mensions or
geometries in which the axiom of parallels is no longer true, why may it
not be that the space in which we make our measurements and on which we
base our mechanics is some one of these "non-Euclidian" spaces? And
indeed many experiments were conducted in search of some clue that this
might be the case. Such experiments in relation to "curved spaces"
seemed particularly alluring, but all have turned out to be fruitless in
results. Failure leads to investigation of the causes of failure. If our
space had been some one of these spaces how would it have been possible
for us to know this fact? The traditional definition of a straight line
has never been satisfactory from a physical point of view. To define it
as the shortest distance between two points is to introduce the idea of
distance, and the idea of distance itself has no meaning without the
idea of straight line, and so the definition moves in a vicious circle.
On the metaphysical side, Lotze (_Metaphysik_, p. 249) and others (Merz,
_History of European Thought in the Nineteenth Century_, Vol. II, p.
716) criticized these attempts, on the whole justly, but the best
interpretation of the situation has been given by Poincare.

Two lines of thought now lead to a recasting of our conceptions of the
fundamental notions of geometry. On the one hand, that very
investigation of postulates that had led to the discovery of the
apparently strange non-Euclidian geometries was easily continued to an
investigation of the simplest basis on which a geometry could be
founded. Then by reaction it was continued with similar methods in
dealing with algebra, and other forms of analysis, with the result that
conceptions of mathematical entities have gradually emerged that
represent a new stage of abstraction in the evolution of mathematics,
soon to be discussed as the dominating conceptions in contemporary
thought. On the other hand, there also developed the problem of the
relations of these geometrical worlds to one another, which has been
primarily significant in helping to clear up the relations of
mathematics in its "pure" and "applied" forms.

Geometry passed through a stage of abstraction like that examined in
connection with arithmetic. Beginning with the discovery of
non-Euclidian geometry, it has been becoming more and more evident that
a line need not be a name for an aspect of a physical object such as the
ridge-pole line of a house and the like, nor even for the mo