A million plus one equals a million. Theory of Relativity of the Natural Series

To the reader

Despite a certain number of smart words and surnames, the article is quite accessible to the perception of nemathematics. Despite the provocativeness of the 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 set theory” was buried by the Russell paradox, the limited axiomatics of Zermelo-Frenkel showed - already in the thirties - Gödel’s incompleteness theorem.

In physics, a revolution was made by the special theory of relativity. 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 the values ​​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 have become commonplace. The situation, when the foundations of science are crumbling, 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, with the addition of speeds, this is just normal, “correct” addition, and our, “usual” arithmetic addition is a construct that is not related to reality? At small values, Lorentzian 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, what if 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, is subject (purely theoretically, with vanishingly small probabilities, but still) to 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 the point is not in some incomprehensible wave-particle duality, but in 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 I was born 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.

... I was kept on the surface by a noteworthy theory of Louis M. Bodisson. Half-jokingly, I mentioned with him what Garcia Bousa told me, and as a possible explanation for this phenomenon, he put forward the theory of some kind of atomic decay that could occur in crowded places. No one ever counted how many people were leaving the River Plate Stadium on Sunday after the match; no one compared this figure to the number of tickets purchased. A herd of five thousand buffaloes that sweeps along a narrow corridor - who knows, as many have run out as they ran in? The constant touching of people against each other on Florida Street quietly erases the sleeves of the coat, the back of the gloves. And when 113,987 passengers cram into crowded trains and shake and rub against each other at every turn or when braking,

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, there was cited the aforementioned story of Cortazar and raised the question of 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:

The process of real counting of physical objects in fairly simple cases is brought to an end, leading to an unambiguously determined result (the number of those present in the hall, for example). It is this situation that the theory of the natural series takes as its basis and in an idealized form distributes it “to infinity”. Roughly speaking, large aggregates are supposed in some sense as readily available for recounting as small ones with the same unequivocal result, even if this recounting was really impossible. In this sense, our idea of ​​the natural series is similar to the visual perception of a panorama, say, a panorama of a historical battle. In the foreground on real earth are real objects: broken cannons, split trees, etc .;

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 desired, anyone 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