CHAPTER III. A GENERAL ORIENTATION OF THE PROBLEM. The foregoing historical sketch brings prominently to view certain matters of great philosophical interest. First of all it has certainly become clear that in so far, at least, as any system of geometry has professed scientific value, or has claimed to be in any sense valid for reality, the whole history of meta-geometry has been, as a matter of fact, one long and very fruitful search for the philosophic foundations of mathematics in general. The same spirit which through the centuries endeavored so earnestly to justify Euclid as an orderly system of necessary and indubitable knowledge by removing the objectionable theory of parallel lines, has finally subjected the fundamentals of arithmetic as well as those of geometry, to the most searching critical testing. The sufficiency, independence, and mutual compatibility of the various adjectives which presumably define our notions of number and space, have become problems of absorbing interest and promise. As is usual in every marked intellectual advance, every existing difficulty removed has opened up new fields of research, new tendencies of thought and methods of investigation, and consequently new and more difficult problems calling for solution. The light thus thrown both directly and indirectly upon the space problem has led to a very great refinement of the space conception which has resulted more and more in restricting the a priori realm and in handing over to the empirical, as possibly contingent and depending ultimately upon the peculiar nature of experience, certain matters which were previously thought to be apodeictically true. Prior to Lobatchewsky "geometry upon the plane at infinity" was considered as being just as well known as the geometry of any portion of the table upon which I am writing, but today the geometer "knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity." He does know, however, that, within the limits of the utmost refinements of instrumentation and observation thus far attained, the assumptions of Euclid are true for small portions of space and perhaps, when all due allowance for probable error is made, even for that immense region which 1 W. K. Clifford: Lectures and Essays, Vol. I., p. 359. London, 1901. We do not subscribe to the naive space-realism latent in these words of Clifford; the passage is quoted because it indicates very clearly the changed point of view regarding the nature of geometry. is swept by telescopic vision. Hence the important question as to what are the necessary and sufficient marks of the category of space, once regarded as settled, takes on decidedly a new interest for speculative thought. Glancing back for a moment over the history of this movement, one can easily trace from a psychological view-point the predominant intellectual and practical interests out of which it has grown. It is often contended that geometry is concerned with ideal objects. At present, this is certainly true; but as a matter of history it has not always been true. Geometry is in reality a complex product of two factors; the one empirical or, if you please, intuitional, and the other logical. Both appear to be necessary to any geometry which would validate its claims to be a bona fide body of systematic knowledge. Our historical sketch shows that these factors have been variable quantities, the intuitional element having been steadily reduced until at present so far as space is concerned it is entirely rejected. Geometry as a part of "pure" mathematics is coming to be regarded as merely a branch of symbolic Logic, which no longer claims to throw direct light upon the nature of space. Nevertheless, in contemplating the pure abstract science which thus claims to be free from all intuitional bias, we should not forget its humbler |