On the Teaching of Mathematics: Arnold

ON THE TEACHING OF MATHEMATICS

Vladimir Igorevič Arnol’d

uspekhi Matematicheskikh Nauk (1998) English translation Viktor Goryunov,Russian Mathematical Surveys, vol.53, no.1, 1998, pp.229-234 transl.it. Sandro Graffi in CRITICAL POINTS NO. 3 (MAY 2000) Local copy PDF

Mathematics is a part of physics.Physics is an experimental science, part of the natural sciences. Mathematics is that part of physics where experiences cost little.

Jacobi’s identity (forcing the heights of a triangle to meet at a point) is a fact experimental in the same way that the earth is round (i.e., homeomorphic to a sphere). But it can be discovered with lower expenditure.

In the mid-twentieth century, attempts were made to divide physics and mathematics. The consequences proved to be catastrophic.Whole generations of mathematicians were formed without knowing half of their science and, of course, in total ignorance of any other science. They began by teaching their ugly school pseudomathematics to their students, and then to schoolchildren (forgetting Hardy’s warning that for ugly math there is no permanent place under the sun).

Since school mathematics cut off from physics is not suitable neither for teaching nor for application to any other science, the result has been universal hatred of mathematicians-both by the poor pupils (some of whom have since become ministers) than of the utili tors.

An ugly edifice, built by poorly trained mathematicians driven to exhaustion by their complex of inferiority and unable to familiarize themselves with physics, we are reminded of the rigorous axiomatic theory of odd numbers.Obviously, it is possible to create this theory and make pupils admire the perfection and the internal consistency of the resulting structure (in which, for example, you can defInish sums of an odd number of terms and products of an odd number of factors). From this point of view sectarian the even numbers could be declared a heresy, or, in time, be introduced into the theory to which we add some “ideal” objects (to obey the needs of physics and the real world).

Unfortunately, it was an ugly and convoluted mathematical construction like the one above that dominated for decades the teaching of mathematics.Originating in France, this perversion quickly spread in teaching the fundamentals of mathematics, first to undergraduate students, and then to the schoolchildren of all grades (first in France, and then in all other countries, including Russia).

When asked “how much is 2+3 ?” a French elementary school student replied, “3+2, because addition it’s commuta tiva.” He didn’t know how much he was adding up, and he didn’t get to understand why he was being asked!

Another French schoolboy (very lucid, in my opinion) defined mathematics as follows: “there is a square, but this has yet to be proven.”

Judging from my teaching experience in France, the idea of college students’ mathematics (even of those who studied at the École Normale Supérieure - I feel sorry for almost all of these kids obviously intelligent but deformed) is just as poor as that of the schoolboy .

For example, these students have never seen a paraboloid and a question about the shape of the surface defmited by the equation xy=z2 causes the astonishment of mathematicians studying at ENS. Drawing a curve defined by parametric equations (such as x=t3-t, y=t4_t2) in the plane is a totally impossible problem for the students (and, probably, for most French mathematics professors as well).

Starting with de l’Hopital’s first textbook of infinitesimal calculus (Calculus for Understanding Curved Lines) and roughly until the Treaty of Goursat, the ability to solve these problems was considered (along with to knowledge of multiplication tables) a necessary part of every mathematician’s art.

“Abstract math” fanatics with mental disorders threw out of teaching all the geometry (through which the connection of mathematics with physics and reality mostly takes place). I infmitesimal calculus texts by Goursat, Hermite, and Picard were recently sent to rubbish by the library for students at the Universities of Paris VI and VII (Jussieu) for being obsolete and therefore harmful (They were saved only because of my intervention).

ENS students who attended differential geometry and algebraic geometry courses (taught by respected mathematicians) turned out to be unaware of the Riemann surface of an elliptic curve of the type y2=x3+ax+b, and not even, in fact, with the topological classification of surfaces (and there is no mention of not even elliptic integrals of the first kind and the group property of an elliptic curve, i.e., the theorem of addition by Euler Abel.) They had only taught him Hodge’s structures and Jacobi’s varieties!

How could this happen in France, which gave the world Lagrange and Laplace, Cauchy and Poincaré, Leray and Thom? Seems to me that a reasonable explanation was given by I. G. Petrovsky, who taught me in 1966: i genuine mathematicians do not form gangs, but the weak need them to survive. they can put themselves together for a variety of reasons (it could be superstrateness, anti-Semitism, or problems “applied and industrial”), but the essence is always the solution to a social problem: survival in a more acculturated.

Incidentally, I would like to recall a warning from L. Pasteur: there never has been and never will be “science applied,” there are only applications of the sciences (and very useful ones!).

At that time I had some doubts about Petrovsky’s words, but now I am increasingly convinced of how much he had reason.A considerable part of the superabstract activity simply comes down to the shameless industrialization of taking discoveries away from their discoverers and systematically attributing them to the their epigonizing-generalizers.In the same way that America does not bear the name of Columbus, the results mathematicians almost never bear the names of their discoverers.

To avoid misunderstanding, I should note that my own acquisitions have never been expropriated in this way for some unknown reason, although this has always happened as much to my masters (Kohnogorov, Petrovsky, Pontryaghin, Rokhlin) than to my al mild. Prof. Michael Berry once formulated the following two principles:

-Arnold’s principle.If a notion bears a proper name, then that name is not that of the discoverer.

• Berry’s principle.Arnold’s principle is applicable to itself.

Let us turn, however, to the teaching of mathematics in France. When I was a first-year student at the Faculty of Mate matics and Mechanics of Moscow State University the infrnitesimal calculus lectures were held by the ensemble topologist L. A. Tu markin, who conscientiously repeated the old classic corsò of french school in Goursat’s version. He told us that you could no perform integrals of a function rational along an algebraic curve if the corresponding Riemann surface was a sphere and that, in general, could not be performed if its gender was higher, and that for sphericity it was enough to have a number sufficiently large of double points on the given degree curve (this makes the unicursal curve; you can draw its real points on the plane projective with a single pen stroke).

These facts capture the imagination to such an extent that (even without any demonstration) they give an ‘idea better and more correct modern mathematics than entire volumes of Bourbaki’s treatise. In fact, here we we realize the existence of a meravi gliosa connection between things that seem completely different: on the one hand, the existence of an explicit expression for the inte grals and the topology of the surface of Riemann corresponding, and, on the other hand, between the number of double points and the genus of the su perficie of Riemann corresponding, which also shows up in the real domain in the form of unicursality.

Jacobi noted as a most fascinating property of mathematics the fact that the same function controls both the rap presentation of an integer as the sum of four squares and the movement effective of a pendulum.

These discoveries of connections between heterogeneous mathematical objects can be compared to the discovery of the connection between elet tricity and magnetism in physics or with the discovery of the similarity between the East Coast of America and the west coast of America in geology.

It is difficult to overestimate the emotional significance on teaching of such discoveries. It is they who teach us to seek and find similar wonderful phenomena of harmony in the universe.

De-geometrization in mathematics education and divorce from physics sever these ties. For example, not only stu denti but all modern algebraic geometers ignore Jacobi’s statement here mentioned: an elliptic integral of the first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system ma.

Rephrasing the famous words about the atom and the electron, it can be said that a hypocycloid is as inexhaustible as a ideal in a ring of polynomials.

But teaching ideals to students who have never seen a hypocycloid is just as ridiculous as teaching to add up fractions to children who have never (at least mentally) cut a cake or an apple into parts equal.

No wonder children prefer to add numerators and denominators together.

From my French friends I have heard that the tendency toward super-abstracted gene ralizations is a trait of theirs national.I don’t completely disagree with the ‘statement that it could be hereditary ma lattia, but I would like to emphasize the fact that I took the esem pio of the cake and the apple from Poincaré.

The construction scheme of a mathematical theory is exactly the same as that of any other natural science. At first we consider some objects and make some observations in special cases. Then we search and find the limits of applicability of our observations, we look for counterexamples that might prevent the exten unwarranted sion of our observations to too large a set of events (example: the number of partitions of the consecutive odd numbers 1, 3, 5, 7, 9 into an odd number of na tural addends gives the succession 1, 2, 4, 8, 16 but then comes 29).

The result is that we formulate the empirical discovery we made (e.g., Fermat’s conjecture or the poincaré conjecture) as clearly as possible. After that comes the difficult period of the control of how reliable the conclusions are.

At this point a special technique has been constructed in mathematics. This technique is sometimes useful if applied to the real world, but it can also sometimes be illusory. This technique is called modeling. In constructing a model, the following idea lization is made: certain facts, known only with some probability or with some degree of accuracy, are considered “absolutely” correct and are accepted as “axioms.” The meaning of the"‘absolute" lies precisely in stating that we allow ourselves to use these “facts” according to the rules of formal logic, declaring “theorems” everything we can make derivative of them through the proce diment of logic.

It is obvious that in any real daily activity it is impossible to rely totally on such deductions. The reason, at least, is that the parameters of the phenomena under study are never known with absolute precision, and a small parameter variation (e.g., the initial conditions of a process) can completely alter the result.For this reason, for example, long-term weather forecasts are impossible and will remain impossible no matter how much more and more powerful computers and devices are built to record the increasingly accurate initial conditions.

Exactly in the same way a small variation in the axioms (of which we cannot be completely safe) could lead to completely different conclusions from those obtained through the deduced theorems from accepted axioms.The longer and com plicated the chain of deductions (“demonstrations”), the less reliable the end result.

Complicated models are rarely useful (except for those who write dissertations on them).

The mathematical technique of modeling is to ignore these messes and talk about your deductive model as if it coincided with reality. The fact that this procedure obviously incorrect from the perspective of the natural sciences, often leads to useful results in physics bears the name of “inconceivable effectiveness of mathematics in the natural sciences” (or “Wigner’s principle”).

Here we can add an observation by I. M. Gel’fand: there is another comparable phenomenon in his inconceivability with the inconceivable effectiveness of mathematics in physics noted by Wigner - that is, the equally inconceivable ineffectiveness of mathematics in biology.

“The subtle poison of mathematical training” (in the words of F. Klein) for a physicist consists precisely of in the fact that the model made absolute separates from reality and no longer compares with it. Here is an example simple: mathematics teaches us that the solution of Malthus’ equation dx/dt = x is uniquely deftnite from the initial conditions (i.e., the corresponding integral curves in the (x,t) plane do not intersect with each other).This conclusion of the mathematical model bears little resemblance to reality.An experiment ment to the calculator shows that all these integral curves have points in common on the negative t semi-axis. For example, in fact, curves with initial conditions x(O)=O and x(O)=1 practically intersect at t=-IO and at t=-IOO: not a single atom can be squeezed between them. The properties of space at such small distances do not are described at all by Euclidean geometry. The application of the uniqueness theorem in this situation obviously goes beyond the accuracy of the model. This must be respected in the practical applications of the model, otherwise one may face serious trouble.

I would note, however, that the uniqueness theorem itself explains why the last stage of the docking of a ship to the pier is manually exe guised: if the approach speed were defined as a regular function (linear) distance, in veering the docking pro cess would require a time interval infinitely long.An alternative is a collision with the pier (cushioned by appropriate nonelastic bodies).Between the other, this problem had to be seriously addressed in the landing of the first spacecraft on the Moon and on Mars, and also in docking with space stations-here the uniqueness theorem works against us.

Unfortunately, none of these examples are encountered in modern mathematics textbooks, even the best ones, and not even a discussion of the danger of turning theorems into fetishes. I even reported the impression that scholastic mathematicians (who co n cognize little physics) believe in an essential difference between axiomatic mathematics and the modeling that is done in the natural sciences and that always requires the subsequent checking of deductions by experiments.

Without even mentioning the relative character of the initial axioms, one cannot forget the inevitability of logic errors in long reasoning (e.g., in the form of computer interruptions due to cosmic rays or to quantum fluctuations.) Every professional mathematician knows that if one does not control oneself (the best way is through examples), after, say, ten pages half the signs in the formulas will be wrong and the two will be passed from denominators to numerators.

The technology to counter these errors is the same control external through experiments and observations in use in every science experimental and should be taught fm now to students in schools.

Attempts to create “pure” axiomatic-deductive mathematics led to the rejection of the scheme in use in physics (observation-model-study-conclusions-testing by observations) and to the its replacement through the scheme: definition theorem-demonstration. It is impossible to understand a definition that is not motivated but this does not stop the axiomatizing-algebraic criminals. For example, they would be ready to defmire the product between natural numbers by means of the long multiplication rule. With this the commutativity of multiplication becomes difficult to prove but it is still possible to deduce it as a theorem from the axioms. It is therefore possible to force poor students to learn this theorem and its proof (for the purpose of raise as much the standard of science as that of the people who teach it.) It is obvious that such definitions and such evidence can only harm teaching and practice.

It is possible to understand the commutativity of multiplication only by counting and recounting soldiers by rows or by columns or by calculating the area of a rectangle in the two ways.Any attempt to do without this interference of physics and reality on mathematics is sectarianism and isolationism that destroys the image of the thematic ma as a useful human activity in the eyes of any right-minded person.

I will now reveal a few more such secrets (in the interest of the poor students). The determinant of a matrix is the (oriented) volume of the parallelepiped whose edges are its columns. If you tell the students this secret (which is carefully hidden in the purified algebraic formation), then the whole theory of deter minants becomes a clear chapter of multilinear forms. If determinants are defined in other way, then any sensible person will forever hate determinants, Jacobians, and the theorem of the implicit functions.

What is a group? Algebraists teach that it should be a set with two operations satisfying a load of properties that are easy to forget.This definition causes a natural protest: why would a person of common sense need these two operations? “Oh, damn the mathematicians”-concludes the student (who may become Minister of Re search Scientific in the future) .

We get a totally different situation if we start not from the group but from the concept of transformation (a one-to-one application of a set on itself), as it was historically. A collection of transformations of a set is called a group if together with any two transformations it contains the result of their consecutive application and an inverse transformation together with each transformation.

That’s all there is to the definition. The so-called “axioms” are really just properties (obvious) groups of transformations.What axiomatizers call “abstract groups” are only groups of transformations of various considered sets short of isomorphisms (which are one-to-one applications that conserve operations.) As Cayley demonstrated, there are no other “abstract groups” in the world. Then why do algebrists keep pestering students with abstract defrnization?

Incidentally, in the 1960s I taught group theory to schoolchildren in Moscow. Avoiding axiomatics and staying as close to physics as possible in half a year I arrived at Abel’s irresolvability theorem by radicals of a general equation of degree five (having meanwhile taught the children the numbers complexes, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the students, V. Alekseev, under the title Abel’s Theorem in problems.

What is a regular variety? I read in a recent American book that Poincaré did not know this notion (which he himself introduced) and that the “modern” definition was given by Veblen around 1930: “a variety is a topological space that satisfies a long series of axioms.”

For what sins must students try to go through all these twists and turns? Actually, in the Analysis Si tus of Poincaré there is an absolutely clear defrnition of regular variety much more useful than the “abstract” one.

A regular subvariety at dimension k of the Euclidean space RN is a subset of it that in the neighborhood of each his point is the gra fic of a regular application of Rk in RN-k (where Rk and RN-k are coordinated subspaces). This is a direct gene,ralization of the most common regular curves in the plane (e.g., of the circumference x2+y2",1) or of curves and surfaces in three-dimensional space.

Regular applications between regular varieties can be defined in a natural way. diffeomorphisms are applications that turn out to be regular, along with their inverses. An “abstract” regular variety is a regular subvarieties of a Euclidean space considered short of diffeomorphisms. There are no varieties in the world finite-dimensional “more abstract” (Whitney’s theorem). Why do we keep pestering students with the abstract definition? Wouldn’t it be better to prove to them the theorem about the explicit classification of le closed two-dimensional varieties (surfaces)?

It is this wonderful theorem (which states, for example, that every compact, connected and oriented surface is a sphere with handles) that gives a ‘correct impression of what modern mathematics is and not the superabstract generalizations of simple sub-varieties of a Euclidean space that do not actually say anything again and are presented as great acquisitions by axiomatizers.

The surface classification theorem is an acquisition of exalted class mathematics, comparable to the discovery of America or the discovery of X-rays. This is a genuine discovery of the mathematics as a science of nature, and it is also difficult to say whether the fact itself is attributable more to the physics than to mathematics. In its meaning, both for applications and for the processing of a Weltanschauung correct, it far surpasses “acquisitions” of mathematics such as the demonstration of Fermat’s last theorem or the demonstration of the fact that any sufficiently large integer can be represented as the sum of three primes.

Often, for publicity purposes, modern mathematicians present these sporting achievements as the last word in their science.Understandably this not only does not contribute to the ‘appreciation of mathematics by part of society but, on the contrary, cau sa a healthy distrust of the need to waste energy on similar practices (of the type free climbing on rocks) with similar exotic issues that no one wants and of no one feels the need for.

The surface classification theorem should have been included in high school math courses (probably without the demonstration) but for some reason you don’t even do that in undergraduate courses in mathematics (from which in France, by the way, all geometry has been banned in recent decades).

The return of math teaching at all levels from school talk to the presentation of this important domain of the natural sciences is a particularly topical issue in France. They are been amazed that all the best and most important books in the methodological approach to the mathematics are almost unknown to students here (and, it seems to me, have not even been translated into French). Between these I count Numbers and Figures by Rademacher and Toeplitz, and Geometry and Intuition by Hilbert and Cohn-Vossen, What is Mathematics by Courant and Robbins, How to Solve It, and Mathematics and Plausible Reasoning by Polya, Development of mathematics in the 19th century by F.Klein.

I remember well what a very strong impression it made on me when i was a student Hermite’s infinitesimal calculus course (which does not exists in Russian translation!).

Riemann surfaces appeared there, I think, in one of the first lectures (the whole analysis was, of course, complex, as do vould be). The asymptotic trend of the integrals was studied by means of the contour deformations on Riemann surfaces as a result of branch point motion (nowadays gior no, we would call this the Picard-Lefschetz theory; Picard, by the way, was Hermite’s son-in-law -the skills mathematics are often transferred to genders: the Hadamard dynasty - P. Levy - L. Schwartz , U. Frisch is a another famous example at the Academy of Sciences in Paris).

Hermite’s “obsolete” course from a century ago (now probably bujtted out of the libraries of the french universities) was much more modern than these boring infinitesimal calculus texts with which the students are being harassed today.

If mathematicians do not return to them, consumers who have retained the need for a mathematical theory modern (in the best sense of the word) as well as immunity (characteristic of every right-thinking person) to the useless axiomatic chatter that will eventually reject the services of uncultured scholasticism both in the secondary schools than in universities.

A mathematics teacher who has not come into contact with at least some of the volumes of Landau’s course and Lifshitz will become a wreck like those who today do not know the difference between open sets and sets closed.

Some Works of V.I. Arnold:

• Geometric methods of the theory of ordinary differential equations
• Mathematical methods of classical mechanics
• Catastrophe theory
• Yesterday and long ago - Me and Math
• Experimental Mathematics
• Lectures and Problems: A Gift to Young Mathematicians
• Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians
• Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals
• The Theory of Singularities and Its Applications