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 7
complished this idea remains the eternal achievement of Lobachevsky, on the one
hand, and of the
Bolyais,[3]
father and son, on the other.
The conviction thus became established among mathematicians that besides the
Euclidean geometry there were still others, logically just as justifiable geometries,
and this obviously led to the question of whether Euclidean geometry and no other
is the necessary basis of physics. The question was also posed in the specific form:
Is it Euclidean geometry or another geometry that is valid in the physical world?
There has been much argument about whether this last question is meaningful.
To see this clearly, one must logically adopt one of the two following
viewpoints.[4]
Either one assumes that the geometrical “body” is realized in principle in the nat-
ural
solids,[5]
provided that certain rules concerning temperature, mechanical
stresses, etc., are observed; this is the practical physicist’s point of view. Then, a
natural object corresponds to geometrical “distance,” and all the propositions of ge-
ometry take on the character of statements about real bodies.
Helmholtz[6]
repre-
sented this point of view especially clearly, and one may add that without him the
formulation of the theory of relativity would have been practically impossible.
Alternatively, one denies in principle the existence of objects corresponding to
the basic concepts of geometry. Then, geometry alone contains no statements about
objects of reality, but only geometry together with physics. This point of view,
which may be the more comprehensive for the systematic presentation of a com-
plete physics, was especially clearly represented by
Poincaré.[7]
From this point of
view the whole content of geometry is conventional; whether a geometry is to be
preferred depends on how “simple” a physics may be constructed with it that would
be in harmony with experience.
Here we shall choose the first point of view as the one better suited to the present
state of our knowledge. From its vantage, our question as to the validity or invalid-
ity of Euclidean geometry has a clear
meaning.[8]
Euclidean geometry, and geom-
etry in general, retains as before the character of a mathematical science, inasmuch
as the derivation of its theorems from the axioms remains a purely logical one, but
at the same time it becomes a physical science, inasmuch as the axioms contain as-
sertions about natural objects whose validity can only be decided by experiment.
But we must always be conscious of the fact that the idealization contained in the
fiction that the rigid (measuring) body actually exists as an object in nature may one
day prove unjustified, or only justified with respect to certain natural
phenomena.[9]
The general theory of relativity has already proved that this concept
is unjustified for spaces of small extension, in an astronomical sense. The theory of
elementary electrical quanta may prove the concept unjustified for an extension of
atomic orders of magnitude. Riemann already realized that both were possible.
[p. 19]