Reconciling Rigor and Intuition
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Reconciling Rigor and Intuition Silvia De Toffoli1 Received: 13 May 2019 / Accepted: 25 May 2020 © Springer Nature B.V. 2020
Abstract Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work (De Toffoli and Giardino in Erkenntnis 79(3):829–842, 2014; Lolli, Panza, Venturi (eds) From logic to practice, Springer, Berlin, 2015; Larvor (ed) Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I address two criticisms that have been raised in Tatton-Brown (Erkenntnis, 2019. https://doi.org/10.1007/s10670-019-00180-92019) against our approach: (1) that it leads to a form of relativism according to which validity is equated with social agreement and (2) that it implies an antiformalizability thesis according to which it is not the case that all rigorous mathematical proofs can be formalized. I reject both criticisms and suggest that our previous case studies provide insight into the plausibility of two related but quite different theses.
1 Introduction Mathematics is very successful. We are proving more and more sophisticated propositions with an ever-increasing cornucopia of techniques, notations, methods, and abstract concepts linking separate areas of the discipline. Certain achievements even reach the ears of non-specialists: Wiles’ proof of Fermat’s Last Theorem and Perelman’s proof of the Poincaré Conjecture are two examples. The number of informal proofs is vastly greater than that of formal proofs and as a result restricting mathematics to what has been formally proven would strip it of many of its exciting successes. As Tom Hales (2008) explains,
* Silvia De Toffoli [email protected] 1
Philosophy Department, Princeton University, 1879 Hall, Princeton, NJ 08540, USA
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proofs are written in a way to make them easily understood by mathematicians. Routine logical steps are omitted. An enormous amount of context is assumed on the part of the reader. (p. 1371) Proofs1 are enthymematic valid deductive arguments—it is because they contain all sorts of shortcuts that the reader must be appropriately trained to check their correctness. Formal proofs, whose steps are either axioms or derived from previous steps with the aid of explicit inference rules, can instead be mechanically checked. Formal proofs were first defined at the end of the 19th century by the fathers of modern logic, Frege, Russell, Hilbert, and others. Moreover, after a turbulent period of foundational uncertainties, in the first half of the 20th century, set theory became widely recognized as able to provide foundations for all mathematics, that is, all provable results should be in principle provable in the language of set theory,2 even if Gödel showed that no single f
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