Page:Popular Science Monthly Volume 67.djvu/652

course of intuitive geometry which does not attempt to be rigorously demonstrative, which emphasizes the sensuous rather than the rational. But in a serious work it is now no longer permissible to have nothing to start from. Wherever rigorous mathematics, there pure logic.

It may be a relief to many that the non-Euclidean geometry has shown the limitations to the arithmetization of mathematics. The opinion that only the concepts of analysis or arithmetic are susceptible of perfectly rigorous treatment Hilbert considers entirely erroneous.

On the contrary, he says, I think that wherever, from the side of the theory of knowledge, or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas.

The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content.

In other words, the world has outgrown Wentworth's geometry. More than this, as Frankland puts it, the possibility of explaining 'mass' (the fundamental property of matter) as a function of 'electric charge' is on the point of banishing both ordinary gross matter and also ether, since the principle of parsimony forbids needless hypothetical entities. Now the relation between the two opposite electricities so closely resembles that between Bolyaian and Riemannean space that, as Clifford adumbrated, we may expect to see matter, ether and electricity banished in favor of space, the various and changing geometries of which will be found adequate to account for all the phenomena of the material world.

Furthermore, these geometries of physical space will be found not to be 'continuous,' but to be the varied and changing 'tactical' arrangements of a discrete, a discontinuous manifold consisting of indivisible units. The notion of continuous extension, so long considered ultimate, will, by this simplification, be subsumed under the finally ultimate notion of juxtaposition, with which Lobachevski begins his great treatise 'Noviya nachala,' in whose very first article he says of it: "This simple idea derives from no other, and so is subject to no further explanation."