Amir Alexander
On the Materiality of Mathematics
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Geometrical figures are like pieces of cloth. The Italian mathematician Bonaventura Cavalieri (1598-1647) thought so, and invited others to think in this way: “plane figures should be conceived by us as pieces of cloth are made up of parallel threads” he wrote in his Geometria Indivisibilibus of 1635. “And solids are like books, which are composed of parallel pages.” Cavalieri’s view counted for much, because he was widely credited as a founder of infinitesimal mathematics, which led ultimately to the Calculus. His teacher Galileo believed that infinitesimals were like threads, which woven together make up a rope, and their English contemporary Thomas Harriot (1560-1621) argued that the mathematical continuum is made up of points in the same way that a physical body is composed of atoms. In the 17th century, geometrical bodies were material objects, complex tapestries of interwoven lines, points, surfaces.
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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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Atomism was a popular doctrine among 17th 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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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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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 materialist mathematics of the 17th 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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Has mathematics today become depoliticized? To some extent, surely. Since the 19th 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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And yet, the politics of mathematics are still with us, bubbling beneath the surface. We sense it in the rule of professional “experts,” champions of order and necessity, who are forever seeking to present their claim in a mathematical mantle. They hope to infuse their arguments with a touch of mathematics’ aura of inevitability. We also sense it in the indifference, and even hostility towards mathematics of students and citizens. They refuse to engage with a field that will impose it absolute truths upon them, and never ask for their considered opinion. We see it in the schools, in the repeated struggles between “new math” and “old math” teaching methods, which map closely onto the political divisions between Left and Right.
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Mathematicians sense it too. The introduction of computers into mathematics has introduced a new materialist challenge into higher mathematics. Can a machine made of wires and silicon chips be allowed in the aetherial world of abstract mathematics? Are proofs accomplished by such material means legitimate? Is mathematics, the realm of pure unadulterated reason, compromised by the brute calculating power of these material beings?
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Mathematics is always about Order, and Order is always political. Hence, the politics of mathematics are always with us.
Amir Alexander works and teaches History of Mathematics in California. He is the author of: Geometrical Landscapes -The Voyages of Discovery and the Transformation of Mathematical Practice. Stanford: Stanford University Press, 2002.
His fields of research are history, philosophy, and the history of Science. Amir sends a package of 10 posts on threads, ropes and other materials in mathematics. This is a reaction on the Gobelins of Ingrid Wiener. In the same time it is a proposal to conceive mathematics politically. You find the posts under:http://www.journalfuerkunstsexundmathematik.ch/
Ingrid Wieners Gobelins you find at:
http://www.journalfuerkunstsexundmathematik.ch/ingrid-wiener/
Links (Amir Alexander)
http://www.math.rutgers.edu/~zeilberg/Opinion54.html
http://www.thalesandfriends.org/en/index.php?option=com_content&task=view&id=49&Itemid=86
Links (Bonaventura Cavalieri)
http://muse.jhu.edu/login?uri=/journals/configurations/v009/9.1alexander.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cavalieri.html
http://www.maths.tcd.ie/pub/HistMath/People/Cavalieri/RouseBall/RB_Cavalieri.html
http://scienceworld.wolfram.com/biography/Cavalieri.html
http://library.thinkquest.org/27694/Bonaventura%20Cavalieri.htm