Since science tries to deal with reality, even the most
precise sciences normally work with more or less
imperfectly understood approximations toward which
scientists must maintain an appropriate skepticism. Thus,
for instance, it may come as a shock to mathematicians to
learn that the Schrodinger equation for the hydrogen atom
is not a literally correct description of this atom, but only
an approximation to a somewhat more correct equation
taking account of spin, magnetic dipole, and relativistic
effects; and that this corrected equation is itself only an
imperfect approximation to an infinite set of quantum
field-theoretical equations. Physicists, looking at the
original Schrodinger equation, learn to sense in it the
presence of many invisible terms in addition to the
differential terms visible, and this sense inspires an
entirely appropriate disregard for the purely technical
features of the equation. This very healthy skepticism is
foreign to the mathematical approach. Mathematics must
deal with well-defined situations. Thus, mathematicians
depend on an intellectual effort outside of mathematics
for the crucial specification of the approximation that
mathematics is to take literally.