Practical reasoning and the witnessably rigorous proof
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Practical reasoning and the witnessably rigorous proof Eric Livingston1 Received: 19 December 2019 / Accepted: 17 September 2020 © Springer Nature B.V. 2020
Abstract This paper introduces an anthropological approach to the foundations of mathematics. Traditionally, the philosophy of mathematics has focused on the nature and origins of mathematical truth. Mathematicians, however, treat mathematical arguments as determining mathematical truth: if an argument is found to describe a witnessably rigorous proof of a theorem, that theorem is considered—until the need for further examination arises—to be true. The anthropological question is how mathematicians, as a practical matter and as a matter of mathematical practice, make such determinations. This paper looks first at the ways that the logic of mathematical argumentation comes to be realized and substantiated by provers as their own immediate, situated accomplishment. The type of reasoning involved is quite different from deductive logic; once seen, it seems to be endemic to and pervasive throughout the work of human theorem proving. A number of other features of proving are also considered, including the production of notational coherence, the foregrounding of proof-specific proof-relevant detail, and the structuring of mathematical argumentation. Through this material, the paper shows the feasibility and promise of a real-world anthropology of disciplinary mathematical practice. Keywords Mathematical practice · Deductive logic · Practical reasoning · Theorem proving
1 Introduction By the late 1800s, an ever growing number of mathematical discoveries, technical problems, and methodological issues gave rise to debates among mathematicians concerning the practices of proving mathematical theorems (Kline 1972; Ewald 1996). This cauldron of activity influenced mathematics thereafter. It also led to the growth of the philosophy of mathematics as a distinct subdiscipline of philosophy, but one
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Eric Livingston [email protected] Sociology - Criminology, University of New England, Armidale, NSW 2351, Australia
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increasingly separated from the practical circumstances and problems faced by mathematicians. Over the course of the twentieth century, philosophers of mathematics predominantly came to view the foundations of mathematics as a problem concerning the nature and origins of mathematical truth (Beracerraf and Putnam 1964; Shapiro 2005; Irvine 2009). From an anthropological point of view, perceived truths of the world are not matters of nature: they belong to the ways of a “tribe” and are generated and sustained by the real-worldly practices of its members. In the case of mathematics, the “tribe” of professional theorem provers, in practice, see written mathematical argumentation as the arbiter of mathematical truth: if the truth of a theorem is questioned, they examine the written argument that describes the claimed proof and try to establish the veracity of its reasoning. In this way, viewed anthropologically, the everyday production of witness
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