Chemistry, once concerned with defining as sharply as possible the qualities of elements, such as valency, weight, affinity and reactivity, is setting to work to get rid of these sensible traits. The elements are held to differ in character according to their derivation from this or that compound. They are represented to be complexes of different units which indeed behave (“actually”) as units of a higher order and are not practically separable but show deep differences in point of radioactivity. Through the emanation of radiant energy degradation is always going on, so that we can speak of the lifetime of an element, in formal contradiction with the original concept of the element and the spirit of modern chemistry as created by Lavoisier. All these tendencies are bringing the ideas of chemistry very close to the theory of Entropy, with its suggestive opposition of causality and destiny, Nature and History. And they indicate the paths that our science is pursuing—on the one hand, towards the discovery that its logical and numerical results are identical with the structure of the reason itself, and, on the other, towards the revelation that the whole theory which clothes these numbers merely represents the symbolic expression of Faustian life.
And here, as our study draws to its conclusion, we must mention the truly Faustian theory of “aggregates,” one of the weightiest in all this form-world of our science. In sharpest antithesis to the older mathematic, it deals, not with singular quantities but with the aggregates constituted by all quantities [or objects] having this or that specified morphological similarity—for instance all square numbers or all differential equations of a given type. Such an aggregate it conceives as a new unit, a new number of higher order, and subjecting it to criteria of new and hitherto quite unsuspected kinds such as “potency,” “order,” “equivalence,” “countableness,” and devising laws and operative methods for it in respect of these criteria. Thus is being actualized a last extension of the function-theory.[[528]] Little by little this absorbed the whole of our mathematic, and now it is dealing with variables by the principles of the Theory of Groups in respect of the character of the function and by those of the Theory of Aggregates in respect of the values of the variables. Mathematical philosophy is well aware that these ultimate meditations on the nature of number are fusing with those upon pure logic, and an algebra of logic is talked of. The study of geometrical axioms has become a chapter of epistemology.
The aim to which all this is striving, and which in particular every Nature-researcher feels in himself as an impulse, is the achievement of a pure numerical transcendence, the complete and inclusive conquest of the visibly apparent and its replacement by a language of imagery unintelligible to the layman and impossible of sensuous realization—but a language that the great Faustian symbol of Infinite space endows with the dignity of inward necessity. The deep scepticism of these final judgments links the soul anew to the forms of early Gothic religiousness. The inorganic, known and dissected world-around, the World as Nature and System, has deepened itself until it is a pure sphere of functional numbers. But, as we have seen, number is one of the most primary symbols in every Culture; and consequently the way to pure number is the return of the waking consciousness to its own secret, the revelation of its own formal necessity. The goal reached, the vast and ever more meaningless and threadbare fabric woven around natural science falls apart. It was, after all, nothing but the inner structure of the “Reason,” the grammar by which it believed it could overcome the Visible and extract therefrom the True. But what appears under the fabric is once again the earliest and deepest, the Myth, the immediate Becoming, Life itself. The less anthropomorphic science believes itself to be, the more anthropomorphic it is. One by one it gets rid of the separate human traits in the Nature-picture, only to find at the end that the supposed pure Nature which it holds in its hand is—humanity itself, pure and complete. Out of the Gothic soul grew up, till it overshadowed the religious world-picture, the spirit of the City, the alter ego of irreligious Nature-science. But now, in the sunset of the scientific epoch and the rise of victorious Skepsis, the clouds dissolve and the quiet landscape of the morning reappears in all distinctness.
The final issue to which the Faustian wisdom tends—though it is only in the highest moments that it has seen it—is the dissolution of all knowledge into a vast system of morphological relationships. Dynamics and Analysis are in respect of meaning, form-language and substance, identical with Romanesque ornament, Gothic cathedrals, Christian-German dogma and the dynastic state. One and the same world-feeling speaks in all of them. They were born with, and they aged with, the Faustian Culture, and they present that Culture in the world of day and space as a historical drama. The uniting of the several scientific aspects into one will bear all the marks of the great art of counterpoint. An infinitesimal music of the boundless world-space—that is the deep unresting longing of this soul, as the orderly statuesque and Euclidean Cosmos was the satisfaction of the Classical. That—formulated by a logical necessity of Faustian reason as a dynamic-imperative causality, then developed into a dictatorial, hard-working, world-transforming science—is the grand legacy of the Faustian soul to the souls of Cultures yet to be, a bequest of immensely transcendent forms that the heirs will possibly ignore. And then, weary after its striving, the Western science returns to its spiritual home.
[1]. Kant’s error, an error of very wide bearing which has not even yet been overcome, was first of all in bringing the outer and inner Man into relation with the ideas of space and time by pure scheme, though the meanings of these are numerous and, above all, not unalterable; and secondly in allying arithmetic with the one and geometry with the other in an utterly mistaken way. It is not between arithmetic and geometry—we must here anticipate a little—but between chronological and mathematical number that there is fundamental opposition. Arithmetic and geometry are both spatial mathematics and in their higher regions they are no longer separable. Time-reckoning, of which the plain man is capable of a perfectly clear understanding through his senses, answers the question “When,” not “What” or “How Many.”
[2]. One cannot but be sensible how little depth and power of abstraction has been associated with the treatment of, say, the Renaissance or the Great Migrations, as compared with what is obviously required for the theory of functions and theoretical optics. Judged by the standards of the physicist and the mathematician, the historian becomes careless as soon as he has assembled and ordered his material and passes on to interpretation.
[3]. In the original, these fundamental antitheses are expressed simply by means of werden and sein. Exact renderings are therefore impossible in English.—Tr.
[4]. The attempts of the Greeks to frame something like a calendar or a chronology after the Egyptian fashion, besides being very belated indeed, were of extreme naïveté. The Olympiad reckoning is not an era in the sense of, say, the Christian chronology, and is, moreover, a late and purely literary expedient, without popular currency. The people, in fact, had no general need of a numeration wherewith to date the experiences of their grandfathers and great-grandfathers, though a few learned persons might be interested in the calendar question. We are not here concerned with the soundness or unsoundness of a calendar, but with its currency, with the question of whether men regulated their lives by it or not; but, incidentally, even the list of Olympian victors before 500 is quite as much of an invention as the lists of earlier Athenian archons or Roman consuls. Of the colonizations, we possess not one single authentic date (E. Meyer. Gesch. d. Alt. II, 442. Beloch. Griech. Gesch. I, 2, 219) “in Greece before the fifth century, no one ever thought of noting or reporting historical events.” (Beloch. I, 1, 125). We possess an inscription which sets forth a treaty between Elis and Heraea which “was to be valid for a hundred years from this year.” What “this year” was, is however not indicated. After a few years no one would have known how long the treaty had still to run. Evidently this was a point that no one had taken into account at the time—indeed, the very “men of the moment” who drew up the document, probably themselves soon forgot. Such was the childlike, fairy-story character of the Classical presentation of history that any ordered dating of the events of, say, the Trojan War (which occupies in their series the same position as the Crusades in ours) would have been felt as a sheer solecism.
Equally backward was the geographical science of the Classical world as compared with that of the Egyptians and the Babylonians. E. Meyer (Gesch. d. Alt. II, 102) shows how the Greeks’ knowledge of the form of Africa degenerated from Herodotus (who followed Persian authorities) to Aristotle. The same is true of the Romans as the heirs of the Carthaginians; they first repeated the information of their alien forerunners and then slowly forgot it.