Arnold on Mathematics Education

From an analytical point of view, from the time of Newton and Leibniz, physics and mathematics have been closely combined; Mathematics in the eyes of ordinary people is probably synonymous with calculus, because of the popularity of calculus, especially being named higher mathematics; On the other hand, each branch of analysis also shows great vitality and becomes an extremely important part of mathematics. I think Arnold's point of view should be understood as follows: mathematics should not cause such a deep gap with physics. This is very reasonable. As mathematics, we should know how things come from and what their original problems are, although it can be regarded as an axiomatic formal system in mathematics; As physics, we should understand the mathematical essence and intuition of the problem, although it is considered to be entirely based on experiments.

However, math and physics have been separated, even myself. Various fields are booming and changing with each passing day because of new discussions. Scholars who are proficient in most fields of mathematics or in all fields of physics are hard to find in the post-Hilbert era-how should we face such a thing? Arnold's point of view reminds us not to separate some basic theories too much, but at least to have a comprehensive understanding.

Of course, we should not ignore the differences. Mathematics should not give up the pursuit. It is an obvious fact that physics has never given up its pursuit. To talk about the pursuit of mathematics, we have to talk about the origin of mathematics, such as the most direct, number theory, Euclidean geometry, which are undoubtedly considered as typical mathematical theories. Especially number theory, not only has produced many problems of its own, but also guided the development of mathematics-in a sense, a basic mathematics department cannot be a good mathematics department without excellent scholars studying number theory. Contemporary mathematics is mainly analysis, algebra and geometry. On the one hand, they have their own problems, on the other hand, they are closely related to the problems of physics, engineering and even social science (to give a few examples: physics is self-evident; Computer is a kind of mathematics from a certain angle, such as mathematical logic, arithmetic geometry and number theory; Many problems in information science are mathematical problems; The application of statistics in social science is well known. These two kinds of problems are the driving force of mathematics.

Let's end with a story about group theory: 19 10, a physics professor and a mathematics professor discussed the curriculum of physics department in Princeton. The physicist thinks that the class of group theory should be cancelled because he really can't see the relationship between group theory and physics. Of course, the course of group theory remained. Dramatically, the development of quantum mechanics made group theory necessary in physics a few years later. There are many similar examples, many of which are considered to be pure mathematical theories, which have shown their talents in many fields. After all, mathematics is the principle? . I think this property is also the reason why mathematics tends to be abstract (and this abstraction is not contradictory to intuition). On the one hand, it forms its own intuition, on the other hand, it is supported by various concrete examples. For example, a starting point of category theory is that many theories have theorems with the same structure, which can be proved together without being done separately, which is intuitive; On the other hand, these theorems also support category theory); Even this is why mathematics is called mathematics.

Perhaps more telling, analysis, algebra and geometry also show a trend of integration. It is said that someone joked that every time I heard algebra, it always came from a topologist, and every time I heard homology, it always came from an algebra expert. In the past, those seemingly unrelated theories that were considered to be independently developed were found to be closely related. It has to be said that this is a very remarkable work, and perhaps it is also an essence.

The following text:

Mathematics is a part of physics. Physics is an experimental science and a part of natural science. And mathematics is the part of physics that only needs to spend less money on experiments. For example, Jacobian identity (ensuring that the three heights of a triangle intersect at one point) is an experimental fact, just like the earth is round (that is, homeomorphism is on a sphere). But finding the former costs less than finding the latter.

In the mid-20th century, people tried to make a strict distinction between physics and mathematics. The consequences are catastrophic. A whole generation of mathematicians grew up knowing nothing about the other half of their science, and of course they knew nothing about other sciences. These people began to teach their ugly academic pseudo-mathematics to students again, and then these ugly pseudo-mathematics were handed over to children in primary and secondary schools (they completely forgot Hardy's warning: ugly mathematics can't have a hiding place in the sun forever).

Because the academic mathematics artificially excavated from physics is not conducive to teaching and useless to other sciences, it is conceivable that people all over the world hate mathematicians (even those children in poor schools who are taught by them and those who use these ugly mathematics). These mathematicians with congenital deficiency are exhausted by their own low-energy syndrome and can't even have basic physical knowledge. An ugly building that people still remember is the "odd number strict axiomatic theory". Obviously, it is entirely possible to create a theory that makes naive pupils afraid of its perfection and the harmony of its internal structure (for example, this theory defines the product of the sum of odd terms and arbitrary factors). From this paranoid and narrow point of view, with the passage of time, even numbers are either considered as a kind of "heresy" or used as a supplement to several "ideal" objects in this theory (in order to meet the needs of physics and the real world). Unfortunately, this theory is only an ugly and deformed structure in mathematics, but it has dominated our mathematics education for decades. It originated in France, and this unhealthy trend soon spread to the teaching of basic mathematics. First, it poisoned college students, then primary and secondary school students, which was inevitable (the disaster area was first France, then other countries, including Russia). If you ask a French pupil, "What is 2+3?" , he (she) will answer: "equal to 3+2, because the addition operation is interchangeable." He (she) has no idea what the total is, or even what you are asking him (her)! Some French pupils will define mathematics like this (at least I think it is very possible): "There is a square, but it has not been proved".

According to my teaching experience in France, students in universities have the same understanding of mathematics as these primary school students (even those who study mathematics in' Advanced Normal School '(ENS)-I feel extremely sorry for these obviously clever but deeply poisoned children).

For example, these students have never seen a paraboloid. The following questions: Describing the shape of the surface given by equation xy = z 2 can make mathematicians studying in ENS stunned for a long time; But the following problem: it is impossible for students (even most French mathematics professors) to draw a curve given by the parametric equation (for example, x = t 3-3t, y = t 4-2t 22). The ability to solve these problems is considered as a basic skill for every mathematician, from the introductory textbook of calculus to the textbook compiled by Goursat.

Fans who like to challenge the so-called "abstract mathematics" in the brain exclude all geometry in mathematics that can often be associated with physics and reality from teaching. Coursat, Hermite, Picard and other calculus courses. Is considered harmful. Recently, it was almost thrown away as garbage by the libraries of University of Paris VI and University of Paris VII, and only after my intervention was it preserved. ENS students who have listened to all the courses of differential geometry and algebraic geometry (taught by different mathematicians) are not familiar with Riemannian surfaces determined by elliptic curves y 2 = x 3+ax+b, nor do they know the topological classification of surfaces (not to mention the group properties of elliptic integrals and elliptic curves of the first kind, that is, Euler-Abel addition theorem). They only learned Hodge structure and Jacobian cluster!

This phenomenon should appear in France! This country has contributed Lagrange, Laplace, Cauchy, Poincare, Le Rey and Thom to the whole world. For me, a reasonable explanation comes from I.G. Petrovskii, who taught me in 1966 that real mathematicians will never form gangs, and only the weak will join gangs in order to survive. They can connect many aspects (they may be super-abstract, anti-Semitic or "applied and industrial" problems), but their essence is always to solve the problem of social survival.

By the way, L Pasteur's advice: There has never been and will never be any so-called "applied science", only the application of science.

I have always had doubts about what Petrovsky said, but now I am more and more sure that what he said is right. A considerable part of those ultra-abstract activities are degenerating into shamelessly plundering the achievements of those discoverers in an industrialized mode, and then systematically organizing and designing them into omnipotent promoters. Just as the new world where Mei Jian lived was not named after Columbus, mathematical achievements were almost never named after its real discoverer.

In order to avoid being thought that I am talking nonsense, I have to declare here that some of my own achievements have been requisitioned for free in the above way for inexplicable reasons. In fact, such things often happen to my teachers (Kolmogorov, Petrovskii, Pontryagin, Rokhlin) and students. Professor M. Berry once put forward the following two principles:

Arnold Principle: If a person's name appears in an idea, it must be the name of the person who discovered the idea.

Berry principle: Arnold principle applies to itself.

However, let's talk about mathematics education in France. When I was a freshman in the Department of Mathematics and Mechanics of Moscow University, L.A. Toumar King, a topologist of set theory, taught us calculus. He talked over and over again about the ancient and classic French calculus course Gulsat in class. He told us that if the Riemann surface corresponding to the algebraic curve is spherical, we can find the integral of rational function along the algebraic curve. Generally speaking, if the genus of a surface is high, such an integral is not available, but for a sphere, as long as there are enough double points on the curve of a given degree (that is, the curve is required to be single-ring: that is, its real points can be drawn on the projection plane).

These facts (even if they have not been proved) have left us a deep impression, and they have given us a very beautiful and correct concept of modern mathematics, and those long-winded works of Beable Baki School are much better. Seriously, we see here that there is an amazing relationship between those seemingly completely different things: on the one hand, the integrals and topologies on the corresponding Riemannian surfaces have clear expressions, on the other hand, there is also an important relationship between the number of two points and the genus of the corresponding Riemannian surfaces.

Such examples are not uncommon. As one of the most fascinating properties in mathematics, Jacoby once pointed out that the same function can not only understand the properties of an integer that can be expressed as the sum of the squares of four numbers, but also describe the motion of a simple pendulum.

The discovery of the connection between these different kinds of mathematical objects is similar to the discovery of the connection between electricity and magnetism in physics and the discovery of the similarity between the east coast of Central America and the west coast of Africa in geology.

The exciting significance of these findings to teaching is immeasurable. It is they who guide us to study and discover the harmonious and wonderful phenomena in the universe.

However, the non-geometrization of mathematics education and the separation from physics cut off this connection. For example, not only students of mathematics, but also most algebraic geometricians don't know the Jacobian fact mentioned below: an elliptic integral of the first kind represents the time of moving along an elliptic phase curve in the corresponding Hamiltonian system.

We know that hypocycloids are as endless as ideals in polynomial rings. But if you want to teach the ideal concept to a student who has never seen any hypocycloid, it is like teaching the addition of fractions to a student who has never divided a cake or an apple equally (at least in his mind). There is no doubt that children will tend to add both numerator and denominator, plus mother.

According to French friends, this super abstract generalization is a traditional feature of their country. If this may be a genetic defect, I won't disagree, but I want to emphasize the fact that "cakes and apples" were borrowed from Poincare.

Mathematical theory is constructed in the same way as other natural sciences. First of all, we should consider some objects and observe some special cases. Then we try to find some limitations in the application of our observations, that is, to find counterexamples to prevent us from inappropriately extending our observations to a wider range of fields. Therefore, we put forward the findings from experience as clearly as possible (such as Fermat conjecture and Poincare conjecture). After that, it will be a difficult stage to test how reliable our conclusion is.

In this respect, mathematics has developed a special set of skills. This technology, when applied to the real world, is sometimes very useful, but sometimes it can also lead to self-deception. This technique is called "modeling". When building the model, we should idealize the following points: some facts that can only be known with a certain probability or accuracy are often considered "absolutely" correct and accepted as "axioms". The significance of this "absoluteness" lies in that we allow ourselves to use these "facts" according to the rules of formal logic in the process of calling all the conclusions we can draw with these fact theorems.

Obviously, in any real daily life, our activities can't completely rely on such reduction. Because at least the parameters of the studied phenomenon can never be known absolutely and accurately, and small changes in parameters (such as small changes in the initial conditions of a process) will completely change the results. Because of this, we can say that any long-term weather forecast is impossible, no matter how advanced the computer is and how sensitive the instrument for recording initial conditions is, it will never be possible. Just like this, although minor changes are allowed to axioms (those we are not completely sure about), generally speaking, theorems derived from those recognized axioms will draw completely different conclusions. The longer and more complex the derivative chain (so-called "proof"), the lower the reliability of the final conclusion. Complex models are almost useless (except those who write boring papers).

Mathematical modeling technology knows nothing about this kind of trouble, and has been boasting about its own model, which seems to be really in line with the real world. In fact, from the perspective of natural science, this practice is obviously incorrect, but it often leads to many physically useful results, which is called "mathematics with incredible effectiveness" (or "Wigner principle").

Here I would like to mention a sentence from Mr. Gelfant: There is another phenomenon that is very similar to the mathematics in physics mentioned by Wigner above, that is, the mathematics used in biology is also incredibly effective.

For a physicist, "the subtle toxic effect caused by mathematics education" (F.Klein's original words) is only reflected in the absolute model drawn from the real world, which is no longer consistent with reality. Let's give a simple example: Mathematical knowledge tells us that the solution of Malthus equation dx/dt = x is uniquely determined by the initial conditions (that is, the corresponding integral curves on the (t-x)- plane do not intersect each other). The conclusion of this mathematical model seems to be irrelevant to the real world. However, computer simulation shows that these integral curves all have common points on the negative semi-axis of T. In fact, the curves with initial conditions of x(0) = 0 and x(0) = 1 intersect at t=- 100. In fact, when t=- 100, you can't insert another atom between two curves. Euclidean geometry does not describe the properties of this space at a small distance. In this case, the application of uniqueness theorem has obviously exceeded the accuracy allowed by the model. In the practical application of the model, we must pay attention to this situation, otherwise it may lead to serious trouble.

I would also like to say that the same uniqueness theorem can also explain why ships must rely on manual operation before berthing: otherwise, if the driving speed is a smooth (linear) function of distance, the whole berthing process will take an infinite time. Another feasible method is to collide with the wharf (of course, there must be a non-ideal elastic object between the ship and the wharf to cushion it). By the way, we must attach great importance to such issues as landing on the moon and Mars and docking with the space station-at this time, uniqueness will give us a headache.

Unfortunately, in modern mathematics textbooks, even better textbooks, there are no examples or discussions about the dangers hidden by this admirable theorem. I even got the impression that those academic mathematicians (who don't know anything about physics) are used to the main difference between axiomatic mathematics and modeling. They think this is very common in natural science, but they only need to control theoretical derivation with later experiments.

I don't think it is necessary to mention the relative characteristics of the initial axiom, and people will never forget that logical errors are inevitable in lengthy discussions (as if cosmic rays or quantum vibrations caused the collapse of calculation). Every mathematician who is still working knows that if you don't control yourself (it is better to use examples), after the discussion on page 10, half of the symbols in all formulas will be wrong.

The technology against this fallacy also exists in any experimental science and should be taught to every junior college student. Trying to create the so-called pure deductive axiomatic mathematics makes us use this method instead of the research method in physics (observation-modeling-model research-drawing conclusions-testing the model with more observations): definition-theorem-proof. It is impossible for people without motivation to understand a definition, but we can't stop these guilty "algebraic-axioms". For example, they always want to use the long multiplication rule to define the product of natural numbers. But it is difficult to prove the commutativity of multiplication in this way, but it is still possible to deduce such a theorem from a bunch of axioms. And it is entirely possible to force those poor students to learn this theorem and its proof (the purpose is nothing more than to improve the social status of the subject and the people who teach it). Obviously, this definition and this proof are harmful to teaching and practical work.

To understand the commutativity of multiplication, the only possible way is to calculate the soldier number of a square in row order and column order, or to calculate the area of a rectangle in two ways. Any attempt to do mathematics without dealing with physics and the real world belongs to sectarianism and isolationism, which will certainly damage the good impression that mathematical creation is regarded as beneficial human activities in the eyes of all sensitive people.

I'll reveal a few more such secrets (poor students are very interested). The determinant of a matrix is the (oriented) volume of a parallel polyhedron, and each side of this polyhedron corresponds to the column of the matrix. If students know this secret (which is carefully hidden in pure algebra education), then the whole determinant theory will become a part of multilinear formal theory. If determinant is defined in other ways, any sensitive person will always hate determinant, Jacobian formula and implicit function theorem.

What is a group? Algebraists will teach this: this is a set of assumptions with two operations that satisfy a set of axioms that are easy to forget. This definition can easily lead to a natural protest: why does any sensitive person need this operation? "Oh, to hell with this kind of mathematics"-this is the student's reaction (he is likely to become a strong man in science in the future).

If our starting point is not a group, but a concept of transformation (1- 1 mapping from a set to itself), then we will definitely get a different situation, which is more like historical development. The set of all transformations is called a group, in which the composition of any two transformations is still in this set, and the inverse transformation of each transformation is also in this set. This is the key to definition. In fact, the so-called "axiom" is nothing more than the obvious nature of the transformation group. The "abstract group" advocated by axiomatic advocates is nothing more than a transformation group that allows different sets in the sense of phase difference isomorphism (1- 1 mapping). As Gloria proved, there is no "more abstract" group in this world. So why are those algebras still torturing these suffering students with abstract definitions?

By the way, I taught group theory to primary and middle school students in Moscow in the 1960s. I avoided any axiom and kept the content as close to physics as possible. For half a year, I taught them Abel Theorem about the insolubility of general quintic equations (similarly, I also taught elementary school students the complex number, Riemann surface, basic group and single-valued group of algebraic functions). The content of this course was later published by one of my listeners, V Alekseev, and it was called Abel Theorem in Problems.

What is a smooth manifold? Recently, I learned from an American book that Poincare was not proficient in this concept (although he introduced it), and the so-called "modern" definition was not given by Van Buren until the 1920s: Manifolds are topological spaces that satisfy a long list of axioms.

What crime did the students commit and must be tortured by these distorted axioms to understand this concept?

In fact, there is an absolutely clear definition of smooth manifold in Poincare's original analysis Situs, which is much more useful than this abstract thing. K-dimensional smooth submanifolds in Euclidean space R n are subsets such that the neighborhood of each point is an image of a smooth mapping from R k to R (n-k) (where R k and R (n-k) are coordinate subspaces). This definition is a direct generalization of the most common smooth curves on a plane (such as circular ring X 2+Y 2 =1) or curves and surfaces in three-dimensional space. Smooth mappings between smooth manifolds are naturally defined. The so-called differential homeomorphism is a smooth mapping and its inverse is also smooth. The so-called "abstract" smooth manifold is a smooth submanifold in the sense of allowing differential homeomorphism difference in Euclidean space. There is no so-called "more abstract" finite-dimensional smooth manifold (Whitney theorem).

Why do we always torture students with abstract definitions? Wouldn't it be better to prove the classification theorem of closed two-dimensional manifold (surface) to students? It is this wonderful theorem (that is, any tightly connected orientable surface is a sphere with several ring handles) that gives us a correct impression of what modern mathematics is. On the contrary, those hyperabstract generalizations of simple submanifolds in Euclidean space actually give nothing new at all, but are only used to show the difference product achieved by axiomatic chemists. The classification theorem of surfaces is a top-level mathematical achievement, comparable to the discovery of American continent or X-ray. This is a real discovery in mathematical science, and it is even difficult for us to tell whether the fact itself contributes more to physics or mathematics. Its extraordinary significance to the application and development of the correct world outlook has now surpassed other "achievements" in mathematics, such as the proof of Fermat's last theorem, and the fact that any integer large enough can be expressed as the proof of three prime numbers. In order to show off, contemporary mathematicians sometimes show some achievements of "sports" and claim that it is the last problem in their discipline. It is conceivable that this practice will not only help the society appreciate mathematics, but also make people wonder: Is it necessary to spend energy on these exercises (such as rock climbing) for such a useless striptease problem? The classification theorem of surfaces should be included in the curriculum of high school mathematics (without proof), but I don't know why it can't even be found in the curriculum of college mathematics (by the way, all geometry courses have been banned in France in recent decades).

At all levels, mathematics education has changed from the characteristics of colleges to the characteristics of showing the importance of natural sciences, which is a problem that France is very concerned about. To my shock, the best and most important well-organized math books are almost unknown to students here (and in my opinion, they have not been translated into French). These books include Numbers and Graphics; Written by Rademacher and To; Plic, Geometry and Imagination written by Hilbert and Cohen-Watson; What is the math written by Courand and Robbins? 》; How to solve it and the mathematics and specious reasoning written by Paulia; F. Klein wrote1the development of mathematics in the 9th century.

I clearly remember how impressed I was at school with the calculus course written by Hermite (with Russian translation). I remember that Riemann surfaces appeared in the first lecture (of course, all the analysis contents are aimed at complex variables, which should be). The content of integral asymptotic is studied by the method of road deformation on Riemannian surface (today we call this method piccard-Lefschetz theory; By the way, piccard is Hermite's son-in-law-mathematical ability is often inherited by his son-in-law: Hadamard-p.levy-L.schwarz-u.frisch is another such example of the Paris Academy of Sciences. The so-called "outdated" course written by Hermite/Kloc more than 0/00 years ago (which may have been thrown away as garbage by the student libraries of French universities) is actually much more modern than the most boring calculus textbook that afflicts students today.

If mathematicians don't wake up, consumers who still need modern (the most active) mathematical theory and are immune to useless axiomatic features (which are the characteristics of any keen person) will not hesitate to sweep the uneducated pedants out of these schools. A math teacher, if he (she) doesn't master at least a few volumes of physics tutorials written by Landau and Livschitz, then he (she) will definitely become a rare survivor in the field of mathematics, just like a person who still doesn't know the difference between open set and closed set.