Arnold is not only the creator of mathematics, but also the creator of mathematicians. He is a link between the past and the future in the Soviet-Russian mathematics school. He believes that mathematics is a part of physics, and the essence of physics is geometry. His masterpiece "Mathematical Methods of Classical Mechanics" is a thoroughly remoulded transformation of classical mechanics with the framework of symplectic geometry. This book is called "the Bible of Geometric Mechanics". In mathematics, he advocates the way of thinking in geometry and physics, but hates axiomatic and formal mathematics and mathematics education. He thinks that this kind of mathematics cuts off the connection with the physical world and eliminates the intuitive feeling, which is ugly pseudo-mathematics. This kind of mathematician is a remnant monster, and this kind of mathematics education is a torture to children and a crime.
In fact, there have always been two traditions in mathematics, and geometry and algebra represent their basic spirit respectively. As attiya, the winner of Fields Prize, said, in modern times, Newton-Poincare-Arnold was regarded as a school of mathematical intuitionism, which emphasized the spirit of physics and geometry. Taking the Leibniz-Hilbert-Bourbaki school as a series, it emphasizes the spirit of axiomatization and formalization. The ups and downs between them have always been the norm in the history of mathematics, and they are all made by the times. This can even be traced back to the geometry of ancient Greeks and algebra of ancient India and Arabs. In any case, Arnold has become the embodiment of that spirit in the mathematical tradition to which he belongs.
The following content comes from Arnold's original text, which can reflect some of his intuitive views on mathematics education.
Mathematics is a part of physics. Physics is an experimental science, which is a part of natural science. And mathematics is the part of physics that only needs to spend less money on experiments. For example, Jacobi's identity (ensuring that three heights of a triangle intersect at one point) is an experimental fact, just like the fact that the earth is round (that is, homeomorphism is in a sphere). But it costs less to find the former than to find the latter.
in the middle of the 2th century, people tried to distinguish between physics and mathematics strictly. The consequences are disastrous. A whole generation of mathematicians grew up ignorant of the other half of the science they were engaged in, and of course, ignorant of other sciences. These people began to teach their ugly academic pseudo-mathematics to their students again, and then these ugly pseudo-mathematics were handed over to the children in primary and secondary schools (they completely forgot Hardy's warning: ugly mathematics can't always have a hiding place in the sun).
Since the academic mathematics artificially dug out from physics is neither helpful for teaching nor useful for 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 low-energy syndrome, and they can't have a basic understanding of physics. An ugly building that people remember vividly is the "strict axiomatic theory of odd numbers".
obviously, it is entirely possible to create such a theory, which makes naive pupils fear its perfection and the harmony of its internal structure (for example, this theory defines the sum of odd terms and the product of arbitrary factors). From this paranoid and narrow point of view, 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) as time goes by. Unfortunately, this theory is just an ugly and abnormal structure in mathematics, but it has dominated our mathematics education for decades. It first originated in France, and this unhealthy trend soon spread to the teaching of basic mathematics. First, it poisoned college students, and then it was inevitable for primary and secondary school students (the disaster area was first in France, and then in other countries, including Russia).
If you ask a French pupil, "What is 2+3?" , he (she) will answer: "It is equal to 3+2, because the addition operation is exchangeable". He (she) has no idea what this sum is, or even what you are asking him (her)!
Some French pupils will define mathematics in this way (at least I think it is possible): "There is a square, but it has not been proved".
According to my teaching experience in France, students in universities have similar knowledge of mathematics to these primary school students (even those who study mathematics in the' Advanced Normal School' (ENS)-I feel extremely sorry for these obviously smart but deeply poisoned children).
For example, these students have never seen a paraboloid, and a question like this: Describing the shape of the surface given by the equation xy = z 2 can make mathematicians who are studying in ENS stay in a daze for half a day; But the following problem: It is impossible for students (even for most French mathematics professors) to draw a curve given by parametric equations (for example, x = t^3-3t, y = t^4-2t^2 2). From the introductory textbook of calculus to the textbook written by Goursat, the ability to solve these problems is considered to be the basic skill that every mathematician should have.
Those enthusiasts who like to challenge the so-called "abstract mathematics" of the brain exclude all geometry that can often be associated with physics and reality in mathematics from teaching. Calculus course written by Goursat, Hermite, Picard, etc. was considered harmful, and was almost thrown away as garbage by the libraries of Paris Universities 6 and 7, but it was only after my intervention that it was preserved.
the students of p>ENS 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 Jacobi cluster!
such a phenomenon should appear in France! This country has contributed to the whole world such as Lagrange, Laplace, Cauchy, Poincaré, Leray and Thom. 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, but their essence is always to solve the problem of social survival.
by the way, I would like to give you L. Pasteur's advice: there has never been and will never be any so-called "applied science", but only the application of science (very useful stuff! )
I have been suspicious of Petrovskii's words for a long time, but now I am more and more sure that what he said is right. A considerable part of those super-abstract activities are degenerating into shamelessly plundering the achievements of those discoverers in an industrialized mode, and then systematically organizing and designing them to become omnipotent promoters. Just as the new continent where America is located is not named after Columbus, mathematical results are almost never named after their real discoverers.
in order to avoid being thought that I am talking nonsense, I have to declare here that some of my own achievements have been expropriated 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 p>M. Berry once put forward the following two principles:
Arnold principle: If a person's name appears in an idea, it must not be the name of the person who discovered the idea.
Berry principle: Arnold principle applies to itself.
However, let's talk about math education in France. When I was a freshman in the Department of Mathematics and Mechanics of Moscow University, L.A. Tumarkin, a topologist of set theory, taught us calculus. He carefully told the ancient and classic Goursat French calculus course over and over again in class. He told us that the integral of rational function along an algebraic curve can be found out if the Riemann surface corresponding to the algebraic curve is a sphere. Generally speaking, if the genus of the surface is higher, such an integral will not be available, but for a sphere, as long as there are enough double points on a curve with a given degree (that is, the curve is required to be unicursal: that is, its real points can be drawn on the projection plane).
what a deep impression these facts have made on us (even without proof). They have given us a very beautiful and correct idea of modern mathematics, which is much better than those long-winded works of Bourbaki school. Seriously, we see here that there is a surprising connection between those seemingly completely different things: on the one hand, there are explicit expressions for the integrals and topologies on the corresponding Riemannian surfaces, and on the other hand, there is also an important connection between the number of double points and the genus of the corresponding Riemannian surfaces.
such an example is not uncommon. As one of the most fascinating properties in mathematics, Jacobi 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 the American continent and the west coast of the African continent in geology.
The exciting and extraordinary significance of these discoveries to teaching is immeasurable. It is they that guide us to study and discover 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 studying mathematics, but also most algebraic geometricians are ignorant of the Jacobi fact mentioned below: an elliptic integral of the first type represents the time of motion along an elliptic phase curve in the corresponding Hamiltonian system.
We know that a hypocycloid is as endless as an ideal in a polynomial ring. But if you want to teach the concept of ideal to a student who has never seen any hypocycloid, it is like teaching the addition of fractions to a student who has never cut a cake or an apple equally (at least in his mind). There is no doubt that children will tend to add numerator and denominator and add mother at the same time.
I heard from my French friends that this super abstract generalization is the traditional feature of their country. If this may be a hereditary defect, I won't disagree, but I would like to emphasize the fact that "cakes and apples" were borrowed from Poincaré.
The way of constructing mathematical theory is no different from 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 observed results, that is, to find counter-examples that prevent us from improperly extending our observed results to a wider range of fields. As a result, we put forward the findings (such as Fermat's conjecture and Poincare's conjecture) that come from experience as clearly as possible. After that, it will be a difficult stage to test how reliable our conclusions are.
at this point, the mathematical community has developed a special set of techniques. 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 constructing a model, we should idealize the following: 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" is precisely that in the process of calling all the conclusions we can draw with these facts theorems, we allow ourselves to use these "facts" according to the rules of formal logic.
obviously, in any real daily life, it is impossible for our activities to rely entirely on such reduction. The reason is at least that the parameters of the studied phenomenon can never be known absolutely and accurately, and a small change in the parameters (such as a small change in the initial conditions of a process) will completely change the results. For this reason, we can say that any long-term weather forecast is impossible, no matter how advanced the computer is or how sensitive the instrument for recording initial conditions is, it will never be possible.
exactly the same as this, although a small change of axioms (those we are not completely sure about) is allowed, generally speaking, theorems deduced from those accepted axioms will lead to completely different conclusions. The longer and more complex the derived chain (so-called "proof"), the lower the reliability of the final conclusion. Complex models are almost useless (except for those who write boring papers).
The technology of mathematical modeling knows nothing about this trouble, and they keep boasting about the models they get, as if they really fit in with the real world. In fact, from the point of view of natural science, this approach is obviously incorrect, but it often leads to many physically useful results called "mathematics with incredible effectiveness" (or "Wigner principle").
I'd like to mention a sentence from Mr. Gelfond here: There is another kind of phenomenon that has incredible validity similar to the mathematics in physics referred to by Wigner above, that is, the mathematics used in biology is also incredibly effective.
for a physicist, the "imperceptible toxic effect caused by mathematics education" (F.Klein's original words) is just reflected in the absolutized model drawn from the real world, and it is no longer consistent with reality. Here is 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 with each other). The conclusion of this mathematical model seems to have nothing to do with the real world. However, computer simulation shows that all these integral curves have common points on the negative semi-axis of T. In fact, the curves with initial conditions x() = and x() = 1 intersect at t=-1. In fact, when t=-1, it is impossible for you to insert another atom between the two curves. Euclidean geometry has no description of the nature 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, this situation must be paid attention to, no