TITLE: Vladimir Arnold on formal thinking
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: 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.