Whitehead's Poetical Mathematics

Whitehead's Poetical Mathematics

(The earth melts into the sea as the sea sinks into the earth).

Heraclitus¹

In this essay, I trace a set of math-poetic figures from Whitehead's Process and Reality in order to understand how he constructs a theory of the world that prehends, feels, and becomes social. My essay centers on two principal questions: How does Whitehead construct a philosophy of process and organism on mathematical intuitions that retains nonetheless all the living qualities of the unbifurcated world? And to what degree and in what manner does he construct his pata-mathematical concepts the way a mathematician constructs and fabricates concepts? Given his philosophical and theological inheritance, Whitehead responds simply and remarkably to some of the most provocative mathematics and mathematical physics of his day: Bertrand Russell's and David Cantor's set theory, and Einstein's general relativity. But if he seems to respond too bluntly in some respects, to what philosophical purposes—not scientific or mathematical—does he set his speculation? In the latter part of this essay, I will try to extend Whitehead's speculation using "lures for feeling" made from measure theory and topological dynamical systems, and will outline a notion of process that does not appeal to objects.² I argue that elaborating Whitehead's speculation a few steps beyond his artfully blunted set theory and general relativity theory yields a way out of the static and atomistic aspects of his metaphysics. Indeed, my amicus curiae should substantially enrich a plenist and process-oriented concept of unbifurcated nature that more readily accommodates local novelty.

I came to Whitehead after thinking with Gilles Deleuze and Félix Guattari's multiplicity, Deleuze's appropriation of Riemannian manifolds, and the a-signifying semiology of Guattari's chaosmosis. I assure you that my reading is not some truth-seeking missile, but a speculative and poetic exercise in thinking through Whitehead's philosophy of process—with the alchemical accompaniment of all those nonhuman, mathematical objects, like the monsters of set theory (pace Alain Badiou), and the point-free topologies of René Thom and Alexander Grothendieck, whose more fertile philosophical consequences have hardly been adequately developed, I believe.

In their Heraclitus seminar, Martin Heidegger and Eugene Fink tried to steer a middle course between a close, closed hermeneutic study of Heraclitus's Fragments and a free-associative "philosophizing" with the putative sense of the Greek text.³ With their fellow readers, they used the texts to develop a process theory that honored what they found in Heraclitus but also extended their phenomenological investigation. It seems worthwhile to read Whitehead in an analogous constructive and productive spirit to develop a topological approach to a process world.

First let me rapidly rehearse Whitehead's ontology as he develops it. In retracing Process and Reality's argument, we can detect, albeit faintly, what a mathematician might recognize as the rhetoric of proof. These features include preliminary motivations established as definitional "assumptions," paradigmatic examples, and a network of lemmas, theorems, and corollaries. He uses such labels almost nowhere because he supplies almost no arguments with the robustness and precision of a mathematical proof.⁴ (There is no call for actual mathematical argument, of course; and in fact, despite the formal precedents of Spinoza's Ethics, Newton's Principia, and Russell and Whitehead's Principia Mathematica, such rigor probably would sink the speculative enterprise.) Whitehead deploys a surfeit of assumptions, rather than finding a minimal model. One can see a paradigmatic example of this in his extravagant development of abstractive sets, about which his assumptions run into the dozens.⁵ One difficulty is that his conceptual edifice is a floating circle of coconstructive notions: actual entity, prehension, concrescence, nexus (society)—and later on, apparently still more abstractly, ovate sets, abstractive sets, strain, duration. But to an archaeologist of mathematics, elements of the mathematical physics of that era figure prominently in Whitehead's construction, and it seems fruitful to understand what philosophical juice he extracts...