Andrew Arana - Purity of Methods
This 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.
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.
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