Re: grinding versus understanding

[John Denker <jsd@MONMOUTH.COM>]

At 02:46 PM 6/25/00 -0400, Stefan Jeglinski wrote:
> Wasn't it one of the famous
> physicists from early in the century who said something to the effect
> of "you should know the answer before writing down the solution"?

I don't know if it was original with him, but Feynman said on innumerable
occasions that you don't really understand an equation unless you can look
at the equation and know the qualitative behavior of the solution, without
grinding out the exact solution.

Remark #1:  That may not be quite the same sentiment as the quote SJ was
asking about.

Remark #2:  Since I don't understand things as well as Feynman, sometimes I
find it necessary to grind out the solution before knowing what it will
look like.  But I keep Feynman's dictum in mind, and I don't pretend that
grinding is the same as understanding.  It is perhaps a step toward
understanding, in the following sense:  after I've ground out the solution,
I go back to the original equation and re-ask the question:  What _should_
I have seen in this equation that would have told me the qualitative
character of the solution?

Remark #3:  There are a lot of people who read Feynman's popular works and
come away with the idea that
  a) he was really good at getting the right answer by means of hand-wavy
qualitative arguments, and
  b) that's all he ever did.
Well, (a) is correct while (b) is spectacularly wrong.  The truth is he had
one of the sharpest mathematical-physics minds of the century.  He didn't
_need_ to make qualitative arguments -- he could grind out exact answers
faster than most people could state the question.  A good example of what
I'm talking about appears in volume I of the _Lectures_, where he says the
electrodynamics of an atom is analogous to treating the electron as a
classical charged mass connected to the nucleus by a classical spring --
the atom is just a charged harmonic oscillator.  Now there is no elementary
reason to think that an atom is well described as a charged harmonic
oscillator.  So why did he choose that qualitative model?  Because he knew
the exact answer, and he had carefully checked that such a model would
parallel the actual behavior.