Men of Mathematics (98 page)

If precisely
n
numbers occur in each of these ordered
n
-tuples in the manifold, the manifold is said to be of
n dimensions.
Thus we are back again talking coordinates with Descartes. If each of the numbers in
(x
₁
, x
₂
, . . . , x
_n
)
is a positive, zero, or negative integer, or if it is an element of any countable set (a set whose elements may be counted off
1, 2, 3, . . .),
and if the like holds for every
n
-tuple in the set, the manifold is said to be
discrete.
If the numbers
x
₁
, x
₂
, . . . , x
_n
,
may take on values
continuously
(as in the motion of a point along a line), the manifold is
continuous.

This working definition has ignored—deliberately—the question of whether the set of ordered n-tuples is “the manifold” or whether something “represented by” these is “the manifold.” Thus, when we say
(x, y)
are the coordinates of a point in a plane, we do not ask what “a point in a plane” is, but proceed to work with these
ordered couples of numbers (x, y)
where
x, y
run through all real numbers independently. On the other hand it may sometimes be advantageous to fix our attention on what such a symbol as (
x, y) represents.
Thus if
x
is the age in seconds of a man and
y
his height in centimeters, we may be interested in the
man
(or the class of all men) rather than in his
coordinates, with which alone the mathematics of our enquiry is concerned.
In this same order of ideas, geometry is no longer concerned with what “space” “is”—whether “is” means anything or not in relation to “space.” Space, for a modern mathematician, is merely a number-manifold of the kind described above, and this conception of space grew out of Riemann's “manifolds.”

Passing on to measurement, Riemann states that “Measurement consists in a superposition of the magnitudes to be compared. If this is lacking, magnitudes can be compared only when one is part of another, and then only the more or less, but not the how much, can be decided.” It may be said in passing that a consistent and useful theory of measurement is at present an urgent desideratum in theoretical physics, particularly in all questions where quanta and relativity are of importance.

Descending once more from philosophical generalities to less mystical mathematics, Riemann proceeded to lay down a definition of
distance,
extracted from his concept of measurement, which has proved to be extremely fruitful in both physics and mathematics. The Pythagorean proposition
that
where
a
is the length of the longest side of a right-angled triangle and
b, c
are the lengths of the other two sides, is the fundamental formula for the measurement of distances in a
plane.
How shall this be extended to a
curved surface?.
To straight lines on the plane correspond geodesics (see chapter 14) on the surface; but on a sphere, for example, the Pythagorean proposition is not true for a right-angled triangle formed by geodesies. Riemann generalized the Pythagorean formula to any manifold as follows:

Let
(x
₁
x
₂
, . . . , x
_n
), (x
₁
+
x
₁
', x
₂
+
x
₂
',
 . . . , x
_n
+
x
_n
')
be the coordinates of two “points” in the manifold which are “infinitesimally near” one another. For our present purpose the meaning of “infinitesimally near” is that powers higher than the second of
X
₁
', x
₂
', . . . , x
_n
',
which measure the “separation” of the two points in the manifold, can be neglected. For simplicity we shall state the definition when
n
= 4—giving the distance between two neighboring points in a space of four dimensions: the distance is the square root of

g
₁₁
x
₁
′
²
+
g
₂₂
x
₂
′
²
+
g
₃₃
x
₃
′
²
+
g
₄₄
x
₄
′
²
+
g
₁₂
x
₁
′x
₂
′
+
g
₁₃
x
₁
′x
₃
′
+
g
₁₄
x
₁
′x
₄
′
+
g
₂₃
x
₂
′x
₃
′
+
g
₂₄
x
₂
′x′
₄
+
g
₃₄
x
₃
′x
₄
′

in which the ten coefficients
g
_n
, . . ., gu
are functions of
x
₁
, x
₂
, x
₃
, x
₄
.
For a particular choice of the
g's,
one “space” is defined. Thus we might have = 1,
g
₁₁
= 1, £33 = 1,
gu
= −1, and all the other
g's
zero; or we might consider a space in which all the g's except
g's
and
¿•34
were zero, and so on. A space considered in relativity is of this general kind in which all the
g's
except
g
₁₁
,
g
₁₁
,
g
₂₂
,
g
₃₃
,
g
₄₄
are zero, and these are certain simple expressions involving
x
₁
, x
₂
, x
₃
, x
₄
.

In the case of an
n
-dimensional space the distance between
neighboring
points is defined in a similar manner; the general expression contains ½
n
(
n
+ 1) terms. The generalized Pythagorean formula for the distance between neighboring points being given, it is a solvable problem in the integral calculus to find the distance between
any
two points of the space. A space whose
metric
(system of measurement) is defined by a formula of the type described is called
Riemannian.

Curvature, as conceived by Riemann (and before him by Gauss; see chapter on the latter) is another generalization from common experience. A straight line has zero curvature; the “measure” of the amount by which a curved line departs from straightness may be the
same for every point of the curve (as it is for a circle), or it may vary from point to point of the curve, when it becomes necessary again to express the “amount of curvature” through the use of infinitesimals. For curved surfaces, the curvature is measured similarly by the amount of departure from a plane, which has zero curvature. This may be generalized and made a little more precise as follows. For simplicity we state first the situation for a two-dimensional space, namely for a surface as we ordinarily imagine surfaces. It is possible from the formula

g
₁₁
x
₁
′
²
+
g
₁₂
x
₁
′
₂
x
₂
′
+
g
₂₂
x
₂
′
²
,

expressing (as before) the square of the distance between neighboring points on a given surface (determined when the functions
g
₁₁
, g
₁₂
, g
₂₂
are given), to calculate the measure of curvature of any point of the surface
wholly in terms of the given functions g
₁₁
, g
₁₂
, g
₂₂
. Now, in ordinary language, to speak of the “curvature” of a space of more than
two
dimensions is to make a meaningless noise. Nevertheless Riemann, generalizing Gauss, proceeded in the same
mathematical
way to build up an expression involving
all
the
g's
in the general case of an
n
-dimensional space, which is of
the same kind mathematically
as the Gaussian expression for the curvature of a
surface,
and this generalized expression is what he called the
measure of curvature
of the space. It is possible to exhibit visual representations of a curved space of more than two dimensions, but such aids to perception are about as useful as a pair of broken crutches to a man with no feet, for they add nothing to the understanding and they are mathematically useless.

Why did Riemann do all this and what has come out of it? Not attempting to answer the first, except to suggest that Riemann did what he did because his daemon drove him, we may briefly enumerate some of the gains that have accrued from Riemann's revolution in geometrical thought. First, it put the creation of “spaces” and “geometries” in unlimited number for specific purposes—use in dynamics, or in pure geometry, or in physical science—within the capabilities of professional geometers, and it baled together huge masses of important geometrical theorems into compact bundles that could be handled easily as wholes. Second, it clarified our conception of space, at least so far as mathematicians deal in “space,” and stripped that mystic nonentity Space of its last shred of mystery. Riemann's achievement has taught mathematicians to disbelieve in
any
geometry, or in
any
space, as a
necessary
mode of human perception. It was the last nail in the coffin of absolute space, and the first in that of the “absolutes” of nineteenth century physics.

Finally, the curvature which Riemann defined, the processes which he devised for the investigation of quadratic differential forms (those giving the formula for the square of the distance between neighboring points in a space of any number of dimensions), and his recognition of the fact that the curvature is an invariant (in the technical sense explained in previous chapters), all found their physical interpretations in the theory of relativity. Whether the latter is in its final form or not is beside the point; since relativity our outlook on physical science is not what it was before. Without the work of Riemann this revolution in scientific thought would have been impossible—unless some later man had created the concepts and the mathematical methods that Riemann created.

I
. If z = x 4- iy, and w = u -f iv, is an analytic function of z, Riemann's equations are