16. Space of Geometry. The nineteenth century
finally and fully created Euclidean space; and the ven-
erable geometry of Euclid finally acquired a space in
which to house its figures and constructs. As a mathe-
matical object, Euclidean space had already been
clearly present in Descartes and very actively so in
Lagrange. But only the nineteenth century created a
duality between Euclidean space and Euclidean struc-
ture, as a particular case of a general duality of “space
and structure.” In the twentieth century this general
mathematical duality conquered and captured basic
physics from within.
Metaphysically this duality revealed itself with the
advent of the Bolyai-Lobachevsky non-Euclidean ge-
ometry in 1829-30; but mathematically it had mani-
fested itself before (Bochner [1969], Ch. 13), in the
differential and descriptive geometries of Gaspar
Monge and the projective geometry of Jean-Victor
Poncelet; and it had been foreshadowed in the work
of Lagrange.
A memorable event occurred in 1854 when Riemann
farsightedly set forth this duality in his renowned
“Habilitationsschrift” (1868); and as an immediate ap-
plication of it he outlined the so-called Riemannian
Geometry, defining it as a duality between a general
manifold and a so-called Riemannian metric. Leaping
into the twentieth century, Riemann stated, in expres-
sions of his, that a manifold is a space which is locally
Cartesian, so that in local neighborhoods it is deter-
mined by a system of ordinary real numbers as known
from ordinary mensuration.
Riemann's paper was published only in 1868, and
one of the first to plumb its depth was the philosopher
and mathematician William Kingdon Clifford. But the
philosophers J. B. Stallo and Bertrand Russell (see
Bibliography) did not appreciate Riemann's visionary
thrust, and Stallo was almost abusively critical. Con-
temporary mathematicians were telling these philoso-
phers that what distinguished nineteenth-century
mathematics was the creation of projective geometry
in which the numerical and metrical aspects are some-
how derived from the descriptive and qualitative ones.
Riemann, however, anticipating twentieth-century
valuations, did in no wise attempt to hide the numeri-
cal behind a facade of the descriptive, and some
philosophers were puzzled and even dismayed.
In the twentieth century Riemann became philo-
sophically unassailable; and his status became enhanced
when his geometry was elected to be the setting for
the General Theory of Relativity, which filled philoso-
phers with awe. As if to make it quite clear who in
the past had been right and who not, the physicist and
philosopher Percy Williams Bridgman, in an introduc-
tion to a 1960-reissue of the book by Stallo says that
“the discussion of transcendental geometry is definitely
the weakest part of the book” (p. xxiii).
In a sense, the most abstractly conceived general
space in the nineteenth century was the phase space
of statistical mechanics, especially in the general ver-
sion of Josiah Willard Gibbs. In the twentieth century
this space developed into the infinitely dimensional
space of quantum mechanics, as a setting for its physi-
cal states and statistical interpretations. This space is
an outright intentional mathematical construction,
pure and simple. Yet by physicists' constant preoccu-
pation with it, this space is gradually being transformed
from a tool in mathematics to a “reality” in nature,
if by “reality” we understand anything that evokes a
sense of being immediate, familiar, inevitable, and
inalienable.
