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the very meaning of “2,” “+,” “3,” “5,” and “=.” Experience could not
possibly refute such truths because their validity is established (as Hume said)
merely by the “relation of ideas.” Even if – ”miraculously” – putting two and
three objects together should on some occasion yield six objects, this would
be a fascinating feature of those objects (rabbits, perhaps); but it would not in
the least tend to refute the purely definitional truths of arithmetic.
The case of geometry is altogether different. Geometry can be either an
empirical science of natural space or an abstract system with uninterpreted
basic concepts and uninterpreted postulates. The latter is the conception
introduced in rigorous fashion by a late 19th-century mathematician, David
Hilbert, and, still later, by a United States geometer, Oswald Veblen. In the
axiomatizations that they developed, the basic concepts, called primitives, are
implicitly defined by the postulates: thus, such concepts as point, straight line,
intersection, betweenness, and plane are related to each other in a merely
formal manner. The proof of theorems from postulates, and with explicit
definitions of derived concepts (such as of triangle, polygon, circle, or conic
section), is achieved by strict deductive inference. Very different, however, is
geometry as understood in practical life, and in the natural sciences and
technologies, in which it constitutes the science of space. Ever since the
development of the non-Euclidean geometries in the first half of the 19th
century, it has no longer been taken for granted that Euclidean geometry is the
only geometry uniquely applicable to the spatial order of physical objects or
events. In Einstein’s general theory of relativity and gravitation, in fact, a
four-dimensional Riemannian geometry with variable curvature was
successfully employed, an event that amounted to a final refutation of the
Kantian contention that the truths of geometry are “synthetic a priori.” With
respect to the relation of postulates to theorems, geometry is thus analytic, like