On the alleged simplicity of impure proof

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim

On the invisibility and impact of Robert Hooke’s theory of gravitation

Robert Hooke’s theory of gravitation is a promising case study for probing the fruitfulness of Menachem Fisch’s insistence on the centrality of trading zone mediators for rational change in the history of science and mathematics. In 1679, Hooke proposed an innovative explanation of planetary motions to Newton’s attention. Until the correspondence with Hooke, Newton had embraced planetary models, whereby planets move around the Sun because of the action of an ether filling the interplanetary space. Hooke’s model, instead, consisted in the idea that planets move in the void space under the influence of a gravitational attraction directed toward the sun. There is no doubt that the correspondence with Hooke allowed Newton to conceive a new explanation for planetary motions. This explanation was proposed by Hooke as a hypothesis that needed mathematical development and experimental confirmation. Hooke formulated his new model in a mathematical language which overlapped but not coincided with Newton’s who developed Hooke’s hypothetical model into the theory of universal gravitation as published in the Mathematical Principles of Natural Philosophy (1687). The nature of Hooke’s contributions to mathematized natural philosophy, however, was contested during his own lifetime and gave rise to negative evaluations until the last century. Hooke has been often contrasted to Newton as a practitioner rather than as a “scientist” and unfavorably compared to the eminent Lucasian Professor. Hooke’s correspondence with Newton seems to me an example of the phenomenon, discussed by Fisch in his philosophical works, of the invisibility in official historiography of “trading zone mediators,” namely, of those actors that play a role, crucial but not easily recognized, in promoting rational scientific framework change