The Meaning of Numbers

Posted: December 18, 2013 in Culture, History, Philosophy, Science
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Let me continue book blogging on The Decline of the West with the following caveat: I’ve struggled to read and understand Spengler’s chapter on the meaning of numbers. His writing style is often opaque and even quasi-mystical. Plus, my familiarity with the times, people, and places cited is sometimes limited. So even after traversing the chapter more than once, I feel my grasp is, well, a little feeble.

Spengler conceals his conclusion until the final paragraphs, namely, that the world-sense derived from mathematical thought, after its initial elucidation, takes approximately 300 years to exhaust itself through refinement, decay, and perhaps replacement. Although he mentions Egyptian, Arabian, Indian, and other mathematics in passing, the bulk of the chapter contrasts the Classical number of Greek antiquity with the modern number of the scientific era commencing just after the Enlightenment. The fullness of each sensibility is limited to adepts in the respective historical period, but the effect is foundational — enough in fact to define the deep culture of an era. This reminds me that in some undeveloped cultures, number sense is limited to 1, 2, a few, and many, and for that matter, how zero was not initially part of many number systems. Spengler does not discuss these facets in detail. The foundational effect also reminds me (à la Edward T. Hall) of divergences in appreciation of time and proxemics from place to place and era to era, which may be merely subsets of the numerical world-sense Spengler describes.

According to Spengler, the ancient Greek number sense is sensuous: it relates directly to surfaces (especially planar ones such as walls and still water) and bodies (forms) readily perceptible and measurable in the immediate environment. Magnitude is the principal theoretical attribute. As a result, negative numbers, most irrational numbers (pi and the square roots of two and three may be exceptions due to their presence, implied or manifest, in basic geometric forms), and exponential powers greater than three lie beyond the scope of Classical Greek thought because they are not embodied in everyday sensual experience and are too abstract.

Modern mathematics, on the other hand, elaborates a wide range of abstract concepts with function being the principal attribute. Numbers are understood primarily in relation to each other, without reference to specific embodiments. Spengler’s best example is the Cartesian coordinate system, which plots points, lines, curves, figures, bodies, etc. in absolute space.

This is essentially the difference between direct and indirect articles: the line, angle, block, or ball vs. a line, angle, block, or ball. Whereas the Greeks were content to think in terms of actuality, moderns think more readily in terms of possibility, not become but becoming. More insidiously, geometric constructs take on idealized forms that lose congruence with their inevitably fixed and flawed embodiments. Even further, numbers themselves are better understood in their abstract representations, such as x2y2 = z2 as opposed to 32 + 42 = 52. Extend the idea of abstract, disembodied forms and numbers to entire cultures and economies and you get fiat currencies that count debt (literally an absence of money, a negative number, or a deficit) as assets and cling stubbornly to the idea of infinite growth on a finite world.

Aside: These subtle sensibilities are exactly the thing I would have loved to study and learn in math class had my teachers and professors been prepared to teach math from a philosophical perspective. But alas, even the upper levels I achieved (through college calculus, all long forgotten — I can only remember algebra and geometry now) were essentially a long series of abstract transformations devoid of applicability and meaning in the world I inhabit. Had I studied engineering or physics, perhaps I would now know how higher math relates to the world in a more direct sense. Thus, I am not an adept and can only glimpse the underlying philosophy. Indeed, that’s probably true of most practitioners, too: they possess technical skills but know nothing of the Spenglerian number sense.

The introduction to The Decline of the West recommended that Spengler be read as poetics, which is curious to me because, like many philosophical works, the tendency to inflate and reify concepts of the writer’s own creation (exacerbated perhaps by the German-language capitalization of nouns) is apparent in universalization and spiritualization of content, such as in the following passage:

… 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. Whatever the guise — infinite series, curves or functions — in which number appears in the particular case, the essence of it is the theory of the limit. 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. [italics in original]

The mysteries of the infinite, known by limits or lack thereof, are thus understood differently from one era to another. According to Spengler, modern spirituality is derived from something quite opposite that of our forebears, which is an argument no one puts forward because although it is intuitively and artistically expressed in a variety of contexts, it nonetheless requires unusual historical and cross-cultural erudition to observe and describe rationally. Spengler also has an interesting albeit brief discussion of multidimensionality, or extrapolation beyond the usual three or four dimensions, which illustrates how our mindspace expands into realms unimaginable without the benefit of modern thought. Discontented physicists and cosmologists press even further into the so-called multiverse, which makes about as much sense as risible phrases like an infinity of infinities. Seriously, how can a universe, which is everything and all, be expanded into multiples of itself? Even if the theory is permitted in the abstract, it bears no relation to our reality — you know, the one that matters. No wonder idiots experts can talk economic and demographic gibberish without anyone batting an eyelash. We’re all too flummoxed by our own abstract cleverness and fail to think concretely enough about what is truly possible in three/four dimensions and one reality.

Spengler goes on as follows:

This mathematics of ours was bound in due course to reach the point at which not merely the limits of the visual itself were felt by theory and by the soul alike as limits indeed, as obstacles to the unreserved expression of inward possibilities — in other words, the point at which the ideal of transcendent extension came into fundamental conflict with the limitations of immediate perception. Mathematical, “absolute” space, we see then, is utterly un-Classical, and from the first, although mathematicians with their reverence for the Hellenic tradition did not dare to observe the fact, it was something different from the indefinite spaciousness of daily experience and customary painting, the a priori space of Kant which seemed so unambiguous and sure a concept. It is the pure abstract, an ideal and unfulfillable postulate of a soul which is ever less and less satisfied with sensuous means of expression and in the end passionately brushes them aside. The inner eye has awakened. [italics in original]

If conclusions to be drawn by this last quote aren’t altogether clear, don’t be alarmed. I’m not sure I get it fully, either. Regardless, it served as a launchpad for me to make some connections between Spengler’s sensibility and how things have developed over another 100 years of history and technological innovation. For example, many people admit to the feeling of being trapped in the sense-world of immediate perception, of being imprisoned in the body, of being tormented by their own uncontrollable emotions, and so yearning to escape, to transcend, to merge with the beyond, to no longer be in the usual ontological sense. We have stopped protesting that we are not in fact numbers (007 of the James Bond franchise long having eclipsed Number Six from The Prisoner in the cool wars). Quite an inversion of value there: disappearing into secrecy and inscrutability vs. struggling to maintain one’s self in the face of an anonymizing onslaught. We now seek refuge in sleep, in fiction, in alcohol and drugs, in gaming, and in virtual reality, all of which extinguish full awareness and consciousness. Significantly, transhumanism is positively slavering to bridge the mind/computer barrier, allowing us each to live forever as some kind of machine consciousness, because, as we all know, evolving into pure thought/light, were such a thing possible, takes too long. We’re impatient.

Baby steps on this path are taken all the time in medicine, biology, robotics, and communications technologies, all of which believe they provide support to humanity but may secretly (even to themselves) desire to supplant it precisely because technocrats don’t really know what humanity is. The more obvious steps are not technological, however, so much as behavioral. By our preoccupations and dispossessed attentions, we have ceded the integrity and autonomy of the self to consumption, to brands, to celebrities, to candidates, to movements, to media and information, to anything and everything that inauthenticates us and instead makes us into zombies. Clearly, this analysis goes well beyond what Spengler argues, but if you have read this far, you might also be aware of themes and arguments that crop up regularly on this blog. It all contributes to something eschatological in the air, a conflagration not-yet-become that has already been fanned into an imbroglio with everyone on pins and needles knowing to expect something wholly discontinuous and unprecedented. So for all the doomer talk of the powers that be (TPTB) doing their best to maintain (and intensify) business as usual (BAU), it seems we’re nonetheless primed for an eventual if overdue terminus to the decline Spengler describes.


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