# numbers (unfinished)

Without superfluous words

pre scriptum: Any comments, clarifications, requirements to clear up more clearly, constructive participation in the discussion are welcomed by all the limbs (limbs, I said, for those who thought not only about the limbs :), although if beautiful ... well, yes okay).

Numbers are different, but real and complex are used in physics. For a strict sequence of construction which requires the numbers natural, integer, rational.

Natural numbers are given by axioms. Most often, Peano's axioms are

1. 0 and this is a natural number.
2. for each positive integer n the following positive integer is defined - s (n)
3. there is no such positive integer m for which s (m) = 0
4. if a! = B, then s (a)! = S (b )
5. if property 0 is true for 0, and if it follows from the fact that property C is true for n that it is true for s (n), then property C is true for all natural numbers.

Note that the notation of points of axioms is not natural numbers. These are just icons to distinguish them. No finite system can be natural numbers, for some element x, s (x), which also lies inside this system, will not be defined in it.

While the natural numbers look like 0, s (0), s (s (0)), s (s (s (0))), ... And they offer such a view of nature.

There is 0 (however, this is just a notation). Something just exists. Well, something exists, in the worst case, nothing exists. Take this something. Further, it is said, but if something is taken, then something else exists. You can also take it, designate the following to the previously taken, something, but this is not the end of our world. Once the first few are taken, there is something different from all of them, which can be taken and designated as the next to the last of the unshaded something.

They are trying to instill confidence in us in the possibility of endlessly taking and taking elements different from previous ones. Is this an intuitive design? Is this so in the world around us? You might think so. Indeed, for each step it is necessary to take one element. But ... The subtlety is that this element should be different from all those that were taken earlier. With enough steps, you need a gigantic repository of information to determine this. How to build it? And how much will it swell when you add another element? Such simple meditation on simple axioms may very well (?) Lead to the hypothesis of a constantly expanding universe.

It would seem that one can not resort to a dynamic interpretation of these axioms. For example, you can say. This set N - just exists. Natural numbers are, like Tao, and exist. But what if they just exist? By the standards of modern physics, this is not enough. It is necessary, for example, for electrons to be able to count Feynman integrals and perform other tricks. And operating with this set is not any easier. If something decides to find the next element for some natural number n, then something will again have to do a cumbersome procedure to search, first all the previous elements, and then search the next one, which is different from all the previous ones.

Naturally, the universe does this instantly (as physicists convince us, easily attributing complex mathematical qualities to any point in space), but the information connections between pairs of natural numbers do not go away from this. Where are they stored? Yes, you can say: just eat, that's all. Like Tao, at every point. And we, together with the universe, know what will be next. Electrons know, protons know. But computers and crows do not know. Crows do not know how to count to 8, and computers to 2 ^ n + 1 with a sufficiently large n. Why are crows worse than humans, and computers are simpler than protons?

This is all strange, but there may be a way out of this confusion of consciousness. Indeed, the axiom application practice is that you need to find some system in which you can poke your fingers on some components and conditions, see that they satisfy the desired axiomatics and solemnly declare this system as a system of natural numbers, for example. But again, it is not clear what in physics satisfies these properties? And in general, anywhere. There are very strict requirements for endless continuation.

By themselves, natural numbers are not needed by anyone. Naturally. I want to multiply and add them. Well, this can be done strictly by defining the operations as follows:

1. n + 0 = n
2. n + 1 = s (n)
3. n + s (m) = s (n + m)

4. n * 0 = 0
5.n * 1 = n
6. n * s (m) = n * m + n The

numbers, as before, are just numbers. Without proof (which is not hard to get) I’ll say right away - from all this we get the usual multiplication and addition in a column, and even division, and the ability to write numbers in number systems. Now 1, 2, 3, ... have become natural numbers. At the same time, the concept, which is written down by the second rule, is very interesting: it turns out that the concept of the following can be replaced by the concept that all natural numbers consist of 1. That is, you can now count on sticks, and the whole world consists of elementary elements.

That is, they show us a homogeneous world. Do the electrons know that the world is homogeneous? It’s strange. In addition, the situation remains with ever-increasing informational complexity. You can take n - a natural number, which, in order to be different from others, is written down in a huge chain of digits, and then try to take s (n). In theory, it is now known in advance that taking such a number is just an addition of 1. But everyone remembers the addition by column? The operation may take a long time. Changes can be non-local, information dependencies are monstrously long. Even if nature does it instantly. Even if this structure with additions exists at every point. Yes, by the way, if it exists at every point, then the universe has already solved all the algorithmic problems, amusing. And crows can only count to 7.

Whole numbers. Okay, is it possible to correct this incident with a swelling of complexity when moving away from zero? Why is 0 better than 2 ^ 65536 - 1, and how is this number better than not understanding how many times longer 2 ^ 654536798234235 - 1? Following Einstein, let’s say - nothing. It should not be so that the structure was infinitely complex, and significantly more complicated when moving away from zero.

But this means that we can now focus on the number n, and write down all the other numbers, assuming that n is the navel of the earth. Then, the complexity of the designs may be limited. Then, n - 1 becomes the number -1, and n + 5 becomes the number +5. It’s normal and consistent enough, until you make attempts to understand, and what is -1 to start natural numbers? Hmm ... But it’s not clear.

Therefore, we make another spiritual effort, perform an act of faith and begin to believe that, and let 0 really not differ from n. And -1 really exists. This is the number before n. And before -1 there must also be something, because everything is uniform with us. All points should have the same structure. And we get a ring of integers.

Let me remind you, only of three ideas about the nature of things.

1. There is always something.
2. You can always take the next item.
3. Everything can be built from one - something elementary and common.
4. Around any element should be equally complex (in fact, I wanted to be equally simple, but it is complex, at every point there is a system of integers) structure.

That's all. But further (another step in the theory of relativity), believing that everything consists of 1, we again force ourselves to believe in another fact. And the unit also consists of something.

Can it be divided into n parts? Thales, it seems, comes and says: Naturally, my disciples. Take a single segment, lay it off from point A n times along one straight line, get the number n (well, yes, Thales is right, indeed, the process of laying off segments on a straight line is exactly the application of the s (n) procedure with the help of a compass, and since it takes place with the necessary properties, then we have natural numbers). Now, from point A draw another straight line, but there is one, this guarantees us Euclid. Now lay aside a unit segment on it, the end of which is connected with the number n on another line, and now, parallel to this connection, build n lines passing through points 1, 2, 3, ... n - 1. And you will see that these lines intersect a unit segment located at an angle, in n equal parts. For again, the axioms of Euclid.

But really. And what is a unit in itself? What is 1? The generator of our universe of natural numbers? Basic information structure? What? Unclear. So, following Einstein’s precepts, let’s say that 1 is conceptually no different from the rest of the elements, you can do the same physics with it as with the number n, or 3, or 2 ^ 5679087 - 1, which means that these numbers can divided by the corresponding number 1, then 1 can be divided by the corresponding number of such segments.

But generally speaking, in algebra 1 has very definite unique properties. For example, for any number x, x * 1 = x. Is there a contradiction with what was written earlier? Not. Because the previously written only says that in itself 1 never walks. There is no point in 1. Well, actually. I will tell you 1. And what will it mean? Further, of course, we can assume that I gave you the natural number 1 and discuss its properties.

The Greeks did not know natural numbers. But they knew what it was 1. 1 for them was a common measure of two segments. Does everyone remember the Euclidean algorithm for finding GCD? So, Euclid did not know what a GCD is, and the algorithm used to find a common measure of two segments (just subtracting (using a compass) each time the shortest of them from a longer one, it determined by eye, naturally). Nevertheless, his intuition did not fail, and this really gave a general measure of the segments.

The fact that five minutes ago it was considered 1 could be commensurate with a 1/4 length segment (twice divided in half using a compass and a ruler a segment), and it immediately became clear that the new measurement scale is 1/4 = 1 ', and 1 is now 4 '.

Complete nihilism and denial of the existence of anything solid. Not only do we have an infinitely complex structure in every element of the universe, but an even more complex structure exists between any pair.

Yeah, and when we start thinking about a common structure linking the three elements, where do we get to? Right, straight to complex numbers. Having gone before it is valid.

You can paint this process, indicate, for example, along the way that sqrt (2) does not exist without a complete construction of real numbers, without this construction, even the Pythagorean theorem cannot be strictly proved, so the existence of real numbers is similar to the existence of 0, in Peano's axioms. They just exist and that's it. As a given, like a Tao?

And the structures there will arise even more complex. Even more sophisticated, because sqrt (2) the notorious contains infinitely a lot of information that is needed to build it using simple movements in space. But, of course, you can declare it as a unit, and do not worry, moving along the sqrt (2) * N / M points. But this way we will never fall into a unit.

The relations between the numbers are VERY complex, if you look at them not in terms of axioms, but trying to find a worldly interpretation of all this many concepts. Moreover, the only way to accept their existence is to believe that they all exist together, immediately and without limit, here and now, in all its complexity and (in general) magnificence.

But at the same time, the situation is very dual. Numbers cannot add themselves. Yes, there may exist a three-dimensional set (x, y, z) with the property that z = x + y. But ... And why then do we puff and add the numbers in a column? Where is this set? How to get access to it? It's time to believe in a single information space in which everything is there, and from which the Indian fakirs draw data when they add awesome numbers in the mind in five seconds.

But you know, this is fake. And that's why. Because addition, even through access to this set, is an operation anyway. Action, moreover, the execution of an action. Therefore, the only interpretation proposed by mathematics: it simply exists, is not true.

At the same time, the situation changes if we begin to perceive mathematics as a language (as it is necessary to perceive it in general). This is just a way to describe (and not even objects, although everything looks exactly like a description of objects) process. For example, Peano's axioms shown to mathematicians told them how they can construct natural numbers, and the agreement with the definitions of the + and * operations allowed us to raise the construction efficiency to a very high level. There are no natural numbers; there are mathematicians who dynamically interpret Peano's axiomatics.

So it is with everything. The axioms just tell us: if you have something in which you see such and such properties, then you can do such and such things with it, and at the same time these properties will be preserved. Well, actually, a processor can perform operations on rings of whole residues (well, operations modulo some kind of). Can. But does this mean that a plurality exists for the processor? That there is mathematics for him in each of his states? And if you break the processor, where will all this complex structure go?

Vobschem here. All this causes the first stage of cognitive dissonance in the perception of modern constructions of theoretical physics. Be it the theory of relativity, which moves along the path outlined for integers and rational numbers, and states that everywhere everything should be monotonous, the same, and we can only move at the speed of light, along pre-created trajectories. And all this miracle is a very complex structure.

Or be it quantum physics, in which obviously divisible and discrete things are trying to describe infinitely complex and infinitely divisible, infinitely incapable of distinguishing states if they are not compared from outside the system with other states (although this formulation is very important for understanding what it is physical experiment). Well, the infinite and magnificent complexity and predetermination of space-time can still be accepted. But the fact that the electron knows about the same things that we know about, putting an experiment on it ... Hmm. It sounds strange until we accept that the electron (well, or any observed phenomenon) is not an elementary element of the universe, but an elementary component of our experiment.

Okay. I really can't write it more precisely. I tried for four days, and nothing comes of it except such a stream of consciousness. Although, if without reference to reality, I can easily explain the basic concepts and conclusions of at least number theory, at least relativity theory, at least quantum chromodynamics (here, however, I need to peek into the textbook).

But to connect everything with what we see around is very difficult. And the main reason after all the experiences (I emphasize, the experiences, not the analysis) associated with these reflections, I can name only one: mathematics operate on static objects. Here, at least crack. Even a Turing machine is best represented as a trajectory (which I will write about later) in a certain state space.

That is why all the math falls apart when we write programs that actively communicate with the outside world. The outside world is changeable, but mathematics is not. It should work anywhere in space and time, at any speed, with extreme accuracy. This happens if at this point in space there is a sufficiently intelligent interpreter or interpreters of the mathematical language. But what if there is no interpreter?

And another question: so why is physics trying to operate only with such timeless constructions, trying to describe an obviously dynamic system? Is that why physicists have paradoxes and 'anti-intuitive' constructions in theories?

Want an example? All the same electron. For physicists, there was significant shaking up of ideas when the Schrödinger equation appeared, which allowed the electron to live only at certain energy levels. And why was this equation diligently sought? To resolve the contradiction in classical electrodynamics, which requires an electron dangling around a proton to quickly lose all energy and crash onto this proton, which does not happen. And it should happen, according to the analysis of trajectories and other mathematical constructions existing outside of time.

But, after all, everything is obvious ... It is only necessary to accept that the electron is mobile. And this is his main property: neither position at the point, nor possession of energy, but mobility. Accordingly, it simply cannot fall onto a proton. It cannot be at a point - this is nonsense. Actually, any elementary particle resists this nonsense. Heisenberg’s uncertainty principle is just about that - fig, you push the electron to a point, because you don’t have enough energy to keep it from fluctuating. This is logical. No nonsense ... But I'll write about the time later. And, I hope, more stringent, because there will be no picking in the elementary foundations of theories.