This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

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 :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor 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 against our approach: that it leads to a form of relativism according to which validity is equated with social agreement and 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. (shrink)

Representations, in particular diagrammatic representations, allegedly contribute to new insights in mathematics. Here I explore the phenomenon of a “free ride” and to what extent it occurs in mathematics. A free ride, according to Shimojima, is the property of some representations that whenever certain pieces of information have been represented then a new piece of consequential information can be read off for free. I will take Shimojima’s framework as a tool to analyse the occurrence and properties of them. I consider (...) a number of different examples from mathematical practice that illustrate a variety of uses of free rides in mathematics. Analysing these examples I find that mathematical free rides are sometimes based on syntactic and semantic properties of diagrams. (shrink)

The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejía-Ramos (Synthese, 2019) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that we should shift the question of explanation (...) away from why-questions and towards a “what’s going on here” question. Additionally, I argue that when we recognize the connection between understanding and explanation we naturally see how more than just proof can be explanatory. I expand this point by detailing how definitions and diagrams can be explanatory. In all, we see that when we take seriously the connection between understanding and explanation, we get a better sense of how explanation arises within mathematics. (shrink)

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of (...) the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification. (shrink)

Mathematicians’ use of external representations, such as symbols and diagrams, constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition and (...) cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians’ use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process. (shrink)