The German philosopher Immanuel Kant brought the discussion
of the nature of mathematics, most notably the nature
of geometry, back in to central focus with his Critique of Pure
Reason (1952). Whereas he affirmed that all axioms and theorems
of mathematics were truths, he held the view that the
nature of perceptual space was Euclidean and that the contents
of Euclidean geometry were a priori understandings of the human
mind. This was in direct opposition to the emerging understandings
of non-Euclidean geometry.
The establishment of the consistency of non-Euclidean geometry
in the mid-1800s finally freed mathematics from the
restrictive yoke of a single set of axioms thought to be the
only model for the external world. The existence of consistent
non-Euclidean geometries showed the power of man's mind to
construct new mathematical structures, free from the bounds
of an externally existing, controlling world (Eves, 1981;Kline,
1972, 1985; Korner, 1960). This discovery, exciting as it was,
brought with it a new notion of "truth," one buried in the acceptance
of an axiom or a set of axioms defining a model for
an area of investigation. Mathematicians immediately began to
apply this new freedom and axiomatic method to the study of
mathematics.