On teaching mathematics

By V.I. Arnold

This is an extended text of the address at the discussion on
teaching of mathematics in Palais de DŽcouverte in Paris on 7 March 1997.

Mathematics is a part of physics. Physics is an experimental
science, a part of natural science. Mathematics is the part of physics where experiments
are cheap.

The Jacobi identity (which forces the heights of a triangle to
cross at one point) is an experimental fact in the same way as that the Earth is
round (that is, homeomorphic to a ball). But it can be discovered with less
expense.

In the middle of the twentieth century it was attempted to divide physics
and mathematics. The consequences turned out to be catastrophic. Whole
generations of mathematicians grew up without knowing half of their science
and, of course, in total ignorance of any other sciences. They first began
teaching their ugly scholastic pseudo-mathematics to their students, then to
schoolchildren (forgetting Hardy's warning that ugly mathematics has no
permanent place under the Sun).

Since scholastic mathematics that is cut off from physics is fit
neither for teaching nor for application in any other science, the result was the
universal hate towards mathematicians - both on the part of the poor schoolchildren
(some of whom in the meantime became ministers) and of the users.

The ugly building, built by undereducated mathematicians who were exhausted
by their inferiority complex and who were unable to make themselves familiar with
physics, reminds one of the rigorous axiomatic theory of odd numbers. Obviously,
it is possible to create such a theory and make pupils admire the perfection and
internal consistency of the resulting structure (in which, for example, the sum
of an odd number of terms and the product of any number of factors are
defined). From this sectarian point of view, even numbers could either be
declared a heresy or, with passage of time, be introduced into the theory
supplemented with a few "ideal" objects (in order to comply with the
needs of physics and the real world).

Unfortunately, it was an ugly twisted construction of mathematics
like the one above which predominated in the teaching of mathematics for
decades. Having originated in France, this pervertedness quickly spread to teaching
of foundations of mathematics, first to university students, then to school pupils
of all lines (first in France, then in other countries, including Russia).

To the question "what is 2 + 3" a French primary school
pupil replied: "3 + 2, since addition is commutative". He did not
know what the sum was equal to and could not even understand what he was asked
about!

Another French pupil (quite rational, in my opinion) defined
mathematics as follows: "there is a square, but that still has to be
proved".

Judging by my teaching experience in France, the university
students' idea of mathematics (even of those taught mathematics at the Ecole
Normale Superieure - I feel sorry most of all for these obviously intelligent
but deformed kids) is as poor as that of this pupil.

For example, these students have never seen a paraboloid and a
question on the form of the surface given by the equation xy = z^2 puts the mathematicians
studying at ENS into a stupor. Drawing a curve given by parametric equations
(like x = t^3 - 3t, y = t^4 - 2t^2) on a plane is a totally impossible problem
for students (and, probably, even for most French professors of mathematics).

Beginning with l'Hospital's first textbook on calculus
("calculus for understanding of curved lines") and roughly until Goursat's
textbook, the ability to solve such problems was considered to be (along with the
knowledge of the times table) a necessary part of the craft of every
mathematician.

Mentally challenged zealots of "abstract mathematics"
threw all the geometry (through which connection with physics and reality most often
takes place in mathematics) out of teaching. Calculus textbooks by Goursat,
Hermite, Picard were recently dumped by the student library of the Universities
Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only
rescued by my intervention).

ENS students who have sat through courses on differential and
algebraic geometry (read by respected mathematicians) turned out be acquainted neither
with the Riemann surface of an elliptic curve y^2 = x^3 + ax + b nor, in fact,
with the topological classification of surfaces (not even mentioning elliptic
integrals of first kind and the group property of an elliptic curve, that is,
the Euler-Abel addition theorem). They were only taught Hodge structures and
Jacobi varieties!

How could this happen in France, which gave the world Lagrange and
Laplace, Cauchy and PoincarŽ, Leray and Thom? It seems to me that a reasonable explanation
was given by I.G. Petrovskii, who taught me in 1966: genuine mathematicians do
not gang up, but the weak need gangs in order to survive. They can unite on
various grounds (it could be super-abstractness, anti-Semitism or "applied
and industrial" problems), but the essence is always a solution of the
social problem - survival in conditions of more literate surroundings.

By the way, I shall remind you of a warning of L. Pasteur: there
never have been and never will be any "applied sciences", there are
only applications of sciences (quite useful ones!).

In those times I was treating Petrovskii's words with some doubt,
but now I am being more and more convinced of how right he was. A considerable part
of the super-abstract activity comes down simply to industrializing shameless
grabbing of discoveries from discoverers and then systematically assigning them
to epigons-generalizers. Similarly to the fact that America does not carry
Columbus's name, mathematical results are almost never called by the names of
their discoverers.

In order to avoid being misquoted, I have to note that my own achievements
were for some unknown reason never expropriated in this way, although it always
happened to both my teachers (Kolmogorov, Petrovskii, Pontryagin, Rokhlin) and
my pupils. Prof. M. Berry once formulated the following two principles: The
Arnold Principle. If a notion bears a personal name, then this name is not the
name of the discoverer. The Berry Principle: The Arnold Principle is applicable
to itself.

Let's return, however, to teaching of mathematics in France.

When I was a first-year student at the Faculty of Mechanics and Mathematics
of the Moscow State University, the lectures on calculus were read by the set-theoretic
topologist L.A. Tumarkin, who conscientiously retold the old classical calculus
course of French type in the Goursat version. He told us that integrals of
rational functions along an algebraic curve can be taken if the corresponding
Riemann surface is a sphere and, generally speaking, cannot be taken if its
genus is higher, and that for the sphericity it is enough to have a
sufficiently large number of double points on the curve of a given degree
(which forces the curve to be unicursal: it is possible to draw its real points
on the projective plane with one stroke of a pen).

These facts capture the imagination so much that (even given
without any proofs) they give a better and more correct idea of modern
mathematics than whole volumes of the Bourbaki treatise. Indeed, here we find
out about the existence of a wonderful connection between things which seem to
be completely different: on the one hand, the existence of an explicit expression
for the integrals and the topology of the corresponding Riemann surface and, on
the other hand, between the number of double points and genus of the
corresponding Riemann surface, which also exhibits itself in the real domain as
the unicursality.

Jacobi noted, as mathematics' most fascinating property, that in
it one and the same function controls both the presentations of a whole number
as a sum of four squares and the real movement of a pendulum.

These discoveries of connections between heterogeneous
mathematical objects can be compared with the discovery of the connection
between electricity and magnetism in physics or with the discovery of the
similarity between the east coast of America and the west coast of Africa in
geology.

The emotional significance of such discoveries for teaching is
difficult to overestimate. It is they who teach us to search and find such wonderful
phenomena of harmony of the Universe.

The de-geometrisation of mathematical education and the divorce
from physics sever these ties. For example, not only students but also modern algebro-geometers
on the whole do not know about the Jacobi fact mentioned here: an elliptic
integral of first kind expresses the time of motion along an elliptic phase
curve in the corresponding Hamiltonian system.

Rephrasing the famous words on the electron and atom, it can be
said that a hypocycloid is as inexhaustible as an ideal in a polynomial ring.
But teaching ideals to students who have never seen a hypocycloid is as ridiculous
as teaching addition of fractions to children who have never cut (at least
mentally) a cake or an apple into equal parts. No wonder that the children will
prefer to add a numerator to a numerator and a denominator to a denominator.

From my French friends I heard that the tendency towards
super-abstract generalizations is their traditional national trait. I do not
entirely disagree that this might be a question of a hereditary disease, but I would
like to underline the fact that I borrowed the cake-and-apple example from PoincarŽ.

The scheme of construction of a mathematical theory is exactly the
same as that in any other natural science. First we consider some objects and make
some observations in special cases. Then we try and find the limits of application
of our observations, look for counter-examples which would prevent unjustified
extension of our observations onto a too wide range of events (example: the
number of partitions of consecutive odd numbers 1, 3, 5, 7, 9 into an odd
number of natural summands gives the sequence 1, 2, 4, 8, 16, but then comes
29).

As a result we formulate the empirical discovery that we made (for
example, the Fermat conjecture or PoincarŽ conjecture) as clearly as possible.
After this there comes the difficult period of checking as to how reliable are
the conclusions.

At this point a special technique has been developed in
mathematics. This technique, when applied to the real world, is sometimes
useful, but can sometimes also lead to self-deception. This technique is called
modeling. When constructing a model, the following idealization is made:
certain facts which are only known with a certain degree of probability or with
a certain degree of accuracy, are considered to be "absolutely"
correct and are accepted as "axioms". The sense of this "absoluteness"
lies precisely in the fact that we allow ourselves to use these
"facts" according to the rules of formal logic, in the process
declaring as "theorems" all that we can derive from them.

It is obvious that in any real-life activity it is impossible to
wholly rely on such deductions. The reason is at least that the parameters of the
studied phenomena are never known absolutely exactly and a small change in
parameters (for example, the initial conditions of a process) can totally
change the result. Say, for this reason a reliable long-term weather forecast
is impossible and will remain impossible, no matter how much we develop
computers and devices which record initial conditions.

In exactly the same way a small change in axioms (of which we
cannot be completely sure) is capable, generally speaking, of leading to completely
different conclusions than those that are obtained from theorems which have
been deduced from the accepted axioms. The longer and fancier is the chain of deductions
("proofs"), the less reliable is the final result.

Complex models are rarely useful (unless for those writing their dissertations).

The mathematical technique of modeling consists of ignoring this trouble
and speaking about your deductive model in such a way as if it coincided with
reality. The fact that this path, which is obviously incorrect from the point
of view of natural science, often leads to useful results in physics is called
"the inconceivable effectiveness of mathematics in natural sciences"
(or "the Wigner principle").

Here we can add a remark by I.M. Gel'fand: there exists yet
another phenomenon which is comparable in its inconceivability with the inconceivable
effectiveness of mathematics in physics noted by Wigner - - this is the equally
inconceivable ineffectiveness of mathematics in biology.

"The subtle poison of mathematical education" (in F.
Klein's words) for a physicist consists precisely in that the absolutised model
separates from the reality and is no longer compared with it. Here is a simple
example: mathematics teaches us that the solution of the Malthus equation dx/dt
= x is uniquely defined by the initial conditions (that is that the
corresponding integral curves in the (t,x)-plane do not intersect each other).
This conclusion of the mathematical model bears little relevance to the
reality. A computer experiment shows that all these integral curves have common
points on the negative t-semi-axis. Indeed, say, curves with the initial conditions
x(0) = 0 and x(0) = 1 practically intersect at t = -10 and at t = -100 you
cannot fit in an atom between them. Properties of the space at such small
distances are not described at all by Euclidean geometry. Application of the
uniqueness theorem in this situation obviously exceeds the accuracy of the
model. This has to be respected in practical application of the model,
otherwise one might find oneself faced with serious troubles.

I would like to note, however, that the same uniqueness theorem explains
why the closing stage of mooring of a ship to the quay is carried out manually:
on steering, if the velocity of approach would have been defined as a smooth
(linear) function of the distance, the process of mooring would have required
an infinitely long period of time. An alternative is an impact with the quay
(which is damped by suitable non-ideally elastic bodies). By the way, this
problem had to be seriously confronted on landing the first descending apparata
on the Moon and Mars and also on docking with space stations - here the uniqueness
theorem is working against us.

Unfortunately, neither such examples, nor discussing the danger of
fetishising theorems are to be met in modern mathematical textbooks, even in
the better ones. I even got the impression that scholastic mathematicians (who
have little knowledge of physics) believe in the principal difference of the
axiomatic mathematics from modeling which is common in natural science and which
always requires the subsequent control of deductions by an experiment.

Not even mentioning the relative character of initial axioms, one
cannot forget about the inevitability of logical mistakes in long arguments (say,
in the form of a computer breakdown caused by cosmic rays or quantum
oscillations). Every working mathematician knows that if one does not control
oneself (best of all by examples), then after some ten pages half of all the
signs in formulae will be wrong and twos will find their way from denominators
into numerators.

The technology of combating such errors is the same external
control by experiments or observations as in any experimental science and it
should be taught from the very beginning to all juniors in schools.

Attempts to create "pure" deductive-axiomatic
mathematics have led to the rejection of the scheme used in physics
(observation - model -investigation of the model - conclusions - testing by
observations) and its substitution by the scheme: definition - theorem - proof.
It is impossible to understand an unmotivated definition but this does not stop
the criminal algebraists-axiomatisators. For example, they would readily define
the product of 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 then possible
to force poor students to learn this theorem and its proof (with the aim of
raising the standing of both the science and the persons teaching it). It is obvious
that such definitions and such proofs can only harm the teaching and practical
work.

It is only possible to understand the commutativity of
multiplication by counting and re-counting soldiers by ranks and files or by
calculating the area of a rectangle in the two ways. Any attempt to do without
this interference by physics and reality into mathematics is sectarianism and isolationism
which destroy the image of mathematics as a useful human activity in the eyes
of all sensible people.

I shall open a few more such secrets (in the interest of poor
students).

The determinant of a matrix is an (oriented) volume of the parallelepiped
whose edges are its columns. If the students are told this secret (which is
carefully hidden in the purified algebraic education), then the whole theory of
determinants becomes a clear chapter of the theory of poly-linear forms. If
determinants are defined otherwise, then any sensible person will forever hate
all the determinants, Jacobians and the implicit function theorem.

What is a group? Algebraists teach that this is supposedly a set
with two operations that satisfy a load of easily-forgettable axioms. This
definition provokes a natural protest: why would any sensible person need such
pairs of operations? "Oh, curse this maths" - concludes the student
(who, possibly, becomes the Minister for Science in the future).

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

This is all the definition there is. The so-called
"axioms" are in fact just (obvious) properties of groups of
transformations. What axiomatisators call "abstract groups" are just
groups of transformations of various sets considered up to isomorphisms (which
are one-to-one mappings preserving the operations). As Cayley proved, there are
no "more abstract" groups in the world. So why do the algebraists
keep on tormenting students with the abstract definition?

By the way, in the 1960s I taught group theory to Moscow schoolchildren.
Avoiding all the axiomatics and staying as close as possible to physics, in
half a year I got to the Abel theorem on the unsolvability of a general
equation of degree five in radicals (having on the way taught the pupils
complex numbers, Riemann surfaces, fundamental groups and monodromy groups of
algebraic functions). This course was later published by one of the audience,
V. Alekseev, as the book The Abel theorem in problems.

What is a smooth manifold? In a recent American book I read that PoincarŽ
was not acquainted with this (introduced by himself) notion and that the
"modern" definition was only given by Veblen in the late 1920s: a
manifold is a topological space which satisfies a long series of axioms.

For what sins must students try and find their way through all
these twists and turns? Actually, in Poincare's Analysis Situs there is an
absolutely clear definition of a smooth manifold which is much more useful than
the "abstract" one.

A smooth k-dimensional submanifold of the Euclidean space R^N is
its subset which in a neighbourhood of its every point is a graph of a smooth
mapping of R^k into R^(N - k) (where R^k and R^(N - k) are coordinate
subspaces). This is a straightforward generalization of most common smooth
curves on the plane (say, of the circle x^2 + y^2 = 1) or curves and surfaces
in the three-dimensional space. Between smooth manifolds smooth mappings are
naturally defined. Diffeomorphisms are mappings which are smooth, together with
their inverses.

An "abstract" smooth manifold is a smooth submanifold of
a Euclidean space considered up to a diffeomorphism. There are no "more
abstract" finite-dimensional smooth manifolds in the world (Whitney's
theorem). Why do we keep on tormenting students with the abstract definition? Would
it not be better to prove them the theorem about the explicit classification of
closed two-dimensional manifolds (surfaces)?

It is this wonderful theorem (which states, for example, that any compact
connected oriented surface is a sphere with a number of handles) that gives a
correct impression of what modern mathematics is and not the super-abstract
generalizations of naive submanifolds of a Euclidean space which in fact do not
give anything new and are presented as achievements by the axiomatisators.

The theorem of classification of surfaces is a top-class mathematical
achievement, comparable with the discovery of America or X-rays. This is a
genuine discovery of mathematical natural science and it is even difficult to
say whether the fact itself is more attributable to physics or to mathematics.
In its significance for both the applications and the development of correct
Weltanschauung it by far surpasses such "achievements" of mathematics
as the proof of Fermat's last theorem or the proof of the fact that any
sufficiently large whole number can be represented as a sum of three prime
numbers.

For the sake of publicity modern mathematicians sometimes present
such sporting achievements as the last word in their science. Understandably this
not only does not contribute to the society's appreciation of mathematics but,
on the contrary, causes a healthy distrust of the necessity of wasting energy
on (rock-climbing-type) exercises with these exotic questions needed and wanted
by no one.

The theorem of classification of surfaces should have been
included in high school mathematics courses (probably, without the proof) but
for some reason is not included even in university mathematics courses (from
which in France, by the way, all the geometry has been banished over the last
few decades).

The return of mathematical teaching at all levels from the
scholastic chatter to presenting the important domain of natural science is an especially
hot problem for France. I was astonished that all the best and most important
in methodical approach mathematical books are almost unknown to students here
(and, seems to me, have not been translated into French). Among these are
Numbers and figures by Rademacher and Toeplitz, Geometry and the imagination 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 strong impression the calculus course by
Hermite (which does exist in a Russian translation!) made on me in my school years.

Riemann surfaces appeared in it, I think, in one of the first
lectures (all the analysis was, of course, complex, as it should be).
Asymptotics of integrals were investigated by means of path deformations on
Riemann surfaces under the motion of branching points (nowadays, we would have called
this the Picard-Lefschetz theory; Picard, by the way, was Hermite's son-in-law
- mathematical abilities are often transferred by sons-in-law: the dynasty
Hadamard - P. Levy - L. Schwarz - U. Frisch is yet another famous example in
the Paris Academy of Sciences).

The "obsolete" course by Hermite of one hundred years
ago (probably, now thrown away from student libraries of French universities)
was much more modern than those most boring calculus textbooks with which
students are nowadays tormented.

If mathematicians do not come to their senses, then the consumers
who preserved a need in a modern, in the best meaning of the word, mathematical
theory as well as the immunity (characteristic of any sensible person) to the
useless axiomatic chatter will in the end turn down the services of the
undereducated scholastics in both the schools and the universities.

A teacher of mathematics, who has not got to grips with at least
some of the volumes of the course by Landau and Lifshitz, will then become a relict
like the one nowadays who does not know the difference between an open and a
closed set.

V.I. Arnold

Translated by A.V. GORYUNOV