From this there follows a fact of decisive importance which has hitherto been hidden from the mathematicians themselves.
There is not, and cannot be, number as such. There are several number-worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number—each type fundamentally peculiar and unique, an expression of a specific world-feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. For indubitably the inner structure of the Euclidean geometry is something quite different from that of the Cartesian, the analysis of Archimedes is something other than the analysis of Gauss, and not merely in matters of form, intuition and method but above all in essence, in the intrinsic and obligatory meaning of number which they respectively develop and set forth. This number, the horizon within which it has been able to make phenomena self-explanatory, and therefore the whole of the “nature” or world-extended that is confined in the given limits and amenable to its particular sort of mathematic, are not common to all mankind, but specific in each case to one definite sort of mankind.
The style of any mathematic which comes into being, then, depends wholly on the Culture in which it is rooted, the sort of mankind it is that ponders it. The soul can bring its inherent possibilities to scientific development, can manage them practically, can attain the highest levels in its treatment of them—but is quite impotent to alter them. The idea of the Euclidean geometry is actualized in the earliest forms of Classical ornament, and that of the Infinitesimal Calculus in the earliest forms of Gothic architecture, centuries before the first learned mathematicians of the respective Cultures were born.
A deep inward experience, the genuine awakening of the ego, which turns the child into the higher man and initiates him into community of his Culture, marks the beginning of number-sense as it does that of language-sense. It is only after this that objects come to exist for the waking consciousness as things limitable and distinguishable as to number and kind; only after this that properties, concepts, causal necessity, system in the world-around, a form of the world, and world laws (for that which is set and settled is ipso facto bounded, hardened, number-governed) are susceptible of exact definition. And therewith comes too a sudden, almost metaphysical, feeling of anxiety and awe regarding the deeper meaning of measuring and counting, drawing and form.
Now, Kant has classified the sum of human knowledge according to syntheses a priori (necessary and universally valid) and a posteriori (experiential and variable from case to case) and in the former class has included mathematical knowledge. Thereby, doubtless, he was enabled to reduce a strong inward feeling to abstract form. But, quite apart from the fact (amply evidenced in modern mathematics and mechanics) that there is no such sharp distinction between the two as is originally and unconditionally implied in the principle, the a priori itself, though certainly one of the most inspired conceptions of philosophy, is a notion that seems to involve enormous difficulties. With it Kant postulates—without attempting to prove what is quite incapable of proof—both unalterableness of form in all intellectual activity and identity of form for all men in the same. And, in consequence, a factor of incalculable importance is—thanks to the intellectual prepossessions of his period, not to mention his own—simply ignored. This factor is the varying degree of this alleged “universal validity.” There are doubtless certain characters of very wide-ranging validity which are (seemingly at any rate) independent of the Culture and century to which the cognizing individual may belong, but along with these there is a quite particular necessity of form which underlies all his thought as axiomatic and to which he is subject by virtue of belonging to his own Culture and no other. Here, then, we have two very different kinds of a priori thought-content, and the definition of a frontier between them, or even the demonstration that such exists, is a problem that lies beyond all possibilities of knowing and will never be solved. So far, no one has dared to assume that the supposed constant structure of the intellect is an illusion and that the history spread out before us contains more than one style of knowing. But we must not forget that unanimity about things that have not yet become problems may just as well imply universal error as universal truth. True, there has always been a certain sense of doubt and obscurity—so much so, that the correct guess might have been made from that non-agreement of the philosophers which every glance at the history of philosophy shows us. But that this non-agreement is not due to imperfections of the human intellect or present gaps in a perfectible knowledge, in a word, is not due to defect, but to destiny and historical necessity—this is a discovery. Conclusions on the deep and final things are to be reached not by predicating constants but by studying differentiæ and developing the organic logic of differences. The comparative morphology of knowledge forms is a domain which Western thought has still to attack.
IV
If mathematics were a mere science like astronomy or mineralogy, it would be possible to define their object. This man is not and never has been able to do. We West-Europeans may put our own scientific notion of number to perform the same tasks as those with which the mathematicians of Athens and Baghdad busied themselves, but the fact remains that the theme, the intention and the methods of the like-named science in Athens and in Baghdad were quite different from those of our own. There is no mathematic but only mathematics. What we call “the history of mathematics”—implying merely the progressive actualizing of a single invariable ideal—is in fact, below the deceptive surface of history, a complex of self-contained and independent developments, an ever-repeated process of bringing to birth new form-worlds and appropriating, transforming and sloughing alien form-worlds, a purely organic story of blossoming, ripening, wilting and dying within the set period. The student must not let himself be deceived. The mathematic of the Classical soul sprouted almost out of nothingness, the historically-constituted Western soul, already possessing the Classical science (not inwardly, but outwardly as a thing learnt), had to win its own by apparently altering and perfecting, but in reality destroying the essentially alien Euclidean system. In the first case, the agent was Pythagoras, in the second Descartes. In both cases the act is, at bottom, the same.
The relationship between the form-language of a mathematic and that of the cognate major arts,[[48]] is in this way put beyond doubt. The temperament of the thinker and that of the artist differ widely indeed, but the expression-methods of the waking consciousness are inwardly the same for each. The sense of form of the sculptor, the painter, the composer is essentially mathematical in its nature. The same inspired ordering of an infinite world which manifested itself in the geometrical analysis and projective geometry of the 17th Century, could vivify, energize, and suffuse contemporary music with the harmony that it developed out of the art of thoroughbass, (which is the geometry of the sound-world) and contemporary painting with the principle of perspective (the felt geometry of the space-world that only the West knows). This inspired ordering is that which Goethe called “The Idea, of which the form is immediately apprehended in the domain of intuition, whereas pure science does not apprehend but observes and dissects.” The Mathematic goes beyond observation and dissection, and in its highest moments finds the way by vision, not abstraction. To Goethe again we owe the profound saying: “the mathematician is only complete in so far as he feels within himself the beauty of the true.” Here we feel how nearly the secret of number is related to the secret of artistic creation. And so the born mathematician takes his place by the side of the great masters of the fugue, the chisel and the brush; he and they alike strive, and must strive, to actualize the grand order of all things by clothing it in symbol and so to communicate it to the plain fellow-man who hears that order within himself but cannot effectively possess it; the domain of number, like the domains of tone, line and colour, becomes an image of the world-form. For this reason the word “creative” means more in the mathematical sphere than it does in the pure sciences—Newton, Gauss, and Riemann were artist-natures, and we know with what suddenness their great conceptions came upon them.[[49]] “A mathematician,” said old Weierstrass “who is not at the same time a bit of a poet will never be a full mathematician.”
The mathematic, then, is an art. As such it has its styles and style-periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes from epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics. In the very deep relation between changes of musical theory and the analysis of the infinite, the details have never yet been investigated, although æsthetics might have learned a great deal more from these than from all so-called “psychology.” Still more revealing would be a history of musical instruments written, not (as it always is) from the technical standpoint of tone-production, but as a study of the deep spiritual bases of the tone-colours and tone-effects aimed at. For it was the wish, intensified to the point of a longing, to fill a spatial infinity with sound which produced—in contrast to the Classical lyre and reed (lyra, kithara; aulos, syrinx) and the Arabian lute—the two great families of keyboard instruments (organ, pianoforte, etc.) and bow instruments, and that as early as the Gothic time. The development of both these families belongs spiritually (and possibly also in point of technical origin) to the Celtic-Germanic North lying between Ireland, the Weser and the Seine. The organ and clavichord belong certainly to England, the bow instruments reached their definite forms in Upper Italy between 1480 and 1530, while it was principally in Germany that the organ was developed into the space-commanding giant that we know, an instrument the like of which does not exist in all musical history. The free organ-playing of Bach and his time was nothing if it was not analysis—analysis of a strange and vast tone-world. And, similarly, it is in conformity with the Western number-thinking, and in opposition to the Classical, that our string and wind instruments have been developed not singly but in great groups (strings, woodwind, brass), ordered within themselves according to the compass of the four human voices; the history of the modern orchestra, with all its discoveries of new and modification of old instruments, is in reality the self-contained history of one tone-world—a world, moreover, that is quite capable of being expressed in the forms of the higher analysis.