2 1 6 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

of antiquity to gods. One gradually accustomed oneself to consider the fundamen-

tal concepts and axioms of geometry as “evident,” i.e., as objects and qualities of

perception given with the human intellect itself, in such a way, that inner objects of

intuition correspond to the fundamental concepts of geometry and that a negation

of a geometrical axiom is in fact actually inconceivable. From this point of view,

the applicability of these fundamentals to real objects becomes a problem—we

may add: becomes precisely the problem from which Kant’s conception of space

emerged.

A second motive for the separation of geometry from its empirical foundation

was furnished by physics. According to its more refined concepts of the nature of

solid bodies and of light, there exist no natural objects that exactly correspond in

their properties to the fundamental concepts of Euclidean geometry. The solid body

is not absolutely rigid, and the ray of light does not strictly embody a straight line,

or indeed any one-dimensional structure. According to modern science, geometry

strictly speaking does not correspond to any experiences, but only geometry togeth-

er with mechanics, optics, etc. Since, moreover, geometry must precede physics,

inasmuch as the latter’s laws cannot be expressed without the former, geometry ap-

pears to be a science logically preceding every experience and every experiential

science. Thus it came about that at the beginning of the nineteenth century the foun-

dations of Euclidean geometry appeared as something absolutely immutable not

only to mathematicians and philosophers, but to physicists as well.

One may add that during the whole nineteenth century this situation presented

itself even more simply, schematically, and rigidly to the physicist who did not di-

rect his attention in particular to epistemology. His unconsciously maintained po-

sition corresponded to two propositions: The concepts and postulates of Euclidean

geometry are evident. Keeping in mind certain limitations, ruled solids realize the

geometrical concept of length, and light rays that of the straight line.

Overcoming this situation was difficult work and required about a century.

Strangely enough, it had its origin in purely mathematical investigations, long be-

fore the garments of Euclidean geometry became too tight for

physics.[2]

One of

the mathematicians’ tasks was to establish geometry on as few axioms as possible.

Now, among the Euclidean axioms was one which seemed to the mathematicians

less immediately obvious than the others, and which they therefore for a long time

tried to reduce to, i.e., to prove from, the others. This was the so-called axiom of

parallels. Since all attempts to provide such a proof failed, the suspicion gradually

had to develop that such a proof is impossible, that is, that this axiom is indepen-

dent of the rest. This could be proved by erecting a consistent logical structure that

differed from Euclidean geometry in that, and only in that, it replaced the axiom of

parallels by another axiom. To have independently conceived and convincingly ac-

[p. 18]