Vladimir Arnold on formal thinking

Question

In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically correct, they understand nothing. If their deduction is correct (as he says), what is wrong with the given solution? What do the students not understand (or where does exactly their mathematical understanding fail?)? How can oneself make sure to "understand properly"? He says:

For example, at a written examination in dynamical systems for fourth-year students at Paris-Dauphine, one problem was to find the limit of the solution of a system of Hamiltonian equations on the phase plane starting with some given initial point when time goes to infinity. The idea was to choose the initial point on a separatrix of a saddle, with the limit being the saddle point.

Preparing the examination problem, I made an arithmetical error, and the phase curve (the energy-level curve containing the initial point) was a closed oval instead of the separatrix. The students discovered this and concluded that there exists a finite time $T$ at which the solution returns to the initial point. Using the unicity theorem, they were able to deduce that for any integer $n$ the value of the solution at time $nT$ is still the initial point. Then came the conclusion: since the limit at infinite time coincides with the limit for any subsequence of times going to infinity, the limit is equal to the initial point! This solution was invented independently by several good students sitting at different places in the examination hall. In all this reasoning, there are no logical mistakes. It is a correct deduction which one may also generate by a computer. It is apparent that the authors understood nothing.

Answer 1

The solution has no limit, of course, because the phase curve is a closed oval.

However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.

The students should have recognized that the solution does not approach a limit, but rather is periodic.

Answer 2

Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.

Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.

If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:

Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.

That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.

Answer 3

He says, in my opinion, that a theory should be studied from its main conclusions and problems it allows to solve backward to the axioms, that is, by making abstraction, that is, by remaking mathematics. People should get their hands dirty with problems and only by banging their head against all problem difficulties they will understand the real appropriateness of axioms of a theory and at least naive mistakes can be avoided. The problem is that the main conclusions can be very numerous (infinity many) or at least much more numerous than the axioms and the theorems that the axioms follows and so it takes a great deal of time and force of will in order to examine them all. So he tries to incite seriuous students through this kind of provoking remarks.

Answer 4

Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.