Richard Hamming: "The inexplicably high efficiency of mathematics"
“This is the deepest essay I have ever seen regarding the philosophy of science; in fact, it is important for all our understanding of thought, knowledge or reality. ”
From the title it is clear that this is a philosophical question. I will not apologize for the philosophy, although I know that many scientists, engineers and mathematicians do not pay attention to it. Instead, I will give this small prologue to substantiate my approach.
')
A person, as far as we know, has always wondered about himself, the world around him and the meaning of life. Many myths tell of why and how God or gods created man and the universe. These are theological explanations. Their distinguishing feature - it makes no sense to ask why things are as they are, if the gods created them so.
Philosophy arose when a person wondered about a world outside this theological framework. For example, ancient philosophers represented the world as a combination of earth, fire, water and air. No doubt they were convinced that the world was created by the gods and there is no need to worry about this.
From these early attempts to explain the world, philosophy gradually emerged, as did modern science. Not that science explains "why" things are as they are - gravity does not explain "why" objects fall, but science gives us such a detailed explanation of "how" this happens that we have the feeling that we know "why" . Let's be clear; a sea of interconnected facts allows science to say "why" the universe is just that.
Mathematics is our main tool for performing long chains of scientific reasoning. It can be defined as a mental tool for this purpose. For centuries, people have asked themselves the question, the same one that is in the headline - “Why is mathematics so unreasonably effective?” In asking this question, we focus on the logical rather than the material side of how the universe works and works.
Mathematical mathematicians are mostly concerned with self-consistency and the limits of the system. They do not care why the world seems to admit a logical explanation. In this sense, I am closer to the early Greek philosophers who were interested in the material side of the question, and my answers in relation to logic are not much better than theirs. But once and somewhere we must begin to explain the phenomenon of the fact that the world is logically structured in accordance with the laws of mathematics, and that mathematics is the language of science and technology.
After I marked the main topic, you need to understand how to convey these ideas to the audience. Experience shows that I am not always good at it. It became clear to me that some preliminary remarks would be able to help.
In some respects, this discussion is theorized. I must touch, at least slightly, the main theories of mathematics, as well as its individual parts. Further, there are various applied theories. So we will move from one theory to another. You may be surprised that I will use an experimental approach for discussion. No matter what the theories should be, or how you imagine them, or what the experts' point of view is, let's apply the scientific approach and find out everything. I understand perfectly well that my words, especially about the nature of mathematics, will annoy many mathematicians. My experimental approach is contrary to their mindset and preconceived beliefs. So be it!
The inspiration for this article was the article with the similar title "The unreasonable effectiveness of mathematics in the natural sciences . " (1. EP Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math., 13 (Feb. 1960) E.P. Wigner. I removed part of the title and do not repeat his article (I just can’t write better. On the other hand, I will spend much more time explaining the question from the title of the article. But when all the explanations are over, the rest will be so great that our question will practically remain unanswered.
In his article, Wigner gives many examples of the effectiveness of mathematics in the physical sciences. Nevertheless, let me draw on my own experience, which is closer to engineering. My first experience of using mathematics to predict was related to the development of atomic bombs during World War II. How did it happen that the numbers that we so patiently calculated on primitive relay computers agreed so well with what happened on the first test in Almagordo? We could not conduct a small experiment to verify the calculations. Later, the experience with guided missiles showed me that this was not an accident - our mathematical predictions are constantly justified in the real world. Working in the Bell System, I did a lot of calculations for telephony and other mathematical work on subjects such as traveling wave tubes, alignment of television lines, stability of complex communication systems, blocking calls through a central telephone - this is only a small part. In order to show off knowledge, I can conduct research on transistors, space flight, and computer-aided design, but almost all science and technology uses an extensive mathematical apparatus with overwhelming success.
Many of you are familiar with the history of Maxwell's equations. Because of symmetry, he used a specific term, but later radio waves were discovered by Hertz. Many other unknown quantities are predicted mathematically; we will not repeat them here.
Wigner points out the fundamental role of invariance. This is the main part of mathematics and science in general. It was the lack of invariance of the Newton equations (the need for an absolute reference system for velocities) that led Lorentz, Fitzgerald, Poincaré and Einstein to the special theory of relativity.
Wigner also notes that the same mathematical concepts appear in completely unexpected connections. For example, trigonometric functions that appear in Ptolemy's astronomy turn out to be functions invariant with respect to shift (temporal invariance). They are also suitable functions for linear systems. The enormous utility of the same parts of mathematics in a variety of situations has no rational explanation (so far).
Moreover, the simplicity of mathematics has long been considered the key to application in physics. Einstein believed it. But even in mathematics itself, simplicity is wonderful, at least for me; the simplest algebraic equations, linear and quadratic, correspond to the simplest geometric objects, straight lines, circles and conic sections. This makes the implementation of analytic geometry possible. How can it be that simple mathematics, which is ultimately a product of the human mind, can be so useful in many different situations?
Because of these successes of mathematics, there is currently a tendency for each of the sciences to be mathematical. This is considered a goal to be achieved, if not today, then tomorrow. For such an audience, I will use physics and astronomy for further examples.
Pythagoras is the first person to say that “mathematics is a way to understand the universe.” He said loudly and clearly: "The number is the measure of all things."
Kepler is another example. He passionately believed that the work of God’s hands could only be understood through mathematics. After twenty years of tedious computation, he found his famous three laws of planetary motion — three relatively simple mathematical expressions describing the complex motions of the planets.
It is Galileo who owns the words: "The laws of nature are written in the language of mathematics." Newton used the results of Kepler and Galileo to deduce the famous laws of motion, which, together with the law of the law, are perhaps the most famous example of the unjustified effectiveness of mathematics in science. They not only predicted where the planets would be, but also successfully predicted the positions of unknown planets, the movements of distant stars, tides, etc.
Science consists of laws that were originally based on a small, carefully selected set of observations, often not very accurately measured; but it was subsequently established that these laws apply over much wider ranges of observations and much more accurately than the original data allowed. Not always, of course, but most often enough to explain the phenomena.
In my thirty years of mathematical practice in industry, I often worried about the predictions I made. From the mathematical calculations that I did in my office, I confidently (at least, others) predicted some future events - if you do this and that, you will see this and that, and usually turned out to be right. How could phenomena know what I predicted (based on human mathematics) to be consistent with my predictions? It's funny to think that this is happening. No, math somehow provides a reliable model for most of what happens in the universe. And since I can only use relatively simple mathematics, how can it be that simple mathematics is enough to predict so much?
I could cite more examples illustrating the unreasonable efficiency of mathematics, but it would be boring. I suspect that many of you know examples that I don’t know. So let me suggest that you give me a very long list of successes, many of which are as impressive as the prediction of a new planet, a new physical phenomenon, or a new artifact. I don’t have much time and I want to spend it on trying to do what I think is left out of Wigner - to give at least some partial answers to the question in the title of the article.
Looking at the effectiveness of mathematics, we need to pay attention to the question: “What is mathematics?” This is the title of the famous book by Courant and Robbins [2. R. Courant and H. Robbins, What is Mathematics? Oxford University Press, 1941]. In it they do not try to give a formal definition, rather they show mathematics through certain examples. In the same way, I will not give an exhaustive definition. But in more detail, than they, I will try to discuss some of the characteristic features of mathematics, as I see them.
Perhaps the best way to approach the question of what math is is to start from the beginning. In the distant prehistoric past, where we should look for the beginning of mathematics, there were already four major facets of mathematics. First, there was the ability to lead long chains of reasoning, which to this day characterize a significant part of mathematics. Secondly, there was geometry leading through the concept of continuity to topology and beyond. Thirdly, there was a number leading to arithmetic, algebra and beyond. Finally, there was an artistic taste that plays such a large role in modern mathematics. Of course, in mathematics there are many different types of beauty. In number theory, this seems to be basically the beauty of almost endless details; in abstract algebra, beauty is mostly in common. Thus, different areas of mathematics have different standards of aesthetics.
Of course, the earliest history of mathematics should be a complete speculation, because now there is not and there is no real convincing evidence of it. It seems, however, that in the very foundations of primitive life arose for nothing more than for the purpose of survival an understanding of cause and effect. As soon as a sign passes from one observation to the sequence "If it is, then this, and then it follows that ..." we are on the path of the first feature of mathematics, about which I spoke, of long chains of reasoning. But it is difficult for me to understand how simple Darwinian survival of the fittest will lead to the selection of the ability to make long chains of reasoning, which, apparently, are required by mathematics and science.
Geometry may have arisen due to the decoration of the human body for various purposes, such as religious rites, communication and engagement of the opposite sex, as well as to decorate the surfaces of walls, pots, dishes and clothing. It also implies a fourth aspect, which I mentioned, aesthetic taste, and this is one of the deep foundations of mathematics. Most textbooks repeat after the Greeks and say that geometry arose from the needs of the Egyptians to explore the land after each flood on the Nile, but I attribute much more to aesthetics than most historians of mathematics, and, accordingly, pay less attention to its applied goals.
The third aspect of mathematics, numbers, arose from counting. As a famous mathematician once said: “God created whole numbers, man did the rest” [3. L. Kronecker, p. 1634. In the section "Mathematics and Mathematics", RE Moritz.]. The whole numbers seem so fundamental to us that we expect to find them wherever we find intelligent life in the Universe. I tried with some success to get some of my friends to understand my surprise at the possibility and usefulness of abstraction of integers for counting. Isn't it wonderful that 6 sheep plus 7 sheep make up 13 sheep; that 6 stones plus 7 stones make up 13 stones? Is it not a miracle that the universe is built in such a way that such a simple abstraction as a number is possible? For me, this is one of the most powerful examples of the unreasonable effectiveness of mathematics. This is strange and inexplicable.
Working with numbers, we came to the conclusion that these countable integers were successfully used to measure how many times a standard length can be used to extract the desired measured length. But it must have happened rather quickly that the integer number of units did not exactly correspond to the measured length, and the gauges were converted to fractions — an additional part that was used to measure the standard length. Shots do not count as numbers; they measure numbers. Due to their general use in measuring fractions, they soon discovered that they obey the same rules of manipulation as whole numbers, with the added advantage that they made possible in all cases (I did not mention zero). Some familiarity with fractions shows that between any two fractions you can put as many as you wish, and that in a sense they are uniformly dense everywhere. But when we expand the concept of number to include fractions, we must abandon the idea of the next number.
This again brings us back to Pythagoras, the first person to prove that the diagonal of a square and the sides of a square do not have a common measure - that they are irrationally related. This observation seems to have produced a profound upheaval in Greece: mathematics. Until that time, a system of discrete numbers and continuous geometry flourished side by side with little conflict. The crisis of incommensurability was replaced by a Euclidean approach to mathematics. It is curious that the early Greeks tried to make mathematics rigorous, replacing the uncertainty of numbers with what they felt was more specific geometry (because of Eudox). This was a great event for Euclid, and as a result you will find in the "Principles" [4. Euclid, Euclidean Principles, T. E. Heath, Dover Publications, New York, 1956.] a lot, as a result of which we now consider number theory and algebra, cast in the form of geometry. Unlike the ancient Greeks, who doubted the existence of a system of real numbers, we decided that there should be a number that measures the diagonal length of a unit square (although we don’t need this), and, no less, we expanded the system of rational numbers by including algebraic numbers. It was a simple desire to measure length. How can anyone deny that there is a number to measure the length of any straight line segment?
Algebraic numbers, which are the roots of polynomials with integral, fractional and, as was later proved, even algebraic numbers as coefficients, were soon under control, simply expanding the same operations that were used in a simpler number system.
However, measuring the circumference with respect to its diameter soon forced us to consider a relation called pi. This is not an algebraic number, since no linear combination of degree pi with integer coefficients will definitely disappear. One length, a circle that is a curved line, and another length, a diameter that is a straight line, makes the existence of a relation less defined than the ratio of the diagonal of a square to its side; but since it seems that there should be such a number, the transcendental numbers gradually fall into the numerical system. Thus, by further suitable expansion of the earlier representations of numbers, transcendental numbers have been consistently introduced into the numerical system, although few students are satisfied with the technical apparatus that we usually use to show the sequence.
Further study of the numerical system introduced both zero and negative numbers. This time the expansion required us to stop dividing for the singular number by zero. It seems that this complements the system of real numbers for us (as long as we restrict ourselves to the process of accepting the restrictions of sequences of numbers and do not allow further operations) - it’s not so that we still have a solid, logical, simple foundation for them; but they say that acquaintance breeds contempt, and we are all more or less familiar with the system of real numbers. Very few of us believe that the specific postulates that some logic came up with, create numbers - no, most of us believe that the real numbers just are and that it was an interesting, fun and important game - to find a good set of postulates to take them into account. But let's not confuse ourselves. The paradox of Zeno is still, even after 2000 years, too fresh in our minds to deceive ourselves with what we understand everything we would like to say about the relationship between the system of discrete numbers and the continuous line we want to model. We know that from a non-standard analysis of logic, postulates can be made that put entities on a real line, but so far few of us have wanted to follow this path. It is fair to say that there are some mathematicians who doubt the existence of the usual system of real numbers. Several computer theorists admit the existence of only "computable numbers."
The next step in the discussion is a complex system of numbers. When I read the story, it was Cardano who first understood them in some real sense. In his "Great Art" or "Rules of Algebra" [5. G. Cardano, Great art or rules of algebra, translation. “Putting aside the mental torture involved in multiplying (5 + sqrt 15) by (5 - sqrt -15) with a score of 25 - (- 15) ...”. Thus, he clearly realized that the same formal operations on symbols for complex numbers would give meaningful results. Thus, the system of real numbers gradually expanded to a system of complex numbers, except that this time the extension required abandoning the ordering property of numbers — complex numbers cannot be ordered in the usual sense.
Obviously, Cauchy arrived at the theory of complex variables with the task of integrating real functions along the real line. He found that by transferring integration to a complex plane, he could solve integration problems in real numbers.
A few years ago I had the pleasure of teaching a course on complex variables. As always happens when I participate in this topic, I left again with the feeling that "God made the universe out of complex numbers." It is clear that they play a central role in quantum mechanics. They are a natural tool in many other applications, such as electrical circuits, fields, etc.
Summarizing, from simple counting using God-given integers, we created various extensions of the idea of a number to include more things. Sometimes extensions were made for aesthetic reasons, and often we abandoned some property of an earlier number system. Thus, we have come to a number system that is unreasonably effective even in mathematics itself; This shows that we have solved many problems of the number theory of the original system with a high discrete estimate using a complex variable.
From the above, we see that one of the main areas of mathematics is expansion, generalization, abstraction - it is more or less the same thing - known concepts for new situations. But note that in the process itself the definitions themselves change subtly. Therefore, what is not so widely known, the old proofs of the theorems can become false proofs. Old evidence no longer covers newly defined items. The miracle is that almost always the theorems are still true; it's just a matter of fixing evidence. A classic example of this correction is the Euclidean Principles [4]. ( , , ), . , ? , , , . . , , , , , . , — ? , , , , , . - .
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Reprinted From: The American Mathematical Monthly Volume 87 Number 2 February 1980
katifa. , magisterludi2016@yandex.ru
Gritsuk , Sirion , qw1 , KirillGuzenko , NeKpoT , .
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Prologue
From the title it is clear that this is a philosophical question. I will not apologize for the philosophy, although I know that many scientists, engineers and mathematicians do not pay attention to it. Instead, I will give this small prologue to substantiate my approach.
')
A person, as far as we know, has always wondered about himself, the world around him and the meaning of life. Many myths tell of why and how God or gods created man and the universe. These are theological explanations. Their distinguishing feature - it makes no sense to ask why things are as they are, if the gods created them so.
Philosophy arose when a person wondered about a world outside this theological framework. For example, ancient philosophers represented the world as a combination of earth, fire, water and air. No doubt they were convinced that the world was created by the gods and there is no need to worry about this.
From these early attempts to explain the world, philosophy gradually emerged, as did modern science. Not that science explains "why" things are as they are - gravity does not explain "why" objects fall, but science gives us such a detailed explanation of "how" this happens that we have the feeling that we know "why" . Let's be clear; a sea of interconnected facts allows science to say "why" the universe is just that.
Mathematics is our main tool for performing long chains of scientific reasoning. It can be defined as a mental tool for this purpose. For centuries, people have asked themselves the question, the same one that is in the headline - “Why is mathematics so unreasonably effective?” In asking this question, we focus on the logical rather than the material side of how the universe works and works.
Mathematical mathematicians are mostly concerned with self-consistency and the limits of the system. They do not care why the world seems to admit a logical explanation. In this sense, I am closer to the early Greek philosophers who were interested in the material side of the question, and my answers in relation to logic are not much better than theirs. But once and somewhere we must begin to explain the phenomenon of the fact that the world is logically structured in accordance with the laws of mathematics, and that mathematics is the language of science and technology.
After I marked the main topic, you need to understand how to convey these ideas to the audience. Experience shows that I am not always good at it. It became clear to me that some preliminary remarks would be able to help.
In some respects, this discussion is theorized. I must touch, at least slightly, the main theories of mathematics, as well as its individual parts. Further, there are various applied theories. So we will move from one theory to another. You may be surprised that I will use an experimental approach for discussion. No matter what the theories should be, or how you imagine them, or what the experts' point of view is, let's apply the scientific approach and find out everything. I understand perfectly well that my words, especially about the nature of mathematics, will annoy many mathematicians. My experimental approach is contrary to their mindset and preconceived beliefs. So be it!
The inspiration for this article was the article with the similar title "The unreasonable effectiveness of mathematics in the natural sciences . " (1. EP Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math., 13 (Feb. 1960) E.P. Wigner. I removed part of the title and do not repeat his article (I just can’t write better. On the other hand, I will spend much more time explaining the question from the title of the article. But when all the explanations are over, the rest will be so great that our question will practically remain unanswered.
Mathematics Efficiency
In his article, Wigner gives many examples of the effectiveness of mathematics in the physical sciences. Nevertheless, let me draw on my own experience, which is closer to engineering. My first experience of using mathematics to predict was related to the development of atomic bombs during World War II. How did it happen that the numbers that we so patiently calculated on primitive relay computers agreed so well with what happened on the first test in Almagordo? We could not conduct a small experiment to verify the calculations. Later, the experience with guided missiles showed me that this was not an accident - our mathematical predictions are constantly justified in the real world. Working in the Bell System, I did a lot of calculations for telephony and other mathematical work on subjects such as traveling wave tubes, alignment of television lines, stability of complex communication systems, blocking calls through a central telephone - this is only a small part. In order to show off knowledge, I can conduct research on transistors, space flight, and computer-aided design, but almost all science and technology uses an extensive mathematical apparatus with overwhelming success.
Many of you are familiar with the history of Maxwell's equations. Because of symmetry, he used a specific term, but later radio waves were discovered by Hertz. Many other unknown quantities are predicted mathematically; we will not repeat them here.
Wigner points out the fundamental role of invariance. This is the main part of mathematics and science in general. It was the lack of invariance of the Newton equations (the need for an absolute reference system for velocities) that led Lorentz, Fitzgerald, Poincaré and Einstein to the special theory of relativity.
Wigner also notes that the same mathematical concepts appear in completely unexpected connections. For example, trigonometric functions that appear in Ptolemy's astronomy turn out to be functions invariant with respect to shift (temporal invariance). They are also suitable functions for linear systems. The enormous utility of the same parts of mathematics in a variety of situations has no rational explanation (so far).
Moreover, the simplicity of mathematics has long been considered the key to application in physics. Einstein believed it. But even in mathematics itself, simplicity is wonderful, at least for me; the simplest algebraic equations, linear and quadratic, correspond to the simplest geometric objects, straight lines, circles and conic sections. This makes the implementation of analytic geometry possible. How can it be that simple mathematics, which is ultimately a product of the human mind, can be so useful in many different situations?
Because of these successes of mathematics, there is currently a tendency for each of the sciences to be mathematical. This is considered a goal to be achieved, if not today, then tomorrow. For such an audience, I will use physics and astronomy for further examples.
Pythagoras is the first person to say that “mathematics is a way to understand the universe.” He said loudly and clearly: "The number is the measure of all things."
Kepler is another example. He passionately believed that the work of God’s hands could only be understood through mathematics. After twenty years of tedious computation, he found his famous three laws of planetary motion — three relatively simple mathematical expressions describing the complex motions of the planets.
It is Galileo who owns the words: "The laws of nature are written in the language of mathematics." Newton used the results of Kepler and Galileo to deduce the famous laws of motion, which, together with the law of the law, are perhaps the most famous example of the unjustified effectiveness of mathematics in science. They not only predicted where the planets would be, but also successfully predicted the positions of unknown planets, the movements of distant stars, tides, etc.
Science consists of laws that were originally based on a small, carefully selected set of observations, often not very accurately measured; but it was subsequently established that these laws apply over much wider ranges of observations and much more accurately than the original data allowed. Not always, of course, but most often enough to explain the phenomena.
In my thirty years of mathematical practice in industry, I often worried about the predictions I made. From the mathematical calculations that I did in my office, I confidently (at least, others) predicted some future events - if you do this and that, you will see this and that, and usually turned out to be right. How could phenomena know what I predicted (based on human mathematics) to be consistent with my predictions? It's funny to think that this is happening. No, math somehow provides a reliable model for most of what happens in the universe. And since I can only use relatively simple mathematics, how can it be that simple mathematics is enough to predict so much?
I could cite more examples illustrating the unreasonable efficiency of mathematics, but it would be boring. I suspect that many of you know examples that I don’t know. So let me suggest that you give me a very long list of successes, many of which are as impressive as the prediction of a new planet, a new physical phenomenon, or a new artifact. I don’t have much time and I want to spend it on trying to do what I think is left out of Wigner - to give at least some partial answers to the question in the title of the article.
What is math?
Looking at the effectiveness of mathematics, we need to pay attention to the question: “What is mathematics?” This is the title of the famous book by Courant and Robbins [2. R. Courant and H. Robbins, What is Mathematics? Oxford University Press, 1941]. In it they do not try to give a formal definition, rather they show mathematics through certain examples. In the same way, I will not give an exhaustive definition. But in more detail, than they, I will try to discuss some of the characteristic features of mathematics, as I see them.
Perhaps the best way to approach the question of what math is is to start from the beginning. In the distant prehistoric past, where we should look for the beginning of mathematics, there were already four major facets of mathematics. First, there was the ability to lead long chains of reasoning, which to this day characterize a significant part of mathematics. Secondly, there was geometry leading through the concept of continuity to topology and beyond. Thirdly, there was a number leading to arithmetic, algebra and beyond. Finally, there was an artistic taste that plays such a large role in modern mathematics. Of course, in mathematics there are many different types of beauty. In number theory, this seems to be basically the beauty of almost endless details; in abstract algebra, beauty is mostly in common. Thus, different areas of mathematics have different standards of aesthetics.
Of course, the earliest history of mathematics should be a complete speculation, because now there is not and there is no real convincing evidence of it. It seems, however, that in the very foundations of primitive life arose for nothing more than for the purpose of survival an understanding of cause and effect. As soon as a sign passes from one observation to the sequence "If it is, then this, and then it follows that ..." we are on the path of the first feature of mathematics, about which I spoke, of long chains of reasoning. But it is difficult for me to understand how simple Darwinian survival of the fittest will lead to the selection of the ability to make long chains of reasoning, which, apparently, are required by mathematics and science.
Geometry may have arisen due to the decoration of the human body for various purposes, such as religious rites, communication and engagement of the opposite sex, as well as to decorate the surfaces of walls, pots, dishes and clothing. It also implies a fourth aspect, which I mentioned, aesthetic taste, and this is one of the deep foundations of mathematics. Most textbooks repeat after the Greeks and say that geometry arose from the needs of the Egyptians to explore the land after each flood on the Nile, but I attribute much more to aesthetics than most historians of mathematics, and, accordingly, pay less attention to its applied goals.
The third aspect of mathematics, numbers, arose from counting. As a famous mathematician once said: “God created whole numbers, man did the rest” [3. L. Kronecker, p. 1634. In the section "Mathematics and Mathematics", RE Moritz.]. The whole numbers seem so fundamental to us that we expect to find them wherever we find intelligent life in the Universe. I tried with some success to get some of my friends to understand my surprise at the possibility and usefulness of abstraction of integers for counting. Isn't it wonderful that 6 sheep plus 7 sheep make up 13 sheep; that 6 stones plus 7 stones make up 13 stones? Is it not a miracle that the universe is built in such a way that such a simple abstraction as a number is possible? For me, this is one of the most powerful examples of the unreasonable effectiveness of mathematics. This is strange and inexplicable.
Working with numbers, we came to the conclusion that these countable integers were successfully used to measure how many times a standard length can be used to extract the desired measured length. But it must have happened rather quickly that the integer number of units did not exactly correspond to the measured length, and the gauges were converted to fractions — an additional part that was used to measure the standard length. Shots do not count as numbers; they measure numbers. Due to their general use in measuring fractions, they soon discovered that they obey the same rules of manipulation as whole numbers, with the added advantage that they made possible in all cases (I did not mention zero). Some familiarity with fractions shows that between any two fractions you can put as many as you wish, and that in a sense they are uniformly dense everywhere. But when we expand the concept of number to include fractions, we must abandon the idea of the next number.
This again brings us back to Pythagoras, the first person to prove that the diagonal of a square and the sides of a square do not have a common measure - that they are irrationally related. This observation seems to have produced a profound upheaval in Greece: mathematics. Until that time, a system of discrete numbers and continuous geometry flourished side by side with little conflict. The crisis of incommensurability was replaced by a Euclidean approach to mathematics. It is curious that the early Greeks tried to make mathematics rigorous, replacing the uncertainty of numbers with what they felt was more specific geometry (because of Eudox). This was a great event for Euclid, and as a result you will find in the "Principles" [4. Euclid, Euclidean Principles, T. E. Heath, Dover Publications, New York, 1956.] a lot, as a result of which we now consider number theory and algebra, cast in the form of geometry. Unlike the ancient Greeks, who doubted the existence of a system of real numbers, we decided that there should be a number that measures the diagonal length of a unit square (although we don’t need this), and, no less, we expanded the system of rational numbers by including algebraic numbers. It was a simple desire to measure length. How can anyone deny that there is a number to measure the length of any straight line segment?
Algebraic numbers, which are the roots of polynomials with integral, fractional and, as was later proved, even algebraic numbers as coefficients, were soon under control, simply expanding the same operations that were used in a simpler number system.
However, measuring the circumference with respect to its diameter soon forced us to consider a relation called pi. This is not an algebraic number, since no linear combination of degree pi with integer coefficients will definitely disappear. One length, a circle that is a curved line, and another length, a diameter that is a straight line, makes the existence of a relation less defined than the ratio of the diagonal of a square to its side; but since it seems that there should be such a number, the transcendental numbers gradually fall into the numerical system. Thus, by further suitable expansion of the earlier representations of numbers, transcendental numbers have been consistently introduced into the numerical system, although few students are satisfied with the technical apparatus that we usually use to show the sequence.
Further study of the numerical system introduced both zero and negative numbers. This time the expansion required us to stop dividing for the singular number by zero. It seems that this complements the system of real numbers for us (as long as we restrict ourselves to the process of accepting the restrictions of sequences of numbers and do not allow further operations) - it’s not so that we still have a solid, logical, simple foundation for them; but they say that acquaintance breeds contempt, and we are all more or less familiar with the system of real numbers. Very few of us believe that the specific postulates that some logic came up with, create numbers - no, most of us believe that the real numbers just are and that it was an interesting, fun and important game - to find a good set of postulates to take them into account. But let's not confuse ourselves. The paradox of Zeno is still, even after 2000 years, too fresh in our minds to deceive ourselves with what we understand everything we would like to say about the relationship between the system of discrete numbers and the continuous line we want to model. We know that from a non-standard analysis of logic, postulates can be made that put entities on a real line, but so far few of us have wanted to follow this path. It is fair to say that there are some mathematicians who doubt the existence of the usual system of real numbers. Several computer theorists admit the existence of only "computable numbers."
The next step in the discussion is a complex system of numbers. When I read the story, it was Cardano who first understood them in some real sense. In his "Great Art" or "Rules of Algebra" [5. G. Cardano, Great art or rules of algebra, translation. “Putting aside the mental torture involved in multiplying (5 + sqrt 15) by (5 - sqrt -15) with a score of 25 - (- 15) ...”. Thus, he clearly realized that the same formal operations on symbols for complex numbers would give meaningful results. Thus, the system of real numbers gradually expanded to a system of complex numbers, except that this time the extension required abandoning the ordering property of numbers — complex numbers cannot be ordered in the usual sense.
Obviously, Cauchy arrived at the theory of complex variables with the task of integrating real functions along the real line. He found that by transferring integration to a complex plane, he could solve integration problems in real numbers.
A few years ago I had the pleasure of teaching a course on complex variables. As always happens when I participate in this topic, I left again with the feeling that "God made the universe out of complex numbers." It is clear that they play a central role in quantum mechanics. They are a natural tool in many other applications, such as electrical circuits, fields, etc.
Summarizing, from simple counting using God-given integers, we created various extensions of the idea of a number to include more things. Sometimes extensions were made for aesthetic reasons, and often we abandoned some property of an earlier number system. Thus, we have come to a number system that is unreasonably effective even in mathematics itself; This shows that we have solved many problems of the number theory of the original system with a high discrete estimate using a complex variable.
From the above, we see that one of the main areas of mathematics is expansion, generalization, abstraction - it is more or less the same thing - known concepts for new situations. But note that in the process itself the definitions themselves change subtly. Therefore, what is not so widely known, the old proofs of the theorems can become false proofs. Old evidence no longer covers newly defined items. The miracle is that almost always the theorems are still true; it's just a matter of fixing evidence. A classic example of this correction is the Euclidean Principles [4]. ( , , ), . , ? , , , . . , , , , , . , — ? , , , , , . - .
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Reprinted From: The American Mathematical Monthly Volume 87 Number 2 February 1980
katifa. , magisterludi2016@yandex.ru
Gritsuk , Sirion , qw1 , KirillGuzenko , NeKpoT , .
:
Source: https://habr.com/ru/post/359056/
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