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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

possible.[1]

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]