numbers (unfinished)
Without many words
pre scriptum: Any comments, clarifications, demands to express more clearly, constructive participation in the discussion are welcomed by all limbs (limbs, I said, for those who thought not only about limbs:), although, if beautiful ... well, okay).
Numbers are different, but real and complex are used in physics. For a strict sequence of construction which requires the numbers natural, whole, rational.
')
Natural numbers are given by axioms. Most often, Peano's axioms
1. 0 exists and it is a natural number.
2. for each natural number n, the following positive integer is defined - s (n)
3. there is no natural number m for which s (m) = 0
4. if a! = B, then s (a)! = S (b)
5. If property C is true for 0, and if the fact that property C is true for n also implies that it is true for s (n), then property C is true for all natural numbers.
Note that the notation for axioms are not natural numbers. These are just icons to distinguish them. No finite system can be natural numbers, for some element x it will not define s (x), which also lies inside this system.
So far, the natural numbers look like 0, s (0), s (s (0)), s (s (s (0))), ... And they offer this view of nature.
There is 0 (however, this is just a designation). There is just something. Well, something exists, in a pinch, nothing exists. Take this something. Further, it is said, but once something is taken, then there is another something. It is also possible to take this, designate it as a follow-up to something previously taken, something, but even this does not exhaust our world. Once the first few things have been taken, there is also something distinct from all of them, which can be taken and indicated as the next to the last of the unshaded something.
People are trying to instill confidence in us in the possibility of infinitely taking and taking elements different from the previous ones. Is this an intuitive design? Is it in the world around us? You might think so. After all, for each step you need to take one element. But ... The subtlety is that this element should be different from all those that were taken earlier. With enough steps, to determine this, you need a giant repository of information. How to build it? And how much will it swell when adding the next item? Such a simple meditation on simple axioms may well (?) Lead to the hypothesis of an ever-expanding universe.
It would seem that one can not resort to the dynamic interpretation of these axioms. For example, you can say. This set of N simply exists. Natural numbers are, as Tao, and exist. But so what, that they just exist? By the standards of modern physics, this is not enough. It is necessary, for example, that electrons are “able” to count Feynman integrals and to craft other stunts. And to operate with this set is not at all easier. If something decides to find the next element for some natural number n, then this something will have to be done again by a cumbersome search procedure, first all previous elements, and then search for the next one, one that is different from all previous ones.
Naturally, the universe does it instantly (as physicists convince us, easily assigning complex mathematical qualities to any point in space), but the informational connections between pairs of natural numbers do not disappear from this. Where are they stored? Yes, you can say: just there, 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 perhaps there is a way out of this confusion of consciousness. After all, the axiomprinting practice is that you need to find a certain system in which you can poke your fingers at some components and states, see that they satisfy the necessary axiomatics and solemnly declare this system a system of natural numbers, for example. But then again it is not clear what satisfies these properties in physics? And generally anywhere else. The requirements for infinite continuation are very strict.
Natural numbers are not needed by anyone. Naturally. I want them to multiply and add. Well, this can be done strictly by defining operations like this:
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 are still just numbers. Without evidence (which is not difficult to get), I will say straight away - from all this, we get the usual multiplication and addition to a bar, and even division, and the ability to write numbers in the number systems. Now, 1, 2, 3, ... become natural numbers. At the same time, the concept is very interesting, which is written down by the second rule: 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 read on sticks, and the whole world consists of elementary elements.
That is, we are shown a homogeneous world. Do electrons know that the world is homogeneous? Strange. In addition, the situation with ever-increasing information complexity still persists. You can take n - a natural number, which, in order to be different from others, is written in a huge chain of numbers, and then try to take s (n). In theory, it is now known in advance that taking such a number is just Appendix 1. But do we all remember the addition by the bar? The operation may take a long time. Changes may be non-local, informational 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 the crows can only count to 7.
Whole numbers. Well, is it possible to correct this incident with a swelling of difficulty at a distance from zero? Than 0 is better than 2 ^ 65536 - 1, and how is this number better than by not understanding how many times longer 2 ^ 654536798234235 - 1? Following Einstein, we say, but nothing. It should not be so that the structure was infinitely complex, and much more complicated at a distance from zero.
But this means that we can now center in 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 will become number -1, and n + 5 - number +5. It is normal and sufficiently coordinated, so far no attempts are made to understand, but what is -1 for the beginning of the natural numbers? Hmm ... But it is not clear.
Therefore, we make another spiritual effort, accomplish an act of faith and begin to believe that, and let 0 really be no different from n. And -1 does exist. This is the number before n. And before -1 should also be something, because everything is uniform. All points should have the same structure. And we get a ring of integers.
Let me remind you of just 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 a unit - something elementary and common.
4. Around each element there must be an equally complex (in fact, we wanted an equally simple, but it is complex, at every point there is a system of integers) structure.
That's all. But then (next step in the theory of relativity), believing that everything consists of one, we again force ourselves to believe in another fact. And the unit also consists of something.
Can it be divided into n parts? It seems Thales comes in and says: naturally, my disciples. Take a single segment, set it aside from the point A n times along one straight line, get the number n (well, yes, Thales is right, indeed, the process of putting the segments on the straight line is exactly the application of the s (n) with a compass, and since takes place with the necessary properties, then we have natural numbers). Now, from point A, draw another straight line, and it is, this is guaranteed by Euclid. Now put on it a single segment, the end of which is connected to the number n on another straight line, and now, parallel to this connection, build n straight lines passing through points 1, 2, 3, ... n - 1. And you will see that these straight lines will cross unit segment, located at an angle, into n equal parts. For again, Euclidean axioms.
But in fact. And what is the unit itself? What is 1? The generator of our universe of natural numbers? Basic information structure? What? Unclear. So, following Einstein's precepts, we say that nothing 1 conceptually differs from other elements, you can do all 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 into the corresponding number of such segments.
But generally speaking, in algebra 1 has well-defined unique properties. For example, for any number x, x * 1 = x. Is there a contradiction with the previously written? Not. Because the previously written says only that by itself 1 never goes. In 1 there is no point. Well, actually. I tell you 1. What will it mean? Then you can certainly assume that I called you a natural number 1 and discuss its properties.
The Greeks did not know the natural numbers. But they knew what 1. 1 was for them was a common measure of two segments. Everyone remembers the Euclidean algorithm for finding the gcd? So, Euclid did not know what a GCD is, and he used the algorithm to find the general measure of two segments (simply subtracting (using compasses) each time the shortest of them from the longer one, he defined by eye, of course). But nevertheless, his intuition did not fail, and it 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 (two times divided by a pair of compasses and a ruler, a segment), and it immediately became clear that the new measurement scale was 1/4 = 1 ', and 1 is now 4 '.
Complete nihilism and denial of the presence of anything solid. Not only do we have an infinitely complex structure in each element of the universe, so an even more complex structure exists between any pair.
Yeah, and when we start to think about the general structure linking the three elements, where will we go? That's right, straight into complex numbers. Having passed before this is valid.
You can paint this process, specify, for example, in the course of the fact 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, how is Tao?
And the structures there will arise even more complex. Even more sophisticated, because the sqrt (2) notorious contains in itself infinitely a lot of information that is needed to build it with the help of simple movements in space. But, of course, you can declare it as a unit, and not worry, moving around the points sqrt (2) * N / M. But so we will never get into the unit.
The relationship between numbers is VERY complex, if you look at them not in terms of axioms, but trying to find an everyday interpretation of this whole set of concepts. At the same time, the only way to accept their existence is to believe that they all exist in a crowd, immediately and without limit, here and now, in all their complexity and (in general) magnificence.
But at the same time, the situation is highly dual. Numbers can't stack themselves. Yes, there can be a three-dimensional set (x, y, z) with the property that z = x + y. But ... And why do we then even puff and put the numbers in a column? Where is this set? How to access it? Now is the time to believe in a single information space, in which everything is there, and from which Indian fakirs draw data when they add fucking numbers in their mind for five seconds.
But you know, it's fake. And that's why. Because addition, even through access to this set, is still an operation. The action, moreover, the execution of the action. Therefore, the only interpretation offered by mathematics is that it simply exists, it does not correspond to reality.
At the same time, the situation changes if we start to perceive mathematics as a language (as it should be perceived in general). This is just a way to describe (not even the objects, although everything looks exactly like the description of the objects) process. For example, Peano's axioms shown to mathematicians told them about how they can construct natural numbers, and agreement with the definitions of the operations + and * allowed us to raise the efficiency of design to a very high level. There are no natural numbers, there are mathematicians dynamically interpreting Peano's axiomatics.
So it is with everything. Axioms we are simply told: if you have something in which you see such and such properties, then you can do this and that with it, and at the same time these properties will be preserved. Well, in fact, the processor can perform operations with rings of entire residues (well, modulo operations for some). Can. But does this mean that the set exists for the processor? That mathematics exists for him in each of his states? And if the processor is broken, 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. Whether it is the theory of relativity, which moves along the path planned for integers and rational numbers, and declares that everything must be the same everywhere, the same, and we can only move at the speed of light, along pre-created trajectories. And this whole miracle is a very complex structure.
Or whether it is quantum physics, in which apparently divisible and discrete things try to describe infinitely complex and infinitely divisible, infinitely incapable of discriminating states, if they are not matched from outside the system with other states (although this wording is very important for understanding what physical experiment). Well, the endless and majestic complexity and prediction of space-time can still be accepted. But the fact that the electron knows all about the same thing that we know, putting an experiment on it ... Hmm. It sounds weird until we accept that the electron (well, or any observable 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 out, except for such a stream of consciousness. Although, if without reference to reality, I can easily explain to you the basic concepts and conclusions, even though number theory, even the theory of relativity, even quantum chromodynamics (here I, however, need to pry 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, experiences, not analysis) associated with these reflections, I can name only one thing: mathematics is operated with static objects. Here, though burst. Even a Turing machine is best represented as a kind of trajectory (which I will write about later) in a certain state space.
That is why all mathematics fall apart when we write programs that actively communicate with the outside world. The outside world is changeable, but math is not. It should work at any point in space and time, at any speed, with extreme precision. This is the case if at this point in space there is a reasonably intelligent interpreter or interpreters of a mathematical language. But what if there is no interpreter there?
And another question: so why is physics trying to operate only with such timeless constructions, trying to describe an obviously dynamic system? Isn't that why physicists have paradoxes and 'anti-intuitive' constructions in theories?
Want an example? All the same electron. For physicists, there was a significant reshuffling of ideas when the Schrödinger equation appeared, which allowed an electron to live only on certain energy levels. And why was this equation diligently sought? To resolve the contradiction in classical electrodynamics, which prescribes the electron dangling around a proton to lose all the energy soon and fall on this proton, which does not happen. And it should happen, according to the analysis of the trajectories and other mathematical constructions that exist outside of time.
But, after all, everything is obvious ... It is only necessary to accept the fact that the electron is mobile. And this is its main property: neither the position at a point, nor the possession of energy, but mobility. Accordingly, it simply cannot fall on a proton. He can not be at the point - this is nonsense. Actually, any elementary particle resists this nonsense. The principle of uncertainty of Heisenberg is just about this - a fig you will crush an electron to a point, because you will not have enough energy for this to contain its fluctuations. This is logical. No nonsense ... But about time I will write later. And, I hope, stricter, because there will be no picking in the elementary foundations of theories.
pre scriptum: Any comments, clarifications, demands to express more clearly, constructive participation in the discussion are welcomed by all limbs (limbs, I said, for those who thought not only about limbs:), although, if beautiful ... well, okay).
Numbers are different, but real and complex are used in physics. For a strict sequence of construction which requires the numbers natural, whole, rational.
')
Natural numbers are given by axioms. Most often, Peano's axioms
1. 0 exists and it is a natural number.
2. for each natural number n, the following positive integer is defined - s (n)
3. there is no natural number m for which s (m) = 0
4. if a! = B, then s (a)! = S (b)
5. If property C is true for 0, and if the fact that property C is true for n also implies that it is true for s (n), then property C is true for all natural numbers.
Note that the notation for axioms are not natural numbers. These are just icons to distinguish them. No finite system can be natural numbers, for some element x it will not define s (x), which also lies inside this system.
So far, the natural numbers look like 0, s (0), s (s (0)), s (s (s (0))), ... And they offer this view of nature.
There is 0 (however, this is just a designation). There is just something. Well, something exists, in a pinch, nothing exists. Take this something. Further, it is said, but once something is taken, then there is another something. It is also possible to take this, designate it as a follow-up to something previously taken, something, but even this does not exhaust our world. Once the first few things have been taken, there is also something distinct from all of them, which can be taken and indicated as the next to the last of the unshaded something.
People are trying to instill confidence in us in the possibility of infinitely taking and taking elements different from the previous ones. Is this an intuitive design? Is it in the world around us? You might think so. After all, for each step you need to take one element. But ... The subtlety is that this element should be different from all those that were taken earlier. With enough steps, to determine this, you need a giant repository of information. How to build it? And how much will it swell when adding the next item? Such a simple meditation on simple axioms may well (?) Lead to the hypothesis of an ever-expanding universe.
It would seem that one can not resort to the dynamic interpretation of these axioms. For example, you can say. This set of N simply exists. Natural numbers are, as Tao, and exist. But so what, that they just exist? By the standards of modern physics, this is not enough. It is necessary, for example, that electrons are “able” to count Feynman integrals and to craft other stunts. And to operate with this set is not at all easier. If something decides to find the next element for some natural number n, then this something will have to be done again by a cumbersome search procedure, first all previous elements, and then search for the next one, one that is different from all previous ones.
Naturally, the universe does it instantly (as physicists convince us, easily assigning complex mathematical qualities to any point in space), but the informational connections between pairs of natural numbers do not disappear from this. Where are they stored? Yes, you can say: just there, 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 perhaps there is a way out of this confusion of consciousness. After all, the axiomprinting practice is that you need to find a certain system in which you can poke your fingers at some components and states, see that they satisfy the necessary axiomatics and solemnly declare this system a system of natural numbers, for example. But then again it is not clear what satisfies these properties in physics? And generally anywhere else. The requirements for infinite continuation are very strict.
Natural numbers are not needed by anyone. Naturally. I want them to multiply and add. Well, this can be done strictly by defining operations like this:
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 are still just numbers. Without evidence (which is not difficult to get), I will say straight away - from all this, we get the usual multiplication and addition to a bar, and even division, and the ability to write numbers in the number systems. Now, 1, 2, 3, ... become natural numbers. At the same time, the concept is very interesting, which is written down by the second rule: 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 read on sticks, and the whole world consists of elementary elements.
That is, we are shown a homogeneous world. Do electrons know that the world is homogeneous? Strange. In addition, the situation with ever-increasing information complexity still persists. You can take n - a natural number, which, in order to be different from others, is written in a huge chain of numbers, and then try to take s (n). In theory, it is now known in advance that taking such a number is just Appendix 1. But do we all remember the addition by the bar? The operation may take a long time. Changes may be non-local, informational 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 the crows can only count to 7.
Whole numbers. Well, is it possible to correct this incident with a swelling of difficulty at a distance from zero? Than 0 is better than 2 ^ 65536 - 1, and how is this number better than by not understanding how many times longer 2 ^ 654536798234235 - 1? Following Einstein, we say, but nothing. It should not be so that the structure was infinitely complex, and much more complicated at a distance from zero.
But this means that we can now center in 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 will become number -1, and n + 5 - number +5. It is normal and sufficiently coordinated, so far no attempts are made to understand, but what is -1 for the beginning of the natural numbers? Hmm ... But it is not clear.
Therefore, we make another spiritual effort, accomplish an act of faith and begin to believe that, and let 0 really be no different from n. And -1 does exist. This is the number before n. And before -1 should also be something, because everything is uniform. All points should have the same structure. And we get a ring of integers.
Let me remind you of just 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 a unit - something elementary and common.
4. Around each element there must be an equally complex (in fact, we wanted an equally simple, but it is complex, at every point there is a system of integers) structure.
That's all. But then (next step in the theory of relativity), believing that everything consists of one, we again force ourselves to believe in another fact. And the unit also consists of something.
Can it be divided into n parts? It seems Thales comes in and says: naturally, my disciples. Take a single segment, set it aside from the point A n times along one straight line, get the number n (well, yes, Thales is right, indeed, the process of putting the segments on the straight line is exactly the application of the s (n) with a compass, and since takes place with the necessary properties, then we have natural numbers). Now, from point A, draw another straight line, and it is, this is guaranteed by Euclid. Now put on it a single segment, the end of which is connected to the number n on another straight line, and now, parallel to this connection, build n straight lines passing through points 1, 2, 3, ... n - 1. And you will see that these straight lines will cross unit segment, located at an angle, into n equal parts. For again, Euclidean axioms.
But in fact. And what is the unit itself? What is 1? The generator of our universe of natural numbers? Basic information structure? What? Unclear. So, following Einstein's precepts, we say that nothing 1 conceptually differs from other elements, you can do all 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 into the corresponding number of such segments.
But generally speaking, in algebra 1 has well-defined unique properties. For example, for any number x, x * 1 = x. Is there a contradiction with the previously written? Not. Because the previously written says only that by itself 1 never goes. In 1 there is no point. Well, actually. I tell you 1. What will it mean? Then you can certainly assume that I called you a natural number 1 and discuss its properties.
The Greeks did not know the natural numbers. But they knew what 1. 1 was for them was a common measure of two segments. Everyone remembers the Euclidean algorithm for finding the gcd? So, Euclid did not know what a GCD is, and he used the algorithm to find the general measure of two segments (simply subtracting (using compasses) each time the shortest of them from the longer one, he defined by eye, of course). But nevertheless, his intuition did not fail, and it 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 (two times divided by a pair of compasses and a ruler, a segment), and it immediately became clear that the new measurement scale was 1/4 = 1 ', and 1 is now 4 '.
Complete nihilism and denial of the presence of anything solid. Not only do we have an infinitely complex structure in each element of the universe, so an even more complex structure exists between any pair.
Yeah, and when we start to think about the general structure linking the three elements, where will we go? That's right, straight into complex numbers. Having passed before this is valid.
You can paint this process, specify, for example, in the course of the fact 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, how is Tao?
And the structures there will arise even more complex. Even more sophisticated, because the sqrt (2) notorious contains in itself infinitely a lot of information that is needed to build it with the help of simple movements in space. But, of course, you can declare it as a unit, and not worry, moving around the points sqrt (2) * N / M. But so we will never get into the unit.
The relationship between numbers is VERY complex, if you look at them not in terms of axioms, but trying to find an everyday interpretation of this whole set of concepts. At the same time, the only way to accept their existence is to believe that they all exist in a crowd, immediately and without limit, here and now, in all their complexity and (in general) magnificence.
But at the same time, the situation is highly dual. Numbers can't stack themselves. Yes, there can be a three-dimensional set (x, y, z) with the property that z = x + y. But ... And why do we then even puff and put the numbers in a column? Where is this set? How to access it? Now is the time to believe in a single information space, in which everything is there, and from which Indian fakirs draw data when they add fucking numbers in their mind for five seconds.
But you know, it's fake. And that's why. Because addition, even through access to this set, is still an operation. The action, moreover, the execution of the action. Therefore, the only interpretation offered by mathematics is that it simply exists, it does not correspond to reality.
At the same time, the situation changes if we start to perceive mathematics as a language (as it should be perceived in general). This is just a way to describe (not even the objects, although everything looks exactly like the description of the objects) process. For example, Peano's axioms shown to mathematicians told them about how they can construct natural numbers, and agreement with the definitions of the operations + and * allowed us to raise the efficiency of design to a very high level. There are no natural numbers, there are mathematicians dynamically interpreting Peano's axiomatics.
So it is with everything. Axioms we are simply told: if you have something in which you see such and such properties, then you can do this and that with it, and at the same time these properties will be preserved. Well, in fact, the processor can perform operations with rings of entire residues (well, modulo operations for some). Can. But does this mean that the set exists for the processor? That mathematics exists for him in each of his states? And if the processor is broken, 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. Whether it is the theory of relativity, which moves along the path planned for integers and rational numbers, and declares that everything must be the same everywhere, the same, and we can only move at the speed of light, along pre-created trajectories. And this whole miracle is a very complex structure.
Or whether it is quantum physics, in which apparently divisible and discrete things try to describe infinitely complex and infinitely divisible, infinitely incapable of discriminating states, if they are not matched from outside the system with other states (although this wording is very important for understanding what physical experiment). Well, the endless and majestic complexity and prediction of space-time can still be accepted. But the fact that the electron knows all about the same thing that we know, putting an experiment on it ... Hmm. It sounds weird until we accept that the electron (well, or any observable 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 out, except for such a stream of consciousness. Although, if without reference to reality, I can easily explain to you the basic concepts and conclusions, even though number theory, even the theory of relativity, even quantum chromodynamics (here I, however, need to pry 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, experiences, not analysis) associated with these reflections, I can name only one thing: mathematics is operated with static objects. Here, though burst. Even a Turing machine is best represented as a kind of trajectory (which I will write about later) in a certain state space.
That is why all mathematics fall apart when we write programs that actively communicate with the outside world. The outside world is changeable, but math is not. It should work at any point in space and time, at any speed, with extreme precision. This is the case if at this point in space there is a reasonably intelligent interpreter or interpreters of a mathematical language. But what if there is no interpreter there?
And another question: so why is physics trying to operate only with such timeless constructions, trying to describe an obviously dynamic system? Isn't that why physicists have paradoxes and 'anti-intuitive' constructions in theories?
Want an example? All the same electron. For physicists, there was a significant reshuffling of ideas when the Schrödinger equation appeared, which allowed an electron to live only on certain energy levels. And why was this equation diligently sought? To resolve the contradiction in classical electrodynamics, which prescribes the electron dangling around a proton to lose all the energy soon and fall on this proton, which does not happen. And it should happen, according to the analysis of the trajectories and other mathematical constructions that exist outside of time.
But, after all, everything is obvious ... It is only necessary to accept the fact that the electron is mobile. And this is its main property: neither the position at a point, nor the possession of energy, but mobility. Accordingly, it simply cannot fall on a proton. He can not be at the point - this is nonsense. Actually, any elementary particle resists this nonsense. The principle of uncertainty of Heisenberg is just about this - a fig you will crush an electron to a point, because you will not have enough energy for this to contain its fluctuations. This is logical. No nonsense ... But about time I will write later. And, I hope, stricter, because there will be no picking in the elementary foundations of theories.
Source: https://habr.com/ru/post/14863/
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