative from the differential. In particular, Cortazar’s aforementioned story was cited there and the question was raised about the possibility that the Natural series (capital letter of the author) is not isomorphic to Peano’s natural numbers. And at the very end of the collection was a short article by P. K. Rashevsky “On the dogma of the natural series”, written in 1973. I wonder if Cortazar read it?

Rashevsky writes:
Процесс реального счета физических предметов в достаточно простых случаях доводится до конца, приводит к однозначно определенному итогу (число присутствующих в зале, например). Именно эту ситуацию берет за основу теория натурального ряда и в идеализированном виде распространяет ее «до бесконечности». Грубо говоря, совокупности большие предполагаются в каком-то смысле столь же доступными пересчету, как и малые и со столь же однозначным итогом, хотя бы реально этот пересчет и был неосуществим. В этом смысле наше представление о натуральном ряде похоже на зрительное восприятие панорамы, скажем, панорамы какого-либо исторического сражения. На первом плане на реальной земле расположены реальные предметы: разбитые пушки, расщепленные деревья и т.п.; затем все это незаметно переходит в раскрашенный холст с точным расчетом на обман даже очень внимательного глаза.
He further discusses the hypothetical properties of the “mathematical theory of a new type”. The article is very short, and those interested can read it personally .

The present


Rashevsky’s predictions (so far) have not come true. The problem, in general, is not to build a “new natural series”, the problem is that this construction be meaningful, lead to some new results or simplify the old ones (metaphorically speaking, “made the relativistic mechanics classical”) . There are no such theories (yet).

However, if you wish, everyone who reads this article can personally get acquainted with "non-classical" natural numbers. At least everyone who reads it in a desktop browser.

Open the developer tools (F12), select the “Console” tab and enter the following there:

var n = Number.MAX_SAFE_INTEGER + 1;
console.log(
    n, 
    Number.isInteger(n),
    n === n + 1
);

The console will answer you:
9007199254740992 true true
- which means that the number n is natural and is equal to itself plus one. Of course, only in terms of JavaScript. And yet, at least in the browser, such a bunch of apples exists.

Notes


1 Self-irony