One million plus one equals one million. Theory of Relativity of the Natural Series

To the reader

Despite a number of clever words and surnames, the article is quite accessible to the perception of nonmathematics. Despite the provocative headline, the article is not fricenic. 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" buried the Russell paradox, the limitations of Zermelo-Fraenkel's axiomatics showed, as early as the thirties, Gödel's incompleteness theorem.
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In physics, the revolution was made a special theory of relativity. The discovery of Einstein, based on the works of Maxwell, Lorentz and other scientists, postulated some counterintuitive properties of physical reality, in particular, the Lorentz addition of speeds. 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 equals 7 m / with. According to relativistic physics, the speed of B will be approximately 6.9999998 m / s. A small difference for practical tasks, but huge - from the point of view of worldview. It turned out that the quantities that were previously folded, like apples, can not be folded in this way.

It cannot be said that in the middle or the end of the twentieth century there were fewer revolutions. Rather, they have become ordinary. The situation when the foundations of science are crumbling has turned from the extraordinary into the expected. Since then, several more “micro-revolutions” have occurred in the foundations of mathematics — for example, Robinson’s “non-standard analysis” (about which I hope to write a separate article). And some revolutions were brewing, but they never happened. About one of them my story will go.

How I almost became a freak

One of my first thoughts after a capturing acquaintance with the SRT was this: what if with addition of speeds is just normal, “correct” addition, and our, “ordinary” arithmetic addition — is a construct that has no relation to reality? On small values, Lorentz addition is almost indistinguishable from usual. What if it works in other areas? For example, if we first pour two liters of water into the bucket and then five more, what if we get not seven liters, but six with six nine and one eight decimal points? Or a different value, depending on what in this case is considered the "speed of light."

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

When I met (again, nodding) with quantum physics, it gave a new ground for my reflections. Under certain conditions, an electron can be in two places at the same time. Perhaps, the point is not in some incomprehensible wave-particle dualism, but in the fact that a unit equals two under certain conditions?

I was very proud of the breadth and non-standard of my thoughts. ^one

Notepad entries

This is the story of Julio Cortazar, written in 1980, before my birth and long before I gained the ability to talk about the properties of natural numbers. It begins with the fact that in the metro of Buenos Aires there was a discrepancy in the number of incoming and outgoing passengers: those who left the metro turned out to be 4 less than those who entered. A thorough search was conducted, but neither the passengers, nor any indication as to how or where they disappeared, was detected. The main character, this circumstance seemed scary.

... I was kept on the surface by a noteworthy theory of Louis M. Bodisson. Somehow half in jest, I mentioned with him what Garcia Bous told me, and as a possible explanation for this phenomenon, he put forward a theory of some kind of atomic decay that could occur in places of large numbers of people. No one ever considered how many people leave the River Plate stadium on Sunday after the match; no one compared this figure with the number of tickets purchased. A herd of five thousand buffaloes rushing along a narrow corridor - who knows, they ran out as many as they ran in? Constant touching of people to each other on Florida Street imperceptibly erase the sleeves of the coat, the back of the gloves. And when 113,987 passengers are crowded into crowded trains and shakes and rubs against each other at every turn or when braking, this can lead (due to the process of disappearing of the individual and dissolving it in plural) to the loss of four units every twenty hours.

I will not retell the further plot of the story, it is beyond the scope of this article. Anyway, I received another confirmation that if an interesting thought occurred to me, then I was definitely not the first to whom it occurred.

About the dogma of the natural series

Literally last night I finished reading V. A. Ouspensky's “Apology of Mathematics”. A very curious collection of articles intended for the humanities, but also concerning the issues of the philosophy of mathematics, which is interesting to a person who distinguishes a derivative from a differential. In particular, the aforementioned story of Cortasar was quoted there and the question was raised about the possibility that the Natural series (the capital letter of the author) was not isomorphic to the natural Peano numbers. And at the very end of the collection there 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 a uniquely definite result (the number of those present in the hall, for example). This situation is based on the theory of the natural series and, in an idealized form, spreads it "indefinitely." Roughly speaking, large aggregates are assumed to be in some sense just as accessible as recalculations, as well as small ones and with an equally unambiguous result, even if this recalculation was really not feasible. In this sense, our understanding of the natural series is similar to the visual perception of a panorama, say, a panorama of a historical battle. In the foreground, real objects are located on real land: broken cannons, split trees, etc .; then all this imperceptibly turns into a painted canvas with the exact calculation of the deception of even a very attentive eye.

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 in person .

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 should be meaningful, lead to some new results or simplify the old ones (metaphorically speaking, “made relativistic mechanics classical”) . There are no such theories (yet).

However, if you wish, anyone who reads this article can personally get acquainted with the "non-classical" natural numbers. At least, everyone who reads it in the 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 at the same time equals itself plus one. Of course, only in terms of JavaScript. And yet, at least in the browser, such a bunch of apples exist.

Notes

¹ self-irony