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Morals

on the Conventionality of Geometry

John
D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

The main
text already lays out Einstein's view on the conventionality of
geometry clearly enough. However, since Einstein did not assert his view
often at length, I report here Einstein's more expansive statement. It
derives from a 1924 review by Einstein of a text, Alfred C. Elsbach, *Kant
und Einstein.*

Elspach sought to develop theses due originally, Einstein
tells us in his review, to Paul Natorp. The fourth thesis incorporates a
conventional view of the metrical geometry of space. Einstein states it
as:

"4. The metric of real space cannot be determined by experiment because space is not real."

Einstein proceeds to explain why he is unsympathetic to this view.

"How one responds to
these theses depends on whether one grants reality to the practically
rigid body. If so, then the concept of distance corresponds to something
that can be experienced. Geometry then contains statements about possible
experiments; it is a physical science directly subject to experimental
testing (standpoint A). If no reality is conceded to the practically rigid
measuring-body, then geometry alone holds no statements about experiences
(experiments), rather only geometry together with the physical sciences
(standpoint B). Physics has hitherto always used the simpler standpoint A
and largely owes to it its productivity; it uses it in all its
measurements. Seen from this standpoint, all the mentioned claims by
Natorp are incorrect; this surely does not need to be elaborated further
in detail. If, however, one assumes standpoint B, which at the current
state of science should be regarded as overly cautious, then geometry on
its own is not experimentally testable. Then geometrical measurements
don’t exist at all. However, one does not have to speak of the “ideality
of space” as a result. “Ideality” is ascribable to all concepts; this is
no more and no less true with respect to space and time. Definite
attribution to sensory experiences is possible only for a *complete*
scientific system of concepts. In my view, Kant influenced developments
unfavorably in that he granted a special place to spatio-temporal concepts
and their relations compared to other concepts.

Seen from standpoint B, though, the choice of geometric concepts and relations is determined only by reasons of simplicity and practicality. Under no condition does the choice of a non-Euclidean geometry presuppose Euclidean geometry as its basis. But then nothing can be discerned empirically about the metric of space; not because “space is not real,” but rather because with this choice of standpoint, geometry is not a complete physical system of concepts but only part of one."

Albert Einstein, "Elsbach's Buch:
*Kant und Einstein*," *Deutsche Literaturzeitung*, 1
(1924): cols. 1685–169. Doc. 321 in *Collected Papers of Albert
Einstein*: *Volume 14: The Berlin Years: Writings &
Correspondence, April 1923-May 1925. *Ed. Diana Kormos
Buchwald *et al.*, Princeton: Princeton University Press, 2015.
Translation from *English Translation Supplement*. p. 326.

Copyright John D. Norton. January 15, 2017.