Chapter II
Synopsis
I- Fundamental notation, page 53
Spengler establishes foundation vocabulary, in particular become and becoming, and the distinction between History and Nature.
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II-Numbers as the sign of delimitation, page 56
Number, as the primal element of mathematics, is science (logic) & mystical & spiritual. Things become defined by extension; number (as notation) like word, allows mechanical demarcation, primitive man can contain sense impressions, thereby controlling them. Used to form picture of the extended world: Nature. Number is elemental, NOT academic mathematics, but the style of a Soul, expressed in art, architecture, engineering & social organization.
III- Every Culture has its own Mathamatic, page 59
The style of mathematics depends on Culture, each is distinct; not actualized in academic treatises but in form such as the Doric temple or Gothic cathedral. Most of Kant’s a priori do not exist, a few universal, most determined by Culture.
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Mathematics exist for each Culture, that Soul’s expression; first articulated by Pythagoras (Classic Age), Descartes (the West); the mathematician is an artist; like tone, line & color the domain of maths is an image of the world form.
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A sequence of mathematics emerges each tied to a Culture. First Egyptian mathematics (lost), then Babylonian; Greek mathematics (Pythagoras) which is fulfilled by 2nd century BC; Arabian mathematics emerges in Alexandria in Roman Imperial period.
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Number as magnitude in the Classical world, page 64.
Classical number meant measure, magnitude, length, surface; an optical symbol, measure of extension, using only natural (positive & whole) numbers, never irrational (pi) or zero; they dreaded the immeasurable, never extended the polis outside city limits or sail beyond sight of land; their religious cults bound by local geography, never abstract ideas (like Christian Trinity); they knew only 3D space, 4th dimension impossible.
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Aristarchus’ heliocentric theory argues against Spengler’s “mathematics”; Spengler dismisses him as a Babylonian influence. Mathematics as extension of the world of phenomena is linked to death & BECOME & this connection is origin of great art: Numbers are symbols of the mortal. Number & world idea of a culture related; number idea is a view of universe & religion. Great mathematicians were religious, like Plato saw number as a window to the divine.
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VII-Diophantus and Arabian Number, page 71
Diophantus discovers Magian number & new number feeling for indeterminateness; his algebra is distant from Classical visible magnitudes. He lived during same era as early Christian sarcophagi & Tetrarchy, both Magian not Classical; indeterminateness is hallmark of Magian number & culminates in Al Khwarzimi.
VIII-Number as Function in the Western Culture, page 74
Descartes’s revolution (1637) & new Geometry frees Western number from Euclid; it does this with the point. It is no longer necessarily 3D; it is purely spatial, reflected in the periodic tables which led to infinite series; pi & Euler's constant (e) creating relations, obliterating links to visual geometry.
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Pythagoras & Descartes establish a new number idea; magnitude & relation. Relation will allow the West to actualize the Infinite via the function. The function relegates Geometry to mere auxiliary. Analysis now looks to the function, employing logarithm, using complex & negative numbers, fractions, breaking link to magnitude.
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The West retained Classic notation of magnitude, but functions do not express magnitude but an infinity of possible positions. Links to magnitude end in the 18th century, the Baroque age, advances became increasingly abstract. It is paralleled in music, Bach is analogous to the Classical age of great sculpture (Praxiteles).
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XI- World-fear and World-longing, page 78
Creativity comes from emotions, first longing, emerging with understanding of number & birth of ego, leads to direction, leading to time (irreversibility) leading to consciousness of the past (BECOME) & intellectual emotion: fear of death. This fear leads to the desire to capture mystery thru names (or scientific labels), practice of binding & conjuring, science, ritual religion & in Cultural Springtime, Beethoven’s quartets & the Pyramids.
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XII- Geometry and arithmetic, page 81
Classical & Western number incompatible: Classic number looks to the visible, single body, boundaries & surface; the West knows disembodied abstract analysis, immeasurable & infinite.
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Number reflected everywhere in Culture (not just mathematical studies), in sculpture & music. Classical number is magnitude & proportion, assuming constancy in single bodies: scaling up or down 1:1 scaling, to the very small, (e.g. the art of the gem). Western function & relation allows transformation of groups, assumes variability. It is reflected in musical composition (exemplified in Tema con Variazioni).
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Classical mathematics is construction of a single visible body using the compass, affirming the optically visible, working with small discrete & concrete individuals & leads to definitive end. . Faustian mathematics is composition (like music), which deny visibility; handles multitudes of group functions, classes, the end is not a result but a course. Since the 18th century it has focused on FORM of mathematics. Faustian problems (convergence of infinity series, transformation of the elliptic, & algebraic integration) are unpopular like Western art (Divine Comedy; Parsifal); in contrast Classical art (Homer, altar of Pergamon) is always popular.
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Faustian number came to focus on the problem of the limit (in curves, series or functions). Through the 18th century Euclid handicapped Faustian number; in the 19th century this relic is removed by having a definitive limit idea, the lower limit of every possible finite magnitude; eventually becoming a variable oscillating between any given number that was not 0; it has become a relation, a process not a result.
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XVI Visual limits transcended; symbolic space worlds, page 86
Geometry is finally free from visual, algebra from magnitude, both untied in the function. Classical constant is dissolved in the variable & geometry of bodies replaced by spatial relations, no visual content, loci replacing Euclidean figures; position replaced by arbitrary co-ordinate system of pure value. Geometry is an equation, the point an abstract number group (x, y, z). So too Baroque architecture moves from Renaissance pure line, flat surface & solids to flows & curves, groups of columns; light plays on the form world of the mature Baroque from Bernini (1650) to the Rococo. Like number, it has become musical.
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XVII Final possibilities, page 87
Faustian number outgrows Greek legacy; the will to power (Nietzsche) led to number moving beyond the visual. Faustian number shows Euclid’s “axioms” were but hypothesis. About 1800 n dimensional space born (a^n); logarithms cut links with sensible surface & solids (a^3); with irrational & complex exponents rule of 3D space ends. Coordinates are now groups of 3 unconditional independent numbers, the point is abstracted onto position & planes, transcendent space remote from sensibility. Now Faustian mathematicians produced results consistent with Faustian physics. The highest minds found hyper complex numbers & infinity taken to the nth power. Actuality not only embraces the sensible, but the spiritual as well as in this the highest most rarefied Faustian number world.
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Developing Functions & groups (a la Dedekind) is the final creative leap of Faustian mathematics: given an n dimensional manifold (or group) & a set of transformations which elements possess invariance during transformations? Both Pythagoras & Descartes created new number ideas; these ideas grew for a century, matured & flourished for 3 centuries, were fulfilled just when Culture became Civilization. Today, like the Greek mathematicians of Alexandria, we will round off, preserve our new number, but NOT create new visions.