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Abstract
I focus on specific practices in twentieth- and twenty-first-century mathematics of articulating, barring, taming, and operating with what mathematicians widely call mathematical monsters. I describe how over centuries the quotidian procedures of the epitome of rational practice – mathematics – have produced beings outside the extant purified categories understood by theorems and proofs, despite, and sometimes as a consequence of, ever greater precision and rigor. However, mathematical monsters stand in a different relation to their makers than socio-economic and moral monsters who are barred from the world. Indeed, we will see how mathematicians have made plural and productive accommodation with their monsters. Although some mathematicians have been persuaded by logicians to bar monsters from the temple of mathematics by simply forbidding the operations that engender beings against nature, most mathematicians have accommodated or simply cohabited with their monsters by extending axioms, definitions, theorems, and methods of proving theorems as a part of quotidianized practice.
disclosure statement
No potential conflict of interest was reported by the author.
Notes
I thank colleagues in the Ontogenetics Process Group, in particular Giuseppe Longo, Adam Nocek, Gaymon Bennett, Stuart Kauffman, Cary Wolfe, Mike Epperson, and Phil Thurtle for gamely responding to my piecemeal thoughts. I thank Niklas Damiris and Helga Wild for extensive conversations over the years, and Gabriele Carotti-Sha for careful feedback.
1 What may seem to be colorful language is actually historically justified, as “rational” derives from the notion of “ratio” in Latin, for numbers that can be written as an integer divided by another integer: such as 5/2 whose decimal form is 2.5, or 355/113 whose decimal form is close to 3.141592. Irrational numbers are numbers that cannot be written as ratios of integers except by approximation, such as the number π = 3.141592653589 … Now one can think of a ratio say 5/2, as a solution to a very simple equation x = 5/2, which can be rewritten as 2x − 5 = 0, so every rational number is the solution to a polynomial equation with integer coefficients, familiar to ancient Greek, Arab, and Chinese mathematicians. However, one can also start with a polynomial equation in the form p[x] = pnxn + pn−1 xn −1 + … + p1 x + p0, where the coefficients pi are integers, and ask for the x that yields p[x] = 0.
2 In Capitalist Sorcery (2011), Philippe Pignarre and Isabelle Stengers identify what they call infernal choices. For making healthcare a scarce resource and then demanding the public or boards of experts to prioritize access, when in fact that scarcity is an artifact of vectoring healthcare through private insurance companies whose interest is orthogonal to caring for people’s health, but to maximize profit and minimize cost due to “risk.” In other words, conflating healthcare with insurance. Another canonical example would be nations in which electoral systems present candidates and parties from such a narrow spectrum of political programs that: (1) the majority of people’s interests are not represented by formal parties; and (2) noise effects can determine the balance of power. (Such a system can come about when parties try to win elections with programs constructed not on principle or argument or even history but purely to maximize statistical appeal. This leads to platforms that converge to the mean, yielding parties whose platforms do not provide strong distinctions.)
I underline that my purpose is not to address specific faults in present-day political economic systems, but to address the sorcery of presenting social, technical, affective appearance as if it were the only possible world. We can propose the amalgamation of incommensurate worlds as an alchemical alternative to rational or transcendental politics, as well as a politics of recurrent negation.
3 These are operations that one would follow in order to derive some quantities solving numeric or geometric problems.
4 See Sha Xin Wei, “Differential Geometrical Performance” and Differential Geometric Performance.
5 From Dardi (1344 (2001): 297) quoted by Roi Wagner (50).
6 Rather than simply repeat what we learn by rote, let’s rehearse a bit of algebraic reasoning that evolved over centuries of mathematical practice between Rome and the Crusades. Applying the notion of proportioning and debt, a minus times a plus yields a minus. (If you owe a debt of 2 to each of 20 people, you have a total debt of 40.) The question is how to interpret the product of two negative numbers.
Let
P := (a + b).
Algebraists already had available the ordinary operation of multiplying two binomials together, yielding:
P2 = (a + b) × (a + b) = a2 + 2a × b + b2.
Applying that with a = 10, b = −2, on the one hand as ordinary integers:
(10 – 2) × (10 – 2) = 8 × 8 = 64
but as binomials, we have:
(10 – 2) × (10 – 2) = 102 + 2 × (10) × (−2) + (−2)2.
So the terms become:
100 − 40 + (−2) × (−2) = 60 + (−2) × (−2).
This implies that the consistent way to interpret how to multiply two negative quantities together is via the rule: minus times minus is a plus.
7 Translation from <https://en.wikipedia.org/wiki/Imaginary_number#cite_note-9>.
8 See an efficient summary of proofs of the Fundamental Theorem of Algebra in <https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Proofs>.
9 The indicator function f(x) is defined by f(x) = 1 if x is rational, of the form p/q where p and q are integers, 0 otherwise. <http://mathworld.wolfram.com/images/eps-gif/DirichletFunction_1000.gif>.
10 Using mathematical measure theory which generalizes the notion of area to arbitrary sets, such “generalized functions” are understood consistently as measures concentrated at a point. This is different from the colloquial sense of the term. See Tao.
11 One can carry this far in the domain of functional analysis, to consider the algebraic structure of the spaces of functions and the spaces of what they act on. See Rudin.
12 One of the richest areas of mathematics over the past 250 years sprang from Plateau’s problem – the problem of finding the surface of least area among all surfaces spanning a given closed curve. First posed by Lagrange in 1760, Plateau’s problem in its classical formulation withstood attack till 1930, and was solved independently by Douglas and Radó. See Almgren, and Harrison and Pugh.
13 A lot of the work requires subtly considering what is meant by “spanning” a boundary, defining notions of regularity (“smoothness”) that make provable sense even for entities that cannot be presumed to exist a priori as classically pointwise-defined functions. Even more profoundly, one needs to extend the notion of “area” when we cannot assume almost no information about entities whose existence has yet to be proven. We would for example need to prove that some a priori arbitrary set of points in fact has the local structure of a k-dimensional surface.
14 In this case, the Horned Sphere is itself defined as the limit of an infinitely iterated process: start with a fairly homogeneously shaped closed surface that is topologically a sphere. (We are describing a topological surface, in which case we do not care about particular metrical shape, only that this starting point can be deformed from a round sphere.) Imagine your body’s skin as the initial topological sphere. Imagine your two arms as the first-generation “protuberances” from the surface of your skin. Then bring your fists close to each other, and pop open your index fingers and thumbs. Arrange them so that they are close but not touching. Now zoom in so that your field of attention is filled by the tips of the index finger and thumb of one hand, and interpret them as the second-generation pair of protuberances. The process actually splits into two sub-processes, one for each hand.
15 That the law crenulates is a matter of observation. Why or how the law proliferates boundlessly is another matter; see, for example, Agamben’s argument about the juridical as a state of exception.
16 The transfinite Axiom of Choice says the same is true even if we consider uncountably many subsets {Uα}. In the countable case, one can imagine some iterative process in which one can mutter, “1, 2, 3 … ” as one is picking out an element from each set. The process takes an infinite time, but at least in any finite duration, one can imagine iterating through some process of selection. However, the transfinite case is more unimaginable, because even in principle one cannot iterate through a discrete series of decisions. One of the most powerful aspects of elementary topological methods is that one can prove limits without resorting to series. For example, a basic theorem is that a continuous function on a closed and bounded domain achieves its maximum and minimum in that domain.
17 In mathematical physics, non-simultaneity is a consequence of the finite speed of propagation of material effect. One could argue that mortality or finitude imply that some relations must be ethical because they cannot be causal, thus not determined. See Sha, Poiesis 9.
18 Deleuze, Difference and Repetition 162.
19 Lian, “Fundamentals of Zermelo–Fraenkel Set Theory,” 23 Aug. 2011. <https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lian.pdf>.
20 Such as approximations to the identity in distribution theory.
21 An example would be the area and co-area formulas in geometric measure theory (Simon).
22 For exemplary work on comparative histories of mathematics and science, see Hart, Chinese Roots and Imagined Civilizations.