14
RICE UNIVERSITY STUDIES
“functions,” “functionals,” “operators”, etc., as if they were entirely
heterogeneous objects [13, passim]. But the 2Oth gradually realized that
they are all functions in a broad sense, if a function y = /(x) is conceived
to be a mapping from a general set X : {x} into a general set Y : {y}, with
no relations between the sets X, Y stipulated. Logicians maintain that some
such broad definition of a function had already been anticipated by the
logician Gottlob Frege in the 1870’s. Maybe so. But even if this was so,
it is most unlikely that working mathematicians were aware of it, or in-
fluenced by it. Rather, they arrived at their insight by their own “hard” way.
Riemann Surfaces. In the realm of complex variables, the pioneering
achievement of Bernhard Riemann (1826-1866) was to characterize certain
classes of functions, which are initially given in the complex plane in terms
of certain expressions, by overall properties of the functions on suitable
compact Riemann surfaces, or, as later developments explicated it, on the
universal covering surfaces of the compact ones. A leading case, in which
the universal covering surface need not be envisaged, is the following.
Consider an equation
F(ιv, z) = O,
in which P(w, z) is an irreducible polynomial of the two variables w and z,
where z varies over the Gaussian sphere S. The solution w = w(z) of the
equation is an л-valued algebraic function (« is the degree of the polynomial
in >v), and with it we can form the class of functions
rɪ = {R(ιv(z), z)},
for all possible rational functions R(w, z). Now, Riemann characterizes
the class F1 as follows. He forms the /г-sheeted Riemann surface T over S,
on which >v(z) is properly defined and memorphic, and he considers the
class F2 of all those functions which are defined and meromorphic on T.
Then,
F2 ≡ F1.
It is easy to see that F1 ⊂ F2, and the burden of the assertion is that also
F2 ⊂ F1 ; for a present-day proof see [14, p. 155], and note that this proof
does not proceed by mere “talking” and “cerebration” but also by a re-
course to the Lagrange interpolation formula. Also, if for some element
t(z) in F2 and some z0 in F over which T is not ramified, the n functional
elements of /(z) over z0 are different from each other, then w(z) is also a
rational function of {∕(z), z}.