Men of Mathematics (95 page)

As a sequel to his philosophical studies with Johann Friedrich Herbart (1776-1841), Riemann came to the conclusion in 1850 (he was then twenty four) that “a complete, well-rounded mathematical theory can be established, which progresses from the elementary laws for individual points to the processes given to us in the plenum ('continuously filled space') of reality, without distinction between gravitation, electricity, magnetism, or thermostatics.” This is probably to be interpreted as Riemann's rejection of all “action at a distance” theories in physical science in favor of field theories. In the latter the physical properties of the “space” surrounding a “charged particle,” say, are the object of mathematical investigation. Riemann at this stage of his career seems to have believed in a space-filling “ether,” a conception now abandoned. But as will appear from his epochal work on the foundations of geometry, he later sought the description and correlation of physical phenomena in the
geometry
of the “space” of human experience. This is in the current fashion, which rejects an existent, unobservable ether as a cumbersome superfluity.

Fascinated by his work in physics, Riemann let his pure mathematics slide for a while and in the fall of 1850 joined the seminar in mathematical physics which had just been founded by Weber, Ulrich, Stern, and Listing. Physical experiments in this seminar consumed the time that scholarly prudence would have reserved for the doctoral dissertation in mathematics, which Riemann did not submit till he was twenty five.

One of the leaders in the seminar, Johann Benedict Listing (18081882), may be noted in passing, as he probably influenced Riemann's thought in what was to be (1857) one of his greatest achievements, the introduction of topological methods into the theory of functions of a complex variable.

It will be recalled that Gauss had prophesied that analysis situs would become one of the most important fields of mathematics, and Riemann, by his inventions in the theory of functions, was to give a partial fulfillment of this prophecy. Although topology (now called analysis situs) as first developed bore but little resemblance to the elaborate theory which today absorbs all the energies of a prolific school, it may be of interest to state the trivial puzzle which apparently started the whole vast and intricate theory. In Euler's time seven bridges crossed the river Pregel in Königsberg, as in the diagram, the shaded bars representing the bridges. Euler proposed the problem of crossing all seven bridges without passing twice over any one. The problem is impossible.

*  *  *

The nature of Riemann's use of topological methods in the theory of functions may be disposed of here, although an adequate description is out of the question in untechnical language. For the meaning of “uniformity” with respect to a function of a complex variable we must refer to what was said in the chapter on Gauss. Now, in the theory of Abelian functions,
multiform
functions present themselves inevitably; an n-valued function of
z
is a function which, except for certain values of
z,
takes precisely
n
distinct values for each value assigned to
z.
Illustrating
multiformity,
or
many-valuedness,
for functions of a real variable, we note that
y,
considered as a function of
x,
defined by the equation
y
²
= x,
is two-valued. Thus, if
x
= 4, we get
y
²
= 4, and hence
y
= 2 or −2; if
x
is any real number except zero or “infinity,”
y
has the two distinct values of
and
In this simplest possible example
y
and
x
are connected by an algebraic equation, namely
y
²
—x = 0.
Passing at once to the general situation of which this is a very special case, we might discuss the
n
-valued function
y
which is defined, as a function of
x,
by the equation

P
₀
(x)y
ⁿ
+
P
₁
(x)y
^n-1
+ . . . +
P
_n-1
(x)y + P
_n
(x)
= 0,

in which the P's are polynomials in
x.
This equation defines
y
as an
n
-valued function of
x.
As in the case of
y
²
—x = 0,
there will be certain values of
x
for which two or more of these
n
values of
y
are equal. These values of
x
are the so-called
branch points
of the
n
-valued function defined by the equation.

All this is now extended to functions of complex variables, and the function
w
(also its integral) as defined by

P
₀
(z)w
ⁿ
+
P
₁
(z)w
ⁿ
–
¹
+ . . . +
P
_n-1
(z)w +
P
_n
(z)
= 0,

in which
z
denotes the complex variable
s
+
it,
where
s, t
are real variables and
The
n
values of
w
are called the
branches
of the function
w.
Here we must refer (chapter on Gauss) to what was said about the representation of
uniform
functions of
z.
Let the variable
z (= s
+
it)
trace out any path in its plane, and let the
uniform
functions
f
(z) be expressed in the form
U + iV,
where
U, V are
functions of
s, t.
Then, to every value of
z
will correspond one, and only one, value for each of
U, V,
and, as
z
traces out its path in the
s,
t
-plane,
f(z)
will trace out a corresponding path in the U, V-plane: the path of
f(z)
will be
uniquely
determined by that of
z.
But if
w
is a
multiform
(many-valued) function of
z,
such that precisely
n
distinct values of
w
are determined by each value of
z
(except at branch points, where several values of
w
may be equal), then it is obvious that
one
w-plane no longer suffices (if
n
is greater than l) to represent the path, the “march” of the function
w.
In the case of a
two
-valued function
w,
such as that determined by
w
²
= z, two
w-planes would be required and, quite generally, for an
n
-valued function
(n
finite or infinite), precisely
n
such
w
-planes would be required.

The advantages of considering
uniform
(one-valued) functions instead of
n
-valued functions (
n
greater than 1) should be obvious even to a non-mathematician. What Riemann did was this: instead of the
n
distinct w-planes, he introduced an rc-sheeted surface, of the sort roughly described in what follows, on which the
multiform
function is
uniform,
that is, on which, to each “place” on the surface corresponds one, and only one, value of the function represented.

Riemann
united,
as it were, all the
n
planes into a
single
plane, and he did this by what may at first look like an inversion of the representation of the
n
branches of the
n
-valued function on
n
distinct planes; but a moment's consideration will show that, in effect, he
restored uniformity.
For he superimposed
n z
-planes on one another; each of these planes, or
sheets,
is associated with a particular branch of the function so that, as long as
z
moves in a particular sheet, the corresponding branch of the function is traversed by
w
(the
n
-valued function of
z
under discussion), and as
z
passes from one sheet to another, the branches are changed, one into another, until, on the variable
z
having traversed all the sheets and having returned to its initial position, the original branch is restored. The passage of the variable
z
from one sheet to another is effected by means of
cuts
(which may be thought of as straight-line bridges) joining branch points; along a given cut providing passage from one sheet to another, one “lip” of the upper sheet is imagined as pasted or joined to the opposite lip of the under sheet, and similarly for the other lip of the upper sheet. Diagrammatically, in cross-section,