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REVIEWS, BIBLIOGRAPHY
This issue publishes excerpts from a recently published book by the famous English teacher and popularizer of mathematics Walter Warwick Sawyer *]. It is clear that it is extremely difficult at a sufficiently high and at the same time accessible level for a wide audience to show a panorama of important sections of modern mathematics without embellishment, without vagueness and distortion.
W. W. Sawyer is one of those authors who succeeds in this. It contains fascinating and so significant content about some of the features of the creative activity of professional mathematicians.
We hope that reading this book will give you satisfaction and give you a glimpse into the world of mathematics and mathematicians. To understand what you read, “the most vague memories of school mathematics are enough.”
The publication was prepared by V. N. Berezin and M. L. Smolyansky.
Almost all mathematical discoveries are based on a very simple idea. Textbooks often hide this fact. They usually contain cumbersome conclusions and thereby create the impression that mathematicians are people who spend their entire lives behind desks and translating tons of paper. This is nonsense. Many mathematicians work very successfully in the bath, in bed, on trains or while riding a bicycle (preferably in light traffic). Mathematical calculations are made before or after opening. The discovery itself will arise from the basic ideas.
Few people realize how vast the scope of modern mathematics is. It would probably be easier to master all existing languages than all the mathematical knowledge currently known. It seems to me that all languages could be learned in one lifetime; but mathematics, of course, does not have everything.
*) W. W. Sawyer, Propositions to Mathematics, M., “Enlightenment”, 1972.
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user’s But mathematics is not an applied science; it is important in itself. What is important is not the usefulness of mathematics, but its harmony. Applied mathematics is the most boring part of mathematics. Look at the people working on number theory that has no practical application - would you offer them a job as an accountant?
All mathematicians are characterized by audacity of mind. A mathematician does not like to be told about anything; he wants to figure out everything himself. Of course, a mature mathematician, having learned about some great discovery, wonders what it is, and does not waste time in discovering it already openly. But I mean young mathematicians, in whom the audacity of their minds is especially pronounced. If, for example, you teach geometry to nine- to ten-year-old children and tell them that it is impossible to divide an angle into three equal parts using a ruler and compass, you will certainly see that one or two boys will stay after lessons and try to find a solution. The fact that the pilot has not solved this problem for two thousand years does not stop them from hoping that they will be able to do it during an hour’s lunch break. This, of course, is not very modest, but it also does not indicate their arrogance. They are ready to take on any challenge. But in fact, it has already been proven that it is impossible to divide an angle into three equal parts using a ruler and compass. Their attempt to find a solution is the same as trying to represent as a rational fraction.
A good student always tries to get ahead of himself. If you explain to him how to solve a quadratic equation by adding to a complete square, he will certainly want to know whether it is possible to solve a cubic equation by adding to a complete cube. The rest of the class does not ask such questions. They’ve had enough of quadratic equations; they don’t feel any polygonal difficulties.
This desire to explore is the second distinctive feature of black mathematics. This is one of the forces that contains the growth of mathematics. A mathematician gets satisfaction from the knowledge he has already mastered and always strives for new knowledge.
I have already spoken about interest in laws - the third necessary quality of a mathematician. Already at the very beginning of arithmetic, patterns are encountered. For example, from four identical stones you can make a square, and from a heel - a kelz. Mathematical and musical abilities manifest themselves very early, from the age of four, and sometimes earlier. One boy once told me: “I like the word September, because it turns out sErtEmbEr.” I myself have never noticed the natural progression of vowels and consonants in this word. They are truly perfectly symmetrical. Such a child will, of course, enjoy arithmetic.
Further, in elementary school, you can develop skills in mathematical laws. Much of Gauss’s early work stemmed from his involvement in doing calculations and analyzing the results. Hermite, one of the greatest French mathematicians, also emphasized how observation often leads to mathematical discoveries. True, one is not enough to become a great mathematician.
Just as being smart at noticing arithmetic patterns helps a student solve arithmetic problems, smartly observing patterns in algebra helps them avoid mistakes or detect errors.
Sometimes, looking at a work of art, we admire its beauty and feel its significant condition, but cannot say to do it. One post opposed the anti-war promise to force children to retell poems in their own words. He said that the only way to learn poetry control is to write better poems. But this, of course, cannot be demanded of children.
But in mathematics, as a rule, things were different. If we encounter a pattern, we must follow the rules: “Why does it occur? What does it mean? And we can usually find answers to these questions. So, whenever a pattern appears, the mathematician feels that he has an obligation to find out why it appears.
The wider the range of questions to which some general principle is applicable, the more often it will help us get out of difficulties. Poincaré said: “Suppose I started a complex calculation and, with great difficulty, finally got the result; but all my efforts will be in vain if they do not help to foresee the result in other analogous calculations, if they do not provide the opportunity to carry them out with confidence, avoiding those errors and forgettings from which I must
was put up for the first time.”
Once generalized, the result becomes more useful. You might be surprised to learn that generalization almost always simplifies the result as well. A more general conclusion is easier to accept than a less general one.
The following trivial problem can serve as an illustration. “One glass contains ten spoons of water, the other contains ten spoons of wine. A spoonful of water from the first glass is poured into the second. The mixture is vigorously shaken, then a spoonful of the resulting mixture is poured into the first glass. The question is, what will be more: wine in the first glass or water in the second?
Obviously, in order to answer this question, we can continue with the following calculation. After a spoonful of water was poured into the second glass, there were 10 spoons of wine and 1 spoon of water, that is, a total of 11 spoons of liquid. Thus, one spoon of the mixture contains spoons of wine and spoons of water. We pour one spoon of the mixture into the first glass: it now contains water, and spoons of wine, in the second glass there will be spoons of water and spoons of wine. So, we see that the amount of wine in the first glass is exactly equal to the amount of water in the second glass.
This result may seem random, but if you change the conditions of the problem, you will find that you will always get the same quantities. If given spoons of water and spoons of wine, then the amount of wine that gets into the first glass will invariably be equal to the amount of water that gets into the second. Next, start with the first quantities of liquid in the glasses, for example, with spoons of water and spoons of wine, and then continue all the procedures as in the original problem, then in the end there will be as much wine in the first glass as there is water in the second.
This is a good example of mathematical tastelessness. In a clear and well-mathematical proof, the result does not appear unexpectedly in the last line, its logical inevitability should be visible throughout the entire proof.
The above trivial task uses a kind of masking. It provides unnecessary information that distracts our attention from the existence of the problem. The original condition is the sentence: “The mixture will be vigorously stirred.” In fact, all that matters is that we pour a spoonful of liquid from the first glass into the second, and then a spoonful of the mixture from the second glass into the first. It doesn’t matter at all what kind of liquid is poured, all that matters is that in the end both glasses contain the same amount of liquid as at the very beginning. And if this is really the case, then as much wine as needed is poured into the first glass,
so that the water would then be pumped up. Of course, the water taken from the first glass will be found in the second. The quantities must be equal, and neither kindness nor algebra have anything to do with it.
In general, this problem looks like this: we have a glass of water and a glass of wine, we perform a number of operations with these liquids, so that in the end the amount of liquid in each glass is equal to the initial one, then the amount of water that gets into the wine must be equal to the amount of wine in the second.
This is so obvious that it is hardly worth mentioning. The general appearance of the city’s problem is simpler than those continued earlier.
The areas of application of the problem in the general form of the city are simpler. You can pour spoons of liquid from one glass to another and back again as many times as you like, and yet the general principle remains the same.
Thus, the study of the problem consists of discarding all unnecessary data and leaving only the essential necessary conditions. The less data, the easier it is to find a solution. A general theorem rarely contains anything confusing; its purpose is to bring important facts to your attention.
In elementary mathematics we find a mixture of all sorts of important details. In higher mathematics we try to separate different elements and study which one separately. In this sense, mathematics came out, please, than elementary.
But one cannot judge the importance of any mathematical research by individual particular problems. Topology can serve as an example of this. Topology is sometimes called “rubber geometry”—the geometry of figures; for example, topology can serve this purpose. In some cases this is true - it actually considers the properties of such figures. But its significance is due to the fact that there is no constant length on an elastic sheet. There cannot be any theorems similar to the Pythagorean theorem. Is it possible to draw conclusions like: “this curve is continuous, and that one is divided into two separate parts.” Continuity is the main property studied by topology. It talks about properties that are preserved under continuous transformations. Since there are so many things that can be transformed in this way, topology covers a wide range of topics. It increasingly attracts both pure mathematicians and engineers; it can be used to prove a number of surprising results. It combines the highest degree of generalization with maximum simplicity.
Everything we talked about above has the goal of expanding the scope of questions that are subject to mathematics. Research, discovery of patterns, explanation of the meaning of each pattern, depiction of new patterns based on the model of already known ones—all these types of activities expand the scope of mathematics. From a practical point of view it becomes extremely difficult to keep track of all the results obtained, and it cannot be said that the accumulation of unrelated theorems represents a welcome solution. Being both business people and artists, mathematicians feel the need to collect all these disparate results.
It is not surprising that the entire history of mathematics consists of alternating processes of “expansion” and “contraction.” For example, some problem attracts the attention of mathematicians, hundreds of articles are written, each of which covers only one side of the truth. The question is growing. Then some genius, relying on all the data, the collection with such difficulty, declares: “Everything that we know will become almost very similar if you look at it from this point of view.” After this, there is no longer any need for anyone, except historians of mathematics, to study hundreds of individual articles. Disparate conclusions are united into one simple doctrine, important facts are separated from the chaff, and the direct path to the desired conclusion is open to all. The amount of information that needs to be studied has been reduced. But this is not the end. After a new method has become worthy, new questions arise for which it is not sufficient to solve, and the search for answers begins again, articles are published again, and the process of “extension” begins again.
If it were possible to lead knowledge to two general laws, the mathematician would not be a Udoble. He would not rest until he had proved that both of these laws were based on the same principle. But even then he would be happy; on the contrary, he would become unhappy, since he would have nothing to do. But the prospect of such a disruption is completely unreal. Life is such that solving one problem always creates a new problem; otherwise life was unbearable. There is and always will be material to study and difficulties to overcome.
And it will always be like this. And if it turned out that all that is essential in mathematics is considering only phenomena with properties A, B and C, mathematicians would immediately ask: “What will happen if some object has only one of them?” And they would plunge back into work.
Let’s take a moment to look at the various patterns from a utilitarian point of view, namely, from the position of a person who simply needs to take an exam. The most important question for the examinee is the character of the examiner: what interests him? What qualities of the examinee is he trying to test?
Many examiners are only interested in the student’s ability to perform formulaic operations: does he know the multiplication tables? Can you use logarithms? Can you use hundreds of other common techniques? There is no doubt that it is necessary to test students’ knowledge of general operations. But since childhood, I find such checks the most boring,
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the dumbest exams.
And yet they are very much loved by mediocre teachers, who believe that their task is simply to drill the class through exercises “from now to now.”
Other examiners try to encourage proactive students who are trying to grasp not only the facts, but also a taste for the subject. These examiners want to test entrepreneurship, imagination and initiative in learning, so they look for problems that will test these qualities.
Some teachers believe that students cannot be prepared for such unpredictable exams. But this is not true. Of course, it is impossible, for example, to predict the course of the battle. But is this why it is impossible to train military personnel? The training of officers is only partly based (or should be based) on general principles common to all branches; the other part should be the development of their own initiative. To this end, the future office worker in the classroom found himself in a number of difficult unforeseen situations, where he must test his determination and initiative. Exactly the same approach is possible to teaching mathematics in the world. There is a lot of promise internationally and in the military.
An exam ticket may contain both series of problems and questions that require communication. How does a student know what kind of task is before him? After all, tasks do not always differ in appearance.
In the tasks that life sets before us, nature itself is the examiner. And here again it is extremely important to reduce the required approach or it has some special property that makes it possible to find a simpler solution.
Let’s look at one example that builds one of the very important uses of translation, namely the presentation of a problem in such a form that the answer can be seen at a glance. Such a translation does not significantly change the task; If we approach it from a purely mathematical point of view, we can say that it does not change anything at all. But for us this translation is very valuable, because it turns the unfamiliar into the familiar.
An example of such a translation is a graphical representation of a function. The graph is easier to understand at a glance. Here we will make a translation from algebra to geometry. Very often we have to do the reverse translation—from geometry to algebra; You will find a translation of this type nearby in this book. You can also translate one geometric problem into another, but in such a way that the unknown type of problem is translated into a known one.
Consider the problem with a chess knight. The knight stands in the center of a square board of 25 fields. The horse must move so as to visit each square once and only once.
Reflecting on this problem, I find it quite difficult to wonder whether our knight gets into a difficult position, say, after board moves; eyes on squares where the knight has not yet been, I cannot say whether they connect into a simple goal of moves. Is it possible to have several
Drawing such a diagram is not as simple as it seems at first glance. You need to be careful enough to ensure that you don’t end up with a too tangled network of lines. The main principle that I adhered to was the observance of symmetry. Square 13 should be placed in the center, and the rest around it, but in such a way as to maintain the symmetry of the chessboard.
Now it is very easy to easily choose the horse’s route. He can, for example, go from 13 to 10, then go along the “inner circle” (19, 22, 11, 2, 9, 20, 23, 16, 7, 4, 15, 24, 17, 6, 3) and go to the “outer circle” (12, 21, 18, 25, 14, 5, 6, 1).
We don’t need to go into theories about the moves of a chess knight. The example was chosen not because the problem is important, but to show how the ability to visualize judgment simplifies finding a solution.