Friday, July 25, 2008

The Meaning of Numbers

UPDATE: Simul-post on UOJ Group.

Some not-so-light reading for your weekend - math as an expression of cultural worldview, both Jewish and Western, and how those world-views brought us to our current situation in the world of high finance.

Decline of the West
Oswald Spengler
Chapter One Excerpts

There are two ways of regarding the entirety of knowledge...One is the world as history, the other the world as nature (these words being used in an unusual sense). It is necessary to study the language of a Culture.

The West has developed, especially concerning economics, accounting and business mathematics, a language of obfuscation. We are seeing the results of that language, which intended to hide the true meanings of words and numbers, on the front pages every day now. It is a language of dishonesty.

A soul seeks to actualize itself in the picture of its outer world. Mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind.

It is a true art.

Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. Number is they symbol of causal necessity. ...The existence of numbers may therefore be called a mystery, and [even] the religious thought of every Culture has felt their impress.

The real secret of all things-become, which are ipso facto things extended (spatially and materially), is embodied in mathematical number as contrasted with chronological number.

The figure, formula, sign, diagram - in short, the number-sign in which one 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.

Ultra Orthodox Judaism fears, with a great fear, the expansive and unknowable millions of years that science has demonstrated the universe has undergone. The ten thousand years of recorded human history is bad enough, but humanity in it's current form is about 60,000 years old (as I learned recently watching a show about "scientific adam" on the history channel, using DNA, mutations and telemers to trace humanity back to a common male ancestor). Mankind as a distinct species is around 100,000 years old, or more. And our own sun is 5 Billion years old, if I recall correctly.

The deep-seated need of Ultra-Orthodox Judaism is first to know and second to control our relationship with Hashem. Numbers on that scale cannot be known, and the events on such a scale cannot be controlled by them. They fear scientific knowledge and scientific numbers because it reveals them for what they are, mere children with a mere child's understanding of Hashem and His Universe.

They insist that all of time be compressed into small numbers, numbers they can apprehend, because it gives them control: control of history, control of knowledge, control of Hashem. In the last couple of decades a fundamentalist insistence upon a "young earth" reflect not anyone's ancient understanding of time or or the scripture, but the current psychological and Cultural need for control of everything. It is also this need for control that has spawned the daily additions and stringencies required in Chereidi communities by the Ravs. Making people change or restrict how they do something is the ultimate manifestation of control - the ultimate rejection of big wide numbers and big wide concepts and big wide tolerances. Their numbers are narrow because their thinking is narrow. And the shtelt based Culture that they have created for themselves allows no other thinking.

The origin of numbers resembles that of the myth. It is by means of names and numbers that the human understanding obtains power over the world.

Nature is the numerable, while History, on the other hand, is the aggregate of that which has no relation to mathematics...exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach.

The mathematical vision and thought that a Culture possesses within itself is as inadequately represented by its written mathematic as its philosophical vision and thought by its philosophical treatises. Number springs from a source that has also quite other outlets. Gothic cathedrals and Doric temples are mathematics in stone.

The great arts are, one and all, modes of interpretation by means of limits based on number.

The world of numbers includes something more than the science thereof.

The mathematic of the Classical soul sprouted almost out of nothingness, the historically constituted Western soul, already possessing the Classical science (not inwardly, but outwardly as a thing learnt), had to win its own by apparently altering and perfecting, but in reality destroying the essentially alien Euclidean system.

The relationship between the form-language of a mathematic and that of the cognate major arts, is in this way put beyond doubt. The temperament of the thinker and that of the artist differ widely indeed, but the expression-methods of the waking consciousness are inwardly the same for each.

The mathematician is only complete insofar as he feels within himself the beauty of the true. Here we feel how nearly the secret of number is related to the secret of artistic creation. Mathematics, then, are an art.

And like all art therefore expresses the inner needs and philosophy of the one so using them. Art itself is interesting in Judaism - "acceptable" art is comprised almost entirely of letters and numbers in some communities - landscapes being considered an old and outdated artform and paintings of people are limited to portraits of old, tallit wearing men. An occasional heavily veiled or faceless woman lighting shabbat candles is permitted. How profoundly these reflected on the shtetl culture are never noticed by most people.

Interestingly, art done outside these communities, done by Reform or Conservative or Secular or Unaffiliated Jews, tends to lean towards the beautifully abstract - a clear rebellion against the confinement and stagnation of the shtetl mentality.

Numbers (accounting, money, economics) are reflected in the shtetl mentality by the fact that they are used - in ways no different from that of the false Western monetary language of obfuscation and deceit - to serve the interests of the shtetl at the expense of the "outside world."

Number is the essence of all things perceptible to the senses.

The worked stone is only a something insofar as it has considered limits and measured form. What it is is what it has become under the sculptor's chisel. Apart from this it is a chaos, something not yet actualized, in fact for the time being a null. The same feeling transferred to the grander stage produces, as an opposite to the state of chaos, that of cosmos, which for the Classical soul implies a cleared-up situation of the external world, a harmonic order which includes each separate thing as a well-defined, comprehensible and present entity.

Anaximander's "apeiron" [note: "indefinate"] is that which possesses no "number" in the Pythagorean sense of the word, no measurable dimentions or definable limits, and therefore no being - the measureless, the negation of form.

Anaximander's fragmentary manuscript is also one of the earliest known surviving writings of science. Ultra Orthodoxy rejects science because they reject the idea that they don't know everything.

The "archi" [note: beginning, origin, outset, prime, principle, start, threshold] [is] optically boundless and formless, which only becomes a something (namely, the world) after being split up by the senses. [Euclid] defines a line as "length without breadth." In our mouths such a definition would be pitiful - in Classical mathematic it was brilliant.

On this account, the idea of irrational numbers - the unending decimal fractions of our notation - was unrealizable within the Greek spirit. ...In fact, it is the idea of irrational numbers that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (pi for example) can never be defined or exactly represented by any straight line.

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 forever." It is the deep metaphysical fear that the sense-comprehensible and present in which the Classical existence had entrenched itself would collapse and precipitate its cosmos (largely created and sustained by art) into unknown primitive abysses. And to understand this far is to understand the final significance of Classical number - that is, measure in contrast to the immeasurable - and to grasp the high ethical significance of its limitation.

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 anumber, 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.

"Nirvana" is not "heaven," it is literally nothingness. The Hindu and Buddhist religions have never had a "heaven" as their goal - only complete absorption into a final "echad" which causes them to cease to exist as individuals, an "echad" which in fact absorbs everything, so that there is nothing left but nothingness. Therefore the concept of "zero" was natural to their thought and their Culture.

Every product of the waking consciousness of the Classical world, then, is elevated to the rank of actuality by way of sculptural definition. That which cannot be drawn is not "number."

The "waking consciousness" of Judaism is that there is no such thing as "nothing" - even if the whole universe was destroyed there would not be "nothing," there would be Hashem. Hashem is therefore quantifiable against the background of what He is not. The whole philosophical mindset still swirls back to being able to know, and therefore control. "My children have defeated me," Hashem cried - deifying themselves. But they do not know everything, not even all their own history, and they certainly do not know Hashem. Their childish need for control drives them farther away from Hashem - every stringency is a rejection of Hashem in its own way. His Torah was inadequate, they needed to fence it and improve on it. They are the real righteous and knowledgeable beings in the universe - not Hashem. Don't believe me, just ask them.

Clearly, out of this no conception of zero as a number could possibly come, for from the point of view of a draughtsman it is meaningless.

And what draughtsmen they are, drawing fences around everything!

We, having minds differently constituted, must not argue from our habits to theirs and treat their mathematic as a "first stage" in the development of "Mathematics." Within and for the purposes of the world that Classical man evolved for himself, the Classical mathematic was a complete thing - it is merely not so for us.

Numbers are images of the perfectly desensualized understanding, of pure thought, and contain their abstract validity within themselves. Their exact application ot the actuality of conscious experience is therefore a problem in itself - a problem which is always being posed anew and never solved - and the congruence of mathematical system with empirical observation is a present anything but self-evident.

Numbers are the symbols of the mortal. Set forms are the negation of life, formulae and laws spread rigidity over the face of all nature - numbers make dead.

The Classical arithmetic, we are always told, was first liberated from all its sense-bondage, widened and extended by Diophantus (CE 25), who did not indeed create algebra (the science of undefined magnitudes) but brought it to expression within the framework of the Classical mathematic that we know - and so suddenly that we have to assume that there was a pre-existent stock of ideas which he worked out. But this amounts not to an enrichment of, but a complete victory over, the Classical world-feeling.

There emerges from under the surface of the Euclidian intention the new limit feeling which [Spengler designates] the "Magian." [Diaphantus] did not widen the idea of number as magnitude, but (unwittingly) eliminated it.

And this Arabian indeterminateness of number is, in its turn, something quite different from the controlled variability of later Western mathematics, the variability of the "function."

In place of the sensuous element of concrete lines and planes - the specific character of the Classical feeling of bounds - there emerged the abstract, spatial, un-Classical element of the point which from then on was regarded as a group of co-ordered pure numbers. [ie the plane coordinate system] The idea of magnitude and of perceivable dimension derived from Classical texts and Arabian traditions was destroyed and replaced by that of variable relation-values between positions in space...and in which no one any longer dreams of actually drawing circles or working out powers.

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 toward the infinite. The symbol of the West is an idea of which no other Culture gives even a hint, the idea of "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.

As to our present predicament in the markets, this Faustian infinity has freed Western accountants to use the ideas of "formula" to remove accounting and economics from any semblance of reality - derivatives, hedges, and so on are insurance against the fall of "real" numbers, the securities and the complex formulas used to analyze stocks, bonds, money and interest are completely divorced from "real," that is, Archimedean and Euclidian mathematics that anyone can understand. They have whatever meanings the analysts want or need them to have. They are infinately malleable. "Lies, damn lies, and stastics," you have probably heard said. By distorting the language of mathematics, numbers can be made to mean anything now.

[This is] a victory over the popular and sensuous [in the academic sense of the word: meaning "able to be sensed," or "apprehended] number-feeling in us, a victory which the new mathematic had to win in order to make the new world-feeling actual.

In all history, so far, there is no second example of one Culture paying to another Culture long extinguished such a reverence and submission in matters of science as ours [Western, Faustian] paid to the classical. ...But though the wish to emulate the Classical was constantly present, every step of the attempt took us in reality further away from the imagined ideal. This history of Western knowledge is thus one of progressive emancipation from Classical thought, an emancipation never willed but enforced in the depths of the unconscious. And so the development of the new mathematic consists of a long, secret and finally victorious battle against the notion of magnitude. The present day sign-language of mathematics perverts its real content.

In fact, directly the essentially anti-Hellenic idea of the irrationals is introduced, the foundations of the idea of numbers as concrete and definite collapse.

Of course, along with this, all Western ideas of anything being concrete and definite collapsed - including but not limited to morality and ethics. Judaism has learned this language, unfortunately, not just resulting in welfare scams, insurance fraud, ponzi schemes, investment schemes, and so on, but the language of falsity and deceit permeate the Orthodox relationship with the rest of the world. It's all about obfuscating and distorting language to make sure the shtetl never "looks bad" to others - and the first casualty is truth. The incident of the Bochur and the 17 hour plane ride continues to linger in my thoughts. He gleefully watched every film on the airline's menu - forbidden films, films depicting sex and violence - because he thought no one was watching. This person has rejected concrete and definite concepts as applying to himself - and others have followed. The introduction of the false language of deception and obfuscation into Judaism has lead to hedonism and decadence which is known to everyone but denied by anyone who is asked. The language of falsity and deceit thus spreads from the top levels of our society to every nook and cranny. When the top brass lies about molestation and scams, lies are normal, accepted and to be expected. This is what the shtetl has become. It can only lead to the death of the community.

Here is the root of our eternal dread of the irrevocable, the attained, the final - our dread of mortality, of the world itself as a thing-become, where death is set as a frontier like birth - our dread of the moment when the possible is actualized, the life is inwardly fulfilled and consciousness stands at its goal.

I have seriously wondered if the Rabbis are not purposefully sabotaging the coming of Mashiach. After all, if this actuality is realized, all of their control vanishes. Hashem Himself returns, fights for us, and the Shekinah returns to the Temple - meaning all questions will now be answered by Hashem Himself, not them. Their need for control simply does not allow them to truly work toward the day when they will be stripped of all control. The power will be restored to the priesthood, of which few of them qualify. This may not be a conscious thought, but it is a deep one, so deep they may not be aware of it in themselves.

The element of direction, too, which is inherent in all "Becoming" is felt owing to its inexorable irreversibility to be something alien and hostile, and the human will-to-understanding ever seeks to bind the inscrutable by the spell of a name. It is something beyond comprehension, this transformation of future into past, and thus time, in contrast with space, has always a queer, baffling, oppressive ambiguity from which no serious man can wholly protect himself.

Like a secret melody that not every ear can perceive, it runs through the form-language of every true artwork, every inward philosophy, every important deed, and although those who can perceive it in that domain are the very few, it lies at the root of the great problems of mathematics. Only the spiritually dead man of the autumnal cities [that is, removed from rural life and sensibilities] loses it or is able to evade it by setting up a secretless "scientific worldview" between himself and the alien.

And Talmud, as it should be obvious, is a manifestation of this secretless "scientific worldview." In it all decisions are made and everything is "known."

We find that every Culture is aware (each in its own special way) of an opposition of time and space, of direction and extension, the former underlying the latter as becoming precedes having become.

The subtlest, as well as the most powerful, form of this defense is causal and systematic knowledge, delimitation by label and number. The world-fear is stilled when an intellectual form-language hammers out brazen vessels in which the mysterious is captured and made comprehensible. The method common to all the great Cultures - the only way of actualizing itself that the soul knows - is the symbolizing of extension.

Our universe of infinite space, whose existence, for us, goes without saying, simply does not exist for the Classical man. It is not even capable of being presented to him. The fact is that the infinite space of our physics is a form of very numerous and extremely complicated elements tacitly assumed, which have come into being only as the copy and expression of our soul, and are actual, necessary and natural only for our type of waking life. The simple notions are always the most difficult. The whole of our mathematics from Descartes onward is devoted to the theoretical interpretation of this great and wholly religious symbol. In the Classical mathematics and physics the content of this word is simply not known.

Here, too, Classical names...have veiled the realities. The mathematics of the West has long ceased to have anything to do with both these forms of defining, but it has not managed to find new names for its own elements - for the word "analysis" is hopelessly inadequate.

Of course, by using the "old words" with an unacknowledged but obviously present "new" interpretation or meaning, we get to pretend that things are as they have always been. Again, a deception wherein we deceive ourselves as well as try and deceive others.

Thus, inevitably, the Classical became by degrees the Culture of the small.

Just as the Greek tongue - again and again we shall note the mighty symbolism of such language phenomena - possessed no word for space, so the Greek himself was destitute of our [Western] feeling of landscape, horizons, outlooks, distances, clouds, and of the idea of the far-spread fatherland embracing the great nation. Classical geometry...concerned itself with the small...the difficulties that arise in establishing figures of astronomical dimensions, which in many cases are not amenable to Euclidean geometry.

All the mathematical ideas that the West found for itself or borrowed from others were automatically subjected to the form-language of the Infinitesimal - and that long before the actual Differential Calculus was discovered. Arabian algebra, Indian trigonometry, Classical mechanics were incorporated as a matter of course in analysis.

The nexus of magnitudes is called "proportion," that of relations is comprised in the notion of "function." The significance of these two words is not confined to mathematics proper.

As we have seen, the form language we employ reflects our Culture and in a feed-back loop extraordinaire helps define it. This applies both to Western and Jewish culture - and Jewish Culture has borrowed heavily from Western, too. The language of obfuscation and deceit has become ours.

The alpha and omega of the Classical mathematic is construction (which in the broad sense includes elementary arithmetic), that is, the production of a single visually present figure.

For the last two centuries - though present day mathemeticians hardly realize the fact - there has been a growing up in the idea of a general morphology of mathematical operations, which we are justified in regarding as the real meaning of modern mathematics as a whole. All this, as we shall perceive more and more clearly, is one of the manifestations of a general tendency inherent in Western Intellect, proper to the Faustian spirit and Culture and found in no other. The great majority of the problems which occupy our mathematic and are regarded as "our" problems...would probably have seemed to the Ancients, who strove for simple and definite quantitative results, to be an exhibition of rather abstruse virtuosity. And so indeed the popular mind regards them even today.

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. 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. 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 [itself], the process, the operation [itself]. It is not a state, but a relation.

This was the grand course of Western number-thought.

And it's how we arrived to a day where financial statements have no real meaning, money is a thing of air and relativity which is based on nothing at all [a "means of exchange, completely infinite, instead of representing a finite amount of real wealth], stock prices are divorced from any underlying reality of the value of a company's assets - and indeed, the whole idea of "value" is now relative and subject to the whim of those stating said "value."

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 tot he unreserved expression of inward possibilities - in other words, the point at which the idea of transcendent extension came into fundamental conflict with the limitations of immediate perception.

It is pure abstract, and ideal and unfulfillable postulate of a soul which is ever less and less satisfied with sensuous [in the academic sense of the word] means of expression and in the end passionately brushes them aside. The inner eye has awakened.

The critical proposition of this geometry, Euclid's axiom of parallels, is as assertion, for which we are quite at liberty to substitute another assertion.

Even the simple axiom that extension is boundless (boundlessness, since Riemann and the theory of curved space, is to be distinguished from endlessness) at once contradicts the essential character of all immediate perception, in that the latter depends upon the existence of light-resistances and ipso facto has material bounds.

Although it was not till about 1800 that the notion of multi-dimensional space (it is a pity that no better word was found) provided analysis with broader foundations, the real first step was taken at the moment when powers - that is, really logarithms - were realized from their original relation with sensually realizable surfaces and solids and, through the employment of irrational and complex exponents, brought within the realm of function as perfectly general relation values.

Once the space-element or point has lost its last persistent relic of visualness and instead of being represented to the eye as a cut in co-ordinate lines, was defined as a group of three independent numbers, there was no longer any inherent objection to replacing the number 3 by the general number "n." The notion of dimension was radically changed.

And the aggregate of all points of n dimensions is called an n-dimensional space. In these transcendent space-worlds, which are remote from every sort of sensualism [that is, apprehension], lie the relations which it is the business of analysis to investigate and which are found to be consistently in agreement with the data of experimental physics. This space of higher degree is a symbol which is through and through the peculiar property of the Western mind.

And here if anywhere it must be understood that actuality is not only sensual actuality. The spiritual is in no wise limited to perception-forms for the actualizing of its idea.

But the Rabbis have laboured hard to make it so. "The Torah is not in a voice from Heaven..." exept that's exactly where it came from. But they can't control voices from Heaven, so a voice from Heaven, Hashem's voice and thus Hashem Himself, must be removed from authority over them.

From this grand intuition of symbolic space-worlds came the last and conclusive creation of Western Mathematic - the expansion and subtilizing of the functoon theory in that of groups. ...and with this culmination of our Western mathematic, having exhausted every inward possibility and fulfilled its destiny as the copy and purest expression of the idea of the Faustian soul, closes its development in the same way as the mathematic of the Classical culture concluded in the third century.

Both, expanding in all beauty, reached their maturity one hundred years later, and both, after flourishing for three centuries, complete the structure of their dieas at the same moment as the Cultures to which they respectively belonged passed over into the phase of megalopolitan Civilization.

[Thus] the time of great mathemeticians is past. Our tasks today are those of preserving, rounding off, refining, selection - in place of big dynamic creation, the same clever detail-work which characterized the Alexandrian mathematic of late Hellenism.

The "preserving, rounding off, refining and selection" of Halacha has reached absurd heights - imposing stringency after stringency - all the while failing to recognize that doing so is the dying gasp of the Civililzation we built on the shtetl mentality and the need for control.

Next: Chapter Two: The problem of world history.

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