TILPS

Lectures

Professor Hacking will deliver one lecture per day. Each lecture will be followed by a discussion period and two invited commentaries, again followed by an open discussion.

Wednesday, 6 October 2010
Lecture I: Why Is there "Philosophy of Mathematics"?
Commentators: Hannes Leitgeb (Bristol) and Mary Leng (Liverpool)

Thursday, 7 October 2010
Lecture II: Meaning and Necessity - and Proof
Commentators: James Conant (Chicago) and Martin Kusch (Vienna)

Friday, 8 October 2010
Lecture III: Roots of Mathematical Reasoning
Commentators: Marcus Giaquinto (London) and Pierre Jacob (Paris)

I. Why Is there "Philosophy of Mathematics"?
First, a historical observation: "mathematics" is not a given. What counts as mathematics has varied greatly in the history of Western mathematics, and indeed Kant's "pure mathematics" is something of a historical artefact. Second, a reflection on why mathematics has so fascinated many of the canonical Western philosophers. Certain experiences of doing mathematics, usually associated with proofs, have prompted philosophical obsessions. The phenomena connected with mathematical demonstration, including the "hardness of the logical must," are the focus of these lectures. And also, the "motley of techniques of proof." Some proofs are primarily calculations, while others are what Wittgenstein called "perspicuous" or "surveyable." These prompt two distinct attitudes to mathematics, the one represented historically by Leibniz, and the other by Descartes. Thus already we discourage the idea that there is one thing, "mathematics," of which there should be a "philosophy."

II. Meaning and Necessity - and Proof
This title is a riff on Carnap's title, Meaning and Necessity (1947). The idea, current in the early Vienna Circle, that logical necessity is grounded on meaning, was already devastated by Quine's Truth by Convention (1936).Yet it contains bold insights that, reworked gingerly and somewhat metaphorically, can be retained. Its fundamental flaw is seldom noticed, namely that it still subscribes to the mediaeval conception of necessary truth as "eternal". It must be replaced by a more dynamic model suggested not only by Wittgenstein, but also by Imre Lakatos's Proofs and Refutations. Kant was correct to say, in his words, that mathematical propositions are "synthetic a priori," but they may become "analytic" in the dialectic of proof. Unlike so much philosophical writing after Kripke, Wittgenstein's thoughts on following a rule are here regarded as a secondary theme, an essential bulwark against a knee-jerk rejection of his primary line of inquiries, rather than their motivation. Scepticism (of a philosophical sort) does not arise.

III. Roots of Mathematical Reasoning
Our picture is somewhat modular: there may be distinct faculties for mathematical reasoning. Kant thought there are two, one arithmetical, and the other geometrical. At any rate there are human capacities, which have emerged in the evolution of our species, and there is the cultural discovery of these capacities, for instance, the discovery, in the Eastern Mediterranean, over 2300 years ago, of the power of demonstrative proofs in geometry. These lectures conclude at the intersection of the evolutionary psychology of the mathematical faculty, and what the historian of ancient mathematics, Reviel Netz, calls its cognitive history. We do not yet well know what a dense cognitive history would look like. The cognitive psychology of the subject is itself evolving rapidly from work as various as that of Susan Carey, Stanislas Dehaene, and Lakoff & Nunez. Here we have a radical revision of Kant's question: "How, as a matter of human prehistory and history, did mathematics become possible?" Yet we should not think we thereby reduce our philosophical questions to sciences and histories, for the bedrock of these lectures is the commonplace, not the esoteric.