Has mathematics today become depoliticized? To some extent, surely. Since the 19^th century mathematics has dissociated itself from the world, creating a universe of wonders all to itself. Absolute universal and necessary truths may still rule in this alternate reality, but they have only tangential bearing on the realities of the world as we know it. We are safe from the tyranny of mathematics, and it is safe from our materialist heresies.

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The materialist mathematics of the 17^th century was a subversive mathematics. Instead of imposing order on a chaotic world, materialist mathematics started with the world as it is, and abstracted from it. Traditional mathematics, exemplified by Euclidean geometry, was a “mathematics from above”, bestowing divine order on a recalcitrant world. The indivisiblist methods of Cavalieri and Harriot were a “mathematics from below,” deriving truth from the world as it is. Hierarchy and order vs. egalitarianism and the risk of chaos; such were the politics of early modern mathematics.

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Im gespateten Vogelholz

kleben Kerbelmeisen mit

angeflanschten Beinen

zwei Augenpunkte im

Selbstschlag als Astobst

und Knotenstock

lärcht das Waldholz

fruchtet

scheucht

oder schwärmt?

Am Werwein lang der
Günderode, Günderode
wieviel Wind
wiegt der Norden
nässt der Talsaft
wer wässert das Schieferschwein
daran vorbei astronautet
in den Wagner-Schlund
Weltende rauchte als trat
die Rakete auf
Sonderbetrug im Schleifenstein
nagt
und bettet
reine gleichmässige
Halme

Ingrid Wiener: Erste Schritte auf dem Weg zu Bayes # 6, Gobelin 2007, 34×50cm nach Blei- und Filzstiftnotizen auf DIN A4

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And if mathematics is itself material? Then the world is turned upside-down. If even this most trusted guarantor of necessity and hierarchy turns out to be based on lowly chaotic matter, then what hope is there for an ordered universe? Matter will rule reason, the body will rule the soul, and all hope for salvation, in this world or the next, vanishes. To those who, like the Jesuit fathers, champion an ordered world and a hierarchical social order, in which everything and everyone have their God-given place, nothing could be more sinister than a material mathematics.

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The Jesuit Doctors were right: materialist mathematics is dangerous. This is because since the time of Plato mathematics was celebrated precisely for being free of matter, and lording over it. Traditional mathematical arguments begin with simple assumptions and proceed step by step through strict logical deduction to a necessary conclusion. They pay no heed to physical circumstances or human desires, but follow their own inevitable road to unerring results. So conceived, mathematical results are not only True, but also Universal: by studying mathematics we learn necessary truths about the world itself. In this way pure mathematical reasoning rules corrupt and chaotic matter and orders it, just as the reasonable soul rules and orders the corrupt body. To the Jesuit fathers and to others through the centuries, mathematical reasoning stood for the triumph of reason, order, and hierarchy over irrationality, disorder, and egalitarianism.

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Atomism was a popular doctrine among 17^th century natural philosophers, but it was also a dangerous one. Church theologians condemned atomism as incompatible with the immortality of the soul, the miracle of transubstantiation, and the proper order of knowledge. And because infinitesimal mathematics was essentially mathematical atomism, it too was viewed with suspicion. In 1632 the authorities of the Collegio Romano, world center of Jesuit learning, issued an edict rejecting the method of indivisibles and forbidding its teaching in Jesuit schools. The charge against mathematics? Materialism!

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Cavalieri called his system “The Method of Indivisibles,” because it was founded on the premise that continuous lines and surfaces were made up of an infinite number of indivisible points or lines. This was practically identical to material atomism, which argued that material bodies are composed of indestructible atoms. No coincidence there – the word “indivisible” is a direct translation of the Greek “atom.” This means that infinitesimal techniques, a pillar of all of modern mathematics, started their way as the doctrine of mathematical atomism.

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