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Spengler’s Meaning of Numbers

I have sometimes talked about numbers as rhetorical devices. I’ve also talked about different concepts and isomorphs (representations of the same “thing”) affording different actions.

Simulation Gap: Discrete (hidden) models in games/software attempt to communicate nuanced experiential feedback to players/users.

Reducing to Numbers: Numbers are rhetorical devices that can take subtle interactions and make them explicit and discrete.

Playing Everything:

Computers are different than people. There are no fictions, there is no “spirit of the law“. There is only the law.

I didn’t think that there was much else to say on the topic beyond that. Numbers-as-rhetoric is a pretty cut-and-dry topic.

As I started looking into Spengler, I discovered that the first chapter of Decline of the West was titled “The Meaning of Numbers.”

I let Spengler do a lot of the talking here although I try to summarize and condense. He wrote about numbers quite a bit. 

 

The Meaning of Numbers

Mathematics, once thought to be the one universal field of knowledge, actually mean wildly different things to the different cultures. Each culture tends to fixate on a particular property or form in the mathematical world that mirrors its aesthetic, its big constraining idea that it will build upon until it expends all of its ideas as a large, baroque, soulless civilization that slowly declines back into barbarism. As with numbers, there is no universal truth of mankind- historical meaning is only really found from within a culture.

Quick link as a refresher: Spengler’s civilization model on Wikipedia.

The essence of Spengler’s vision, remember, is that cultures have lifespans and common growth patterns, and each one iterates on a singular essence that guides them from a simple culture into a high culture and eventually into a civilization before they decline.

 

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 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. … The style of any mathematic which comes into being, then, depends wholly on the Culture in which its 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.” [pdf]

 

Classical culture focused on the number as a measure, developed geometric/arithmetic thought. They lacked or were otherwise repulsed by concepts of infinity, irrational numbers, complex numbers, or limiting processes. Towards this end, they built a science and aesthetics around concepts like matter and form. Their architecture began as a realization of this view of the world. Their development as a culture was a fight against these constraints in how they saw the world. They matured through the creative work of Plato, Archytas (mathematical mechanics), and Eudoxis’ conic sections. The classical passion for this particular view of mathematics was never rekindled as the culture naturally decayed, but some of their knowledge survived them to be re-appropriated by others, in the same way (Spengler says) that Indian culture developed the concept of zero in it’s cultural autumn: that excellent idea survived them even though the sacredness associated with zero didn’t. As the limitations of their sacred view of mathematics were reached, they fought through wild byzantine attempts to dig deeper than their notation and conceptual toolkit found easy to deal with effectively. The final, concluding thoughts at the end of classical culture regarding mathematics were capped by Euclid, Apollonius, and Archimedes.

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 physical bodies, Western mathematics to describe infinite space.

 

“Magian” (Arabian) culture focused on the number as a variable, developing algebra. They escaped the Classical path (although they understood geometry, it had no sacred value to them- algebra did). In lieu of forms and matter, their study developed around chemistry (or alchemy and proto-chemistry, let’s say).  Arabian mathematical thought peaked with number theory and spherical trigonometry, concluding its thought in 10th century algebraics, astronomers, and engineers.

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 starting point) 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 [Magian] algebra is to Classical arithmetic and Western analysis, so is the cupola-church to the Doric temple and the Gothic cathedral.

Briefly, on the blending of the cultures:

[The Magian] used deeply thought-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.

As a note: if it seems like Spengler is repeating himself, it’s because he very much is. His perspective, methodology, and truth claims were often extremely alien to his audience, and his book is occasionally written in polemic tone. He is meaning to convince, repeating his conclusions over and over and constructing different arguments that coincide on the same points.

 

“Faustian” (Western) culture, birthed near the end of the first millenium, focused on the number as a function, developing calculus: Descartes, Pascal, Fermat, Newton and Leibnitz. The creative peak for Western culture regarding mathematics, to Spengler, is found in Euler, Lagrange, and Laplace’s infinitesimal problem. From there, creativity dies but profusion increases within the fields that have been defined. Western mathematical thought was capped in the 19th century- Gauss, Cauchy, and Riemann. Dynamics are to Western culture as Statics were to the classical culture and Alchemy was the Arabic culture.

The idea of Euclidean geometry is actualized in the earliest forms of Classical ornament, and that of infinitesimal calculus in the earliest forms of Gothic architecture, centuries before the first learned mathematicians of the respective Cultures were born

[…].

The modern mathematic, though ‘true’ only for the Western spirit, is undeniably a master-work of that spirit; and yet to Plato it would have seemed a ridiculous and painful aberration from the path leading to the ‘true’ –to wit the Classical — mathematic.

 

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.

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