Reconciling Rigor and Intuition

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. 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.