THE MEANING OF NUMBERS

THE DECLINE OF THE WEST BY OSWALD SPENGLER VOLUME ONE FORM AND ACTUALITY pp 53-90.

Quote from: Spengler vol I p.53

The popular distinction current also in philosophy between "being" and "becoming" seems to miss the essential point in the contrast it is meant to express. An endless becoming "action," "actuality" will always be thought of also as a condition (as it is, for example, in physical notions such as uniform velocity and the condition of motion, and in the basic hypothesis of the kinetic theory of gases) and therefore ranked in the category of "being." On the other hand, out of the results that we do in fact obtain by and in consciousness, we may, with Goethe, distinguish as final elements "becoming" and "the become" (Das Werden, das Gewordne). In all cases, though the atom of humanness may lie beyond the grasp of our powers of abstract conception, the very clear and definite feeling of this contrast fundamental and diffused throughout consciousness is the most elemental something that we reach. It necessarily follows therefore that "the become" is always founded on a "becoming" and not the other way round. Throughout the book Spengler compares the Western/Faustian, the Classic/Appolinian and the Arabic/Magian cultures because these are the only Cultures that we have
enough historical knowledge about. Mathematics is not limited to scientific mathematics and it can as well be expressed in architecture.

Quote from: Spengler vol I p.56-58

In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world to show, that is, in how far Culture in the "become" state can express or portray an idea of human existence I have chosen number, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. [...] Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as-nature. [...] But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the number-sign which he thinks, speaks or writes exactly, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the "alien," in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. [...] Nature is the numerable, while History, on the other hand, is the aggregate of that which has no relation to mathematics hence the mathematical certainty of the laws of Nature, the astounding Tightness of Galileo's saying that Nature is "written in mathematical language," and the fact, emphasized by Kant, that exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach.Numbers as representatives of different Cultures.

Quote from: Spengler vol I p.59

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. [...] 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.Instruments as representatives of different souls/Cultures.

Quote from: Spengler vol I p.62

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 tonecolours 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 sface-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. The Classical/Appolinian world is a world of visable physical bodies and that is reflected in Classical mathematics.

Quote from: Spengler vol I p.64-65

Classical number is a thought-process dealing not with spatial relations but with visibly limitable and tangible units, and it follows naturally and necessarily that the Classical knows only the "natural" (positive and whole) numbers, which on the contrary olay in our Western mathematics a quite undistinguished part in the midst of complex, hypercomplex, non-Archimedean and other number-systems. On this account, the idea of irrational numbers the unending decimal fractions of our notation was unrealizable within the Greek spirit. Euclid says and he ought to have been better understood that incommensurable lines are "not related to one another like numbers." In fact, it is the idea of irrational number that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (π, for example) can never be defined or exactly represented by any straight line. Moreover, it follows from this that in considering the relation, say, between diagonal and side in a square the Greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the Classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. There is a singular and significant late-Greek legend, according to which the man who first published the hidden mystery of the irrational perished by shipwreck, "for the unspeakable and the formless must be left hidden for ever."

Quote from: Spengler vol I p.66

For the transformation of a series of discrete numbers into a continuum challenged not merely the Classical notion of number but the Classical world-idea itself, and so it is understandable that even negative numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone zero as a number, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. Negative magnitudes have no existence.Classical mathematics is a way to measure physicas bodies, Western mathematics to describe infinite space.

Quote from: Spengler vol I p.69

They used deeplythought-out (and for us hardly understandable) methods of integration, but these possess only a superficial resemblance even to Leibniz's definite-integral method. They employed geometrical loci and co-ordinates, but these are always specified lengths and units of measurement and never, as in Fermat and above all in Descartes, unspecified spatial relations, values of points in terms of their positions in space.Late Greek and Roman mathematician were not part of the Classical culture but the first representatives of the new Arabic/Magian Culture. In mathematics and architecture Western culture is represented by analysis and the Gothic cathedral, Classic culture by geometry and the Doric temple and the Arabic culture by algebra and the cupola-church.

Quote from: Spengler vol I p.72-73

In Diophantus, unconscious though he may be of his own essential antagonism to the Classical foundations on which he attempted to build, there emerges from under the surface of Euclidean intention the new limit-feeling which I designate the "Magian." He did not widen the idea of number as magnitude, but (unwittingly) eliminated it. No Greek could have stated anything about an undefined number a or an undenominated number which are neither magnitudes nor lines whereas the new limit-feeling sensibly expressed by numbers of this sort at least underlay, if it did not constitute, Diophantine treatment. [...] The Classical spirituality, which reached its final phase in the cold intelligence of the Romans and of which the whole Classical Culture with all its works, thoughts, deeds and ruins forms the "body," had been born about 1100 B.C. in the country about the Aegean Sea. The Arabian Culture, which, under cover of the Classical Civilization, had been germinating in the East since Augustus, came wholly out of the region between Armenia and Southern Arabia, Alexandria and Ctesiphon, and we have to consider as expressions of this new soul almost the whole "late-Classical" art of the Empire, all the young ardent religions of the East Mandaeanism, Manichaeism, Christianity, Neo-Platonism, and in Rome itself, as well as the Imperial Fora, that Pantheon which is the first of all mosques. That Alexandria and Antioch still wrote in Greek and imagined that they were thinking in Greek is a fact of no more importance than the facts that Latin was the scientific language of the West right up to the time of Kant and that Charlemagne "renewed" the Roman Empire. In Diophantus, number has ceased to be the measure and essence of plastic things. [...] Diophantus does not yet know zero and negative numbers, it is true, but he has ceased to know Pythagorean numbers. And this Arabian indeterminateness of number is, in its turn, something quite different from the controlled variability of the later Western mathematics, the variability of the function. The Magian mathematic we can see the outline, though we are ignorant of the details advanced through Diophantus (who is obviously not a startingpoint) boldly and logically to a culmination in the Abbassid period (9th century) that we can appreciate in Al-Khwarizmi and Alsidzshi. And as Euclidean geometry is to Attic statuary (the same expression-form in a different medium) and the analysis of space to polyphonic music, so this algebra is to the Magian art with its mosaic, its arabesque (which the Sassanid Empire and later Byzantium produced with an ever-increasing profusion and luxury of tangible-intangible organic motives) and its Constantinian high-relief in which uncertain deep-darks divide the freely-handled figures of the foreground. As algebra is to Classical arithmetic and Western analysis, so is the cupola-church to the Doric temple and the Gothic cathedral. Descartes as a purely western mathematician.

Quote from: Spengler vol I p.74

The decisive act of Descartes, whose geometry appeared in 1637, consisted not in the introduction of a new method or idea in the domain of traditional geometry (as we are so frequently told), but in the definitive conception of a new number-idea, which conception was expressed in the emancipation of geometry from servitude to optically-realizable constructions and to measured and measurable lines generally. With that, the analysis of the infinite became a fact. [...] The word "geometry" has an inextensible Apollinian meaning, and from the time of Descartes what is called the "new geometry" is made up in part of synthetic work upon the position of points in a space which is no longer necessarily three-dimensional (a "manifold of points"), and in part of analysis, in which numbers are defined through point-positions in space. And this replacement of lengths by positions carries with it a purely spatial, and no longer a material, conception of extension.The function as the central concept in western mathematics. (Important to understand the role of credit as money in our Western Culture.)

Quote from: Spengler vol I p.75

At the moment exactly corresponding to that at which (c. 540) the Classical Soul in the person of Pythagoras discovered its own proper Apollinian number, the measurable magnitude, the Western soul in the persons of Descartes and his generation (Pascal, Fermat, Desargues) discovered a notion of number that was the child of a passionate Faustian tendency towards the infinite. Number as pure magnitude inherent in the material presentness of things is paralleled by numbers as pure relation, 1 and if we may characterize the Classical "world," the cosmos, as being based on a deep need of visible limits and composed accordingly as a sum of material things, so we may say that our world-picture is an actualizing of an infinite space in which things visible appear very nearly as realities of a lower order, limited in the presence of the illimitable. The symbol of the West is an idea of which no other Culture gives even a hint, the idea of Function. The function is anything rather than an expansion of, it is complete emancipation from, any pre-existent idea of number. With the function, not only the Euclidean geometry (and with it the common human geometry of children and laymen, based on everyday experience) but also the Archimedean arithmetic, ceased to have any value for the really significant mathematic of Western Europe. Henceforward, this consisted solely in abstract analysis. For Classical man geometry and arithmetic were self-contained and complete sciences of the highest rank, both phenomenal and both concerned with magnitudes that could be drawn or numbered. For us, on the contrary, those things are only practical auxiliaries of daily life. Addition and multiplication, the two Classical methods of reckoning magnitudes, have, like their sister geometrical-drawing, utterly vanished in the infinity of functional processes.Limits as the logically secure infinitesimal calculus.

Quote from: Spengler vol I p.86

Thus, finally, the whole content of Western number-thought centres itself upon the historic limit-problem of the Faustian mathematic, the key which opens the way to the Infinite, that Faustian infinite which is so different from the infinity of Arabian and Indian world-ideas. [...] This limit is the absolute opposite of the limit which (without being so called) figures in the Classical problem of the quadrature of the circle. Right into the 18th Century, Euclidean popular prepossessions obscured the real meaning of the differential principle. The idea of infinitely small quantities lay, so to say, ready to hand, and however skilfully they were handled, there was bound to remain a trace of the Classical constancy, the semblance of magnitude, about them, though Euclid would never have known them or admitted them as such. Thus, zero is a constant, a whole number in the linear continuum between +1 and -1 ; and it was a great hindrance to Euler in his analytical researches that, like many after him, he treated the differentials as zero. Only in the 19th Century was this relic of Classical number-feeling finally removed and the Infinitesimal Calculus made logically secure by Cauchy's definitive elucidation of the limit-idea; only the intellectual step from the "infinitely small quantity" to the "lower limit of every possible finite magnitude" brought out the conception of a variable number which oscillates beneath any assignable number that is not zero. A number of this sort has ceased to possess any character of magnitude whatever: the limit, as thus finally presented by theory, is no longer that which is approximated to, but the approximation, the process, the operation itself. It is not a state, but a relation. And so in this decisive problem of our mathematic, we are suddenly made to see how historical is the constitution of the Western soul. Faustian mathematics liberated from the Classical.

Quote from: Spengler vol I p.86-87

The liberation of geometry from the visual, and of algebra from the notion of magnitude, and the union of both, beyond all elementary limitations of drawing and counting, in the great structure of function-theory this was the grand course of Western number-thought. The constant number of the Classical mathematic was dissolved into the variable. Geometry became analytical and dissolved all concrete forms, replacing the mathematical bodies from which the rigid geometrical values had been obtained, by abstract spatial relations which in the end ceased to have any application at all to sense-present phenomena. [...] Number, the boundary of things-become, was represented, not as before pictorially by a figure, but symbolically by an equation. "Geometry" altered its meaning; the co-ordinate system as a picturing disappeared and the point became an entirely abstract number-group.