V. I. Arnold on the perils of “pure” deductive-axiomatic mathematics

The mathematician V. I. Arnold delivered an interesting (and quite opinionated) speech in 1997 (a). Here are two excerpts with his views about how math has been taught in recent decades:

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”).

And:

Attempts to create “pure” deductive-axiomatic mathematics have led to the rejection of the scheme used in physics:

$$\small{\text{Observation → Model → Investigation of the Model → Conclusions → Testing by Observations}}$$Observation → Model → Investigation of the Model → Conclusions → Testing by Observations

And its substitution by the scheme:

$$\small{\text{Definition → Theorem → Proof}}$$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 recounting 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.

The Indian mathematician Kapil Hari Paranjape wrote a thoughtful response to Arnold (a).

Finally, let me also quote two insightful comments on Arnold’s speech at Hacker News:

marcelluspye: I feel there needs to necessarily be a separation of the doing of mathematics and the teaching of mathematics in these kinds of matters. In the teaching of mathematics, especially in the more ‘abstract’ areas, there is not nearly enough driving of intuition, and the ‘problems’ students are given often are unrelated to the ‘problems’ the theory they are learning about was created to solve. Pushing things in the concrete direction is probably the right direction for pedagogy.

tnecniv: Indeed. By far the best teachers I’ve had for math courses spent a good deal of time discussing the history of the topic and motivating its creation.