Re: derivations, formal and otherwise

[John Denker <jsd@MONMOUTH.COM>]

At 11:16 AM 1/23/00 -0500, Ludwik Kowalski wrote:

> > ... But, as you "proved"...

> The word "proved"
> was quoted to emphasize what you wrote yesterday about differences
> between proving things in math and in science.

At 11:49 AM 1/23/00 -0500, Ludwik Kowalski wrote:

> why do we emphasize formal derivations in physics if the
> only acceptable validation is experimental?

Good question.  IMHO the answer depends on the distinction between
"derivation" and "formal derivation".  I will take the latter to be
synonymous with "formal proof".

In physics, the proof of the pudding is in the eating.  But of course we
*do* calculate (i.e. informally derive) the amount of milk and the amount
of tapioca to put in the pot.  These calculations don't "prove" anything,
but they sure are useful.

Obviously, I posted a highly _informal_ derivation of the wave
equation.  I'm not sure a formal derivation would be possible given the
incomplete description of the medium in question (the slinky).  One could
make the derivation less informal (but still not really formal) by
considering various junk effects including friction, nonlinearities in the
spring, et cetera.  One need not calculate such things in gory detail, but
the rules require at least estimating a bound on them.  There's a big
difference between a pulled-out-of-the-air approximation and a controlled
approximation.

In physics (and in real math, for that matter) the published "derivations"
rarely conform to the nominal rules of validity.  The practical rule for
publication is to make the readership happy.  Very few "proofs" published
in math journals would stand up to computer checking, not only because they
leave out intermediate steps but often because they don't state the
proposition carefully enough to rule out nitpicky counterexamples.  As long
as the nits and missing steps are boring, nobody minds -- the readers
supply the missing details as needed.

------

There is a subfield called mathematical physics in which formal derivations
are done.  One starts with a formal model and rigorously calculates the
consequences.  I don't recall ever seeing mathematical physics even
mentioned on this list, let alone exhibited.

Examples of mathematical physics questions include:
 * Is Dirac's matrix mechanics formally identical to Schrödinger's
differential equation?  (Note that because this is mathematical physics, we
do not consider the question of whether either one matches experiment.)
 * Do the equations of general relativity in free space admit solutions
that are sharp shock waves?
 * What are the critical exponents for such-and-such Ising model?
 * Can one build a cellular automaton that (in some limit) leads to the
Navier-Stokes equations?