Andrew Arana

Andrew Arana
labex
Eurias

dates de séjour

15/09/2014 - 17/07/2015

discipline

Philosophie

Fonction d’origine

Professeur

Institution d’origine

Departments of philosophy and mathematics, University of Illinois at Urbana-Champaign (États-Unis)

Fonction actuelle

Maître de conférences

Institution actuelle

Département de philosophie, Université Paris 1 Panthéon-Sorbonne et IHPST (France)

pays d'origine

France

projet de recherche

Purity of Methods

My project aims at the production of a research monograph on purity of methods, a key issue in the philosophy of mathematics. A chief aim of mathematical practice is proving theorems. A minimal constraint on proof is logical soundness, to ensure that it justifies belief in the truth of the theorem in question. But mathematicians often employ other constraints toward other goals, such as seeking efficiency of proof.

This project focuses on one such constraint, purity of methods. Roughly, a proof of a theorem is "pure" if it draws only on what is "close" or "intrinsic'' to that theorem. Purity concerns go back to antiquity: Aristotle states a version of the constraint for his demonstrations. They have been widely cited in discussions of analytic geometry, Descartes' method for transferring problems from geometry to algebra and then back again so that both algebraic and geometric methods can be employed in this solution procedure. They remain live today, in discussions of analytic number theory and algebraic geometry.

In each case the question is whether the knowledge produced by proofs is impacted, positively or negatively, by the "mixing" of mathematical areas. Though mathematicians are highly conscious of this constraint, with some actively favoring it and others against it, they have difficulty explaining their reasons for their views. This is because this question requires philosophical analysis, and locates preciselywhat my proposed project contributes.

My project thus has two focal questions: 1) how distance between proof and theorem should be measured; and 2) why mathematicians value purity (and whether they are rational in doing so).

Crucial to answering these questions is both work on meaning in mathematics, and careful case studies that reveal more closely how this constraint is employed in practice. This project thus fits neatly into the emerging area of research called the philosophy of mathematical practice, which aims at analysis and reflection upon mathematics as a human activity from antiquity through today. Work in this area, like mine, is essentially interdisciplinary, bringing together work from philosophy, history, mathematics, computer science, and psychology, among others. Until now, though, there has been no sustained monograph-length treatment of this topic.

My proposal then aims to remedy this gap by producing a monograph that can serve as the first place to look for those interested in this topic. The expected output of this fellowship, then, is such a monograph, to be submitted to a major press for publication.

biographie

Ancien professeur associé à l'Université de l'Illinois à Urbana-Champaign (2012-2015) et à l'Université d'Etat du Kansas (2011-2012), ses recherches s'inscrivent dans le champ de la logique, de la philosophie et l'épistémologie des mathématiques, de l'histoire des mathématiques, de la philosophie de la connaissance, ainsi que de la philosophie et l'épistémologie des sciences. Il est titulaire d'un doctorat en philosophie et mathématiques de l'Université Notre Dame, Indiana, Etats-Unis (2004).

 

Recrutement

Andrew Arana a été recruté au terme de sa résidence à l'IMéRA et collabore avec l'IHPST (Institut d’histoire et de philosophie des sciences et des techniques).

institut

01/05/2007