V. I. Arnold explaining the physical intuition behind some math concepts

The mathematician V. I. Arnold delivered an interesting (and quite opinionated) speech in 1997 (a). In it he tried to provide simpler explanations for a few mathematical concepts: determinants, groups, and smooth manifolds.

Determinants:

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.

Groups:

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?

Smooth manifolds:

What is a smooth manifold? In a recent American book I read that Poincaré 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 Poincaré’s Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the “abstract” one.

A smooth $\small{k}$k-dimensional submanifold of the Euclidean space $\small{\reals^N}$RN is its subset which in a neighbourhood of its every point is a graph of a smooth mapping of $\small{\reals^k}$Rk into $\small{\reals^{N-k}}$RN−k (where $\small{\reals^k}$Rk and $\small{\reals^{N-k}}$RN−k are coordinate subspaces). This is a straightforward generalization of most common smooth curves on the plane (say, of the circle $\small{x^2 + y^2 = 1}$x2+y2=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.