i Mathematics is concerned with implications, not applications. Such terms as "applied mathematics," "mathematical physics,° and the like, are indeed in common use, but the signification, rightly understood, is always that of a mixed doctrine, a doctrine that is thor oughly analyzable into two disparate parts: one of these consists of determinate concepts formally combined in accordance with the established canons of ratiocination, i.e., it is pure mathematics and not natural science; the other is matter of observation and experi ment, i.e., it is natural science and not mathe matics. No fibre of either component is a filament of the other. Whether the behavior of natural phenomena is or is not exactly descriptible by mathematical formula; can never be ascertained, for the means to "natural knowledge,° viz., observation and experiment, are fallible by nature, and, however refined or prolonged, are incapable of yielding absolute exactness or certainty. Of any so-called law of nature, "the most, the last, the best that can be said° is that its agreement with the facts is so nearly perfect that every discrepance, if any there be, has escaped detection.
Cannot the like be said of mathematics? In the foregoing conception of mathematics it has indeed been tactily assumed that some ideas are completely determinable, that there are possible systems of postulates absolutely free alike from obscurity and from interior contra diction, that the postulates of such a system import a perfectly definite body of ascertain able implications, and that there is a perfect standard of logic quite independent of time and place and of the defects and idiosyncrasies of individual reasoners. May not these assump tions, some or all of them, be incorrect? There are some grounds, historical and biological, for suspecting that such may be the case. Such an admission, however, whether tentative or unconditioned, has by no means the effect of undefining mathematics or of dethroning the science from its commanding position among the knowledges. It would indeed leave man without the possession, even without the hope, of absolute knowledge, but, among all the forms of actual or potential proximate knowledge, mathematics would still rightly rank as highest. It would indeed be marked by a degree of un certainty or indeterminateness or relativity or inexactness, viz., that degree of inexactness which by supposition would belong to all mean ings and standards. Hence it would not be peculiar to mathematics, but would be common to it and every other science. But every such "other science° has an additional mark of in determinateness, the characteristic inexactness of all possible observation and experiment In any case, then, the observational and experi mental sciences are, in respect to certainty and exactness, hopelessly inferior to mathematics.
But in all ages it has been the faith of the mathematician that complete determination of concepts and absolute rigor of demonstration are in fact attainable, not, however, in the realm of observation and experiment, but in the world of pure thought. It is, then, in that world, where all entia dwell, where is every type of order and every manner of correlation and every variety of relationship and every form of implication, it is in this infinite en semble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being,— it is there that the spirit of mathesis has itz home and its life.
Is it a restricted home, a narrow life, static and cold and gray with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of solar light, it is not clad in the colors that glorify the things of sense, but it is an illuminated world, and over it all and everywhere through out are hues and tints transcending sense. painted there by radiant pencils of psychic light, the radiance in which it lies.
It is a silent world. Nevertheless, in respect to the highest principle of art — the inter penetration of content and form, the perfect fusion of mode and meaning—it even rivals music.
In a sense, it is a static world, like those of the sculptor and the architect. The figures. however, which reason constructs and the mathetic vision beholds, transcend the temple and the statue, alike in i simplicity and in in tricacy, in delicacy and n grace, in symmetry and in poise.
Not only are this home and this life, thus rich in msthetic interests, really controlled and sustained by motives of a sublimed and super sensuous art, but the religious aspiration, too. finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks — the changeless in the midst of change, abiding things in a world of flux, configurations that remain the same despite the swirl and stress of countless hosts of curious transformations.
The literature having for its object the exposition of the nature and prin ciples of mathematics is extensive. By far the most important recent contribution, which at „the same time serves to introduce the reader to the chief memoirs in the field, is B. Russell's The Principles of Mathematics,' Vol. I. For an excellent critical review of the principal modern attempts to define mathematics, the reader may be referred to Prof. M. Bocher's The Fundamental Conceptions and Methods of Mathematics' ((Bull. of the American Math. Soc.,) Vol. XI). Consult also Whitehead, A. N., and Russell, B. A. W., 'Principia Mathe matica' (Cambridge 1910-) ; Whitehead, A. N., 'Introduction to Mathematics) (Home Univer sity Library).