A/logos: An Anomalous Episode in the History of Number

“Mathematics is the science of the infinite” (Weyl, “The Current Epistemological Situation” 123).¹ With this pronouncement, the German mathematician Hermann Weyl opens his 1925 diagnostic essay, “The Current Epistemological Situation in Mathematics” (Die heutige Erkenntnislage in der Mathematik), which takes the measure of a contemporary situation identified elsewhere by Weyl as a “foundational crisis” (“On the New Foundational Crisis of Mathematics”). At issue for Weyl, as for many of his fellow mathematicians—whose positions he attempts to reconcile in his paper —is the ongoing battle for dominion over an infinitely divisible continuum. That unbroken flow of experiential time and space, within which, as the Presocratic philosopher Anaxagoras had already observed by the 5^th century BC, there is no smallest part of what is small, since there is always a smaller (Curd 38, fragment 3), poses seemingly insurmountable difficulties for an attempt to analyze the world into discrete, and thus calculable or measurable units. On the one hand: to eschew all such analysis is to relinquish the possibility of science, and perhaps even of language, since both activities would appear to depend on an intellectual capacity for delimiting particular phenomena. On the other hand: the contradictions generated by ignoring Anaxagoras’ prohibition against a carved-up continuum (“the things in the one kosmos have not been separated from one another, nor hacked apart with an axe” [Curd 51, fragment 8]) have dogged both philosophers and mathematicians for millenia. Weyl, who provides a thumbnail history of this “dogging” that runs from Zeno to the traumata of modern set theory, sums up the situation as follows: “One can see how deeply mathematics is tied in its foundations to the general problems of knowledge. The old opposites of realism and idealism, of the Being of Parmenides and the Becoming of Heraclitus, are here again dealt with in a most pointed and intensified form” (“The Current Epistemological Situation” 141).²

Weyl is right to insist (and he is far from alone in doing so) that modern set theoretical arguments are intimately tied to ancient debates about the nature of being and knowledge. His “here again” is deceptive, however, because something actually happens over the course of the late 19th and early 20th centuries—the very something, indeed, that precipitates the “crisis” about which Weyl is writing—that had never happened before in the history of thought. The definition of number undergoes during this period a profound metamorphosis, which calls forth an equally profound metamorphosis in the definition of logical relation. The following article will argue that this double transformation in the definitions of arithmos and logos hangs together with a shift in the mathematical understanding of the role played by anomalous phenomena—and by the concept of the anomalous per se—in the construction of a proper domain for thought.³ The anomaly starts out, via a tradition that harks back to the ancient Greeks and extends well into the 19^th century, as a rule-flouting, definition-resisting exception to be explained away or excluded from the system of self-consistent commensurable truths. It then becomes, for a brief, quasi-Hegelian moment prior to the early 20^th century axiomatization of set theory, the antithesis that drives the mathematical “thesis” of number toward the telos of an all-encompassing synthesis. From this point onward it transforms again—via a powerful deflation of the Idealist model—to produce the situation that still governs much of contemporary mathematical, biological, and philosophical thought: the “anomalous” itself becomes associated with nature, while within this default state of chaos and entropy, the orderly systems of reason and life emerge as improbable, negentropic exceptions.⁴

I

The question of what it means to count has always been bound up, also terminologically, with the question of what it means to think. And the question of what it means to be “anomalous” with respect to both activities—to be non-homologous or asymmetrical to the homologies and symmetries of mathematical reason (anomaly, from Gr. anomalia = “inequality, irregularity,” from an “not” + omalos “even, regular,” from homos “one and the same”)—has always been...