CartLogical Time and the Production of the Intelligible Instructor: Matt HareProgram: Critical Philosophy, History, Design & WorldmakingCredit(s): 2Date: May 17th, 24th, 31st, June 7th, 14th, 21th, 28th, July 5th.Time: 09:00-11:30 ET
DESCRIPTION: In Jean Cavaillès’s final, posthumous “prison manuscript,” written in 1942-1943 and published in 1947, we find two central theses: that “[s]cience moves outside of time—if by time we mean reference to the lived experience of a consciousness,” and that “the fact that everything can not be at once has nothing to do with history, but is the characteristic of the intelligible.” These two theses form the core of a theory of logical time, in which mathematics is defined by its form of historicity, i.e., by a particular relationship between earlier and later theories. Cavaillès’s term for this relation, imported from Cartesian treatises on method, was “mathematical concatenation.” This Seminar will be devoted to studying this notion through the 20th century French mathematical philosophy represented by Cavaillès as well as Léon Brunschvicg, Albert Lautman, Jean-Toussaint Desanti, and others. We will use the short text of On Logic and the Theory of Science (1947) as a guide for investigating this broader intellectual history.
In the first part of the Seminar, we will situate Cavaillès within a tradition of thought surrounding the production of the intelligible, with roots in a particular Platonic-Cartesian “rationalist” history of philosophy constructed by the previous generation in France. The framework of “logical time” implied by this tradition informs a paradoxical notion, led by questions such as: if time is “produced,” in what medium can this production occur, if not that of time itself? Cavaillès’s assertion that “the intelligible” should be defined by its internal non-simultaneity should be situated in the long durée of this philosophical problem. We will read his writings in terms of how to refuse the purportedly “all at once” character of knowledge, a problem that extends from Plato to contemporary computational complexity theory. Cavaillès resumed the stakes of this classical problem in light of transformations in the notions of evidence and proof from Descartes through 19th century mathematics, leading to a fundamental methodological assertion: self-evidence had been replaced by provability, and thus “intuition” should be seen as being produced by concatenation.
In the second part of the Seminar, we will consider how these transformations set the stage for a research program in the historiography of mathematics. We will take Cavaillès to have advanced a speculative interpretation in which conjunctural results from mathematical logic in the 1920s and 1930s concerning the so-called “limitations of formalism” (Gödel, Skolem, Church-Turing) were taken to be positive discoveries concerning the nature of the intelligible. What is at stake in this part is a metaphysics of mathematical production derived from a reading of the actual history of mathematics. The core methodological requirement in this tradition of treating each mathematical work as “a regional and specific product” (Desanti) relates to an underlying metaphysical distinction between production and what is produced (or, in Cavaillès’s language, between the effectuating and the effectuated). We will examine the various philosophical and mathematical sources for this doctrine.
The general aim of the Seminar will be both historical and programmatic. Cavaillès understood contemporary mathematical logic to have shown the ruin of any philosophy of transcendental constitution, and thus of post-Kantian approaches to the philosophy of science, including phenomenology. Against such philosophies of the transcendental subject, Cavaillès (paradoxically) proposed a theory of the act without a subject. We shall conclude the Seminar by considering the origins and future of this doctrine in the philosophy of mathematics, broadly conceived.
Session 1: Two Perspectives on The “Production” of Mathematical Theories and the Transtheoretical Character of Mathematical Objects.
Session 2: A “Cartesian” Opposition in the Theory of Proof.
Session 3: Logical Time as a Problem in the French Philosophical Field.
Session 4: The Necessary Generation of New Concepts, or 19th Century Mathematics Through a Cartesian Lens.
Session 5: Rationalist Nominalism as a Research Programme in the Historiography of Mathematics.
Session 6: The Method of Recursion and the Horizon of Effective Computability.
Session 7: The Act Without a Subject.
Session 8: Old and New Problems.
IMAGE: George Nees, Untitled, 1965
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