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One result of this Classicizing tendency has been to prevent us from finding the new notation proper to our Western number as such. The present-day sign-language of mathematics perverts its real content. It is principally owing to that tendency that the belief in numbers as magnitudes still rules to-day even amongst mathematicians, for is it not the base of all our written notation?
But it is not the separate signs (e.g., χ, π, ς) serving to express the functions but the function itself as unit, as element, the variable relation no longer capable of being optically defined, that constitutes the new number; and this new number should have demanded a new notation built up with entire disregard of Classical influences. Consider the difference between two equations (if the same word can be used of two such dissimilar things) such as 3x + 4x = 5x and xn + yn = zn (the equation of Fermat’s theorem). The first consists of several Classical numbers—i.e., magnitudes—but the second is one number of a different sort, veiled by being written down according to Euclidean-Archimedean tradition in the identical form of the first. In the first case, the sign = establishes a rigid connexion between definite and tangible magnitudes, but in the second it states that within a domain of variable images there exists a relation such that from certain alterations certain other alterations necessarily follow. The first equation has as its aim the specification by measurement of a concrete magnitude, viz., a “result,” while the second has, in general, no result but is simply the picture and sign of a relation which for n>2 (this is the famous Fermat problem[[66]]) can probably be shown to exclude integers. A Greek mathematician would have found it quite impossible to understand the purport of an operation like this, which was not meant to be “worked out.”
As applied to the letters in Fermat’s equation, the notion of the unknown is completely misleading. In the first equation x is a magnitude, defined and measurable, which it is our business to compute. In the second, the word “defined” has no meaning at all for x, y, z, n, and consequently we do not attempt to compute their “values.” Hence they are not numbers at all in the plastic sense but signs representing a connexion that is destitute of the hallmarks of magnitude, shape and unique meaning, an infinity of possible positions of like character, an ensemble unified and so attaining existence as a number. The whole equation, though written in our unfortunate notation as a plurality of terms, is actually one single number, x, y, z being no more numbers than + and = are.
In fact, directly the essentially anti-Hellenic idea of the irrationals is introduced, the foundations of the idea of number as concrete and definite collapse. Thenceforward, the series of such numbers is no longer a visible row of increasing, discrete, numbers capable of plastic embodiment but a unidimensional continuum in which each “cut” (in Dedekind’s sense) represents a number. Such a number is already difficult to reconcile with Classical number, for the Classical mathematic knows only one number between 1 and 3, whereas for the Western the totality of such numbers is an infinite aggregate. But when we introduce further the imaginary (√-1 or i) and finally the complex numbers (general form a + bi), the linear continuum is broadened into the highly transcendent form of a number-body, i.e., the content of an aggregate of homogeneous elements in which a “cut” now stands for a number-surface containing an infinite aggregate of numbers of a lower “potency” (for instance, all the real numbers), and there remains not a trace of number in the Classical and popular sense. These number-surfaces, which since Cauchy and Riemann have played an important part in the theory of functions, are pure thought-pictures. Even positive irrational number (e.g., √2) could be conceived in a sort of negative fashion by Classical minds; they had, in fact, enough idea of it to ban it as ἄῤῥητος and ἄλογος. But expressions of the form x + yi lie beyond every possibility of comprehension by Classical thought, whereas it is on the extension of the mathematical laws over the whole region of the complex numbers, within which these laws remain operative, that we have built up the function theory which has at last exhibited the Western mathematic in all purity and unity. Not until that point was reached could this mathematic be unreservedly brought to bear in the parallel sphere of our dynamic Western physics; for the Classical mathematic was fitted precisely to its own stereometric world of individual objects and to static mechanics as developed from Leucippus to Archimedes.
The brilliant period of the Baroque mathematic—the counterpart of the Ionian—lies substantially in the 18th Century and extends from the decisive discoveries of Newton and Leibniz through Euler, Lagrange, Laplace and D’Alembert to Gauss. Once this immense creation found wings, its rise was miraculous. Men hardly dared believe their senses. The age of refined scepticism witnessed the emergence of one seemingly impossible truth after another.[[67]] Regarding the theory of the differential coefficient, D’Alembert had to say: “Go forward, and faith will come to you.” Logic itself seemed to raise objections and to prove foundations fallacious. But the goal was reached.
This century was a very carnival of abstract and immaterial thinking, in which the great masters of analysis and, with them, Bach, Gluck, Haydn and Mozart—a small group of rare and deep intellects—revelled in the most refined discoveries and speculations, from which Goethe and Kant remained aloof; and in point of content it is exactly paralleled by the ripest century of the Ionic, the century of Eudoxus and Archytas (440-350) and, we may add, of Phidias, Polycletus, Alcamenes and the Acropolis buildings—in which the form-world of Classical mathematic and sculpture displayed the whole fullness of its possibilities, and so ended.
And now for the first time it is possible to comprehend in full the elemental opposition of the Classical and the Western souls. In the whole panorama of history, innumerable and intense as historical relations are, we find no two things so fundamentally alien to one another as these. And it is because extremes meet—because it may be there is some deep common origin behind their divergence—that we find in the Western Faustian soul this yearning effort towards the Apollinian ideal, the only alien ideal which we have loved and, for its power of intensely living in the pure sensuous present, have envied.