Why should we accept the idea that laws in the differential form are always
more fundamental than laws which describe integral effects. In most cases
experimentally verifiable relations are integral.
Consider the Biot-Savart formula (giving dB produced by a selected current
element I*dL at any location). This simple relation can be used to dervive
(by integration) the magnetic field produced by a wire of arbitrary shape.
The opposite was much more difficult; it had to be guessed. Yes, from strictly
mathematical point of view the "fundamental" nature of the B-S formula is
quite obvious. But the formula can never be directly verified (I am referring
to wires, not particles) because a single current element is an abstraction
we can not produce in a laboratory. Measuring dB produced by single charged
particles is not possible in common student labs.
Why should the non-directly-verifiable relations be considered as "more
fundamental" from the philosophical point of view? Why shouldn't we say that
"what students find through measurements is more fundamental than what they
must guess-and-verify"? Aren't we trying to make physics as difficult as it
can possibly be? I know that there are many definitions of "simplicity" but
some are better than others. And what was simple to Feynman is not always
simple to me.
Ludwik Kowalski