D O C . 2 2 0 N O N - E U C L I D E A N G E O M E T R Y 2 1 5
220. “Non-Euclidean Geometry and Physics”
[Einstein 1925g]
Completed March 1924
Published January 1925
In: Die Neue Rundschau 36, no. 1 (1925): 16–20.
Thinking about the connection between non-Euclidean geometry and physics
necessarily leads to the question of the connection between geometry and physics
in general. I will consider this latter one first, and shall try to avoid controversial
philosophical questions as much as
In the most ancient eras, geometry was without doubt a semi-empirical science,
a kind of primitive physics. A point was a body whose extension one ignored, a
straight line was roughly defined by points that could be brought to coincide opti-
cally in the direction of one’s gaze or by using a taut thread. The concepts involved
(as is always the case with concepts) did not, indeed, originate directly from expe-
rience—that is, were not logically determined by experience—but they were nev-
ertheless seen as directly related to the objects of experience. Propositions concern-
ing points, lines, and the equality of lengths or angles were to this mode of
understanding equivalent to propositions concerning certain experiences with nat-
ural objects.
The geometry that was thus understood became a mathematical science through
the recognition that most of its propositions could be derived by a purely logical
path from fewer propositions, the so-called axioms. For mathematics comprises ev-
ery science that concerns itself exclusively with logical relations between given en-
tities according to given rules. The derivation of these relations then became the
sole interest, because the independent construction of a logical system—uninflu-
enced by the uncertain external experience, which depends on chance—has always
possessed an irresistible charm for the human spirit.
The basic concepts—point, line, distance, etc.—and the so-called axioms re-
mained in the geometrical system the only ones logically irreducible, or as witness-
es of its empirical origins. One attempted to confine the number of these logically
irreducible fundamental concepts and axioms to a minimum. The attempt to lift all
of geometry out of the murky sphere of the empirical now led imperceptibly to a
mental reorientation that is somewhat analogous to the promotion of revered heroes
[p. 16]
[p. 17]
Previous Page Next Page