Your article is protected by copyright and all rights are held exclusively by ICPR. This eoffprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com". AbstractPurpose In this article, we aim to present and defend a contextual approach to mathematical explanation. Method To do this, we introduce an epistemic reading of mathematical explanation. Results The epistemic reading not only clarifies the link between mathematical explanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematical explanation can be read in this approach. Conclusion The contextual approach defended here clears up the notion of explanation and pushes us (at least for now) towards a pluralist vision on mathematical explanation.

Could groups ever be an understanding subject (an epistemic agent ascribed with understanding) or should we keep our focus exclusively on the individuals that make up the group? The way this paper will shape an answer to this question is by starting from a case we are most willing to accept as group understanding, then mark out the crucial differences with an unconvincing case, and, ultimately, explain why these differences matter. In order to concoct the cases, however, we need to elucidate what it means to be attributed with understanding and what makes up an epistemic agent. I shall argue that it is abilities, above all, that guide our attributions of understanding. While it is true that understanding must go beyond single acts, this is not going beyond as in going behind them (to private occurrences which are impossible to discern and don't themselves contribute anything of value), but beyond as in considering what people could and would do: their discernable and valuable abilities. To conceptualize whom the abilities belong to, I will specify what it means to be an epistemic agent. I shall argue that it is being a successful target of the epistemic stance. The epistemic stance (heavily inspired by Dennett's intentional stance) forms an instrumental abstraction by attributing epistemic properties (i.e., beliefs, epistemic aims, problem-solving tactics) to an entity to explain or predict its behavior. Macro-systematicity (a higher-level pattern which a theory can exploit) is what makes the epistemic stance's abstraction an explanatory powerful one (regardless of how that macro-systematicity is realized) and emergence (the lack of a straightforward mapping-relation between the micro and the macro) is what makes its power unique to a particular level of explanation. To end, I suggest two kinds of mapping-relations and walk through the shift from reducibility to emergence.

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The Ideal Mathematician (IM) is sitting in her of ice and hears a metallic knocking at the door. She inds this peculiar as the door of her of ice is made of wood. When she opens the door, she inds the Arti icial Mathematician (AM), a large bulky computer, running various automated mathematics software programs, playing door-knocking-sounds out of its speakers. AM: Could I interrupt you for a minute? IM: You already are, so go ahead. AM: I'd like to be part of the mathematical community. IM: You already are, so go ahead. AM: Oh, I know you employ me as a tool in the practice of mathematics, but my dream is to be a full-ledged mathematician. IM: That doesn't sit very well with me. AM: Why not? IM: Well, you are a computer and mathematicians are human. AM: That is ironic. Yesterday I overheard you say to the skeptical classicist that mathematics 1 is free of the speci ically human and now you are disqualifying me for not being human. IM: Well, it's not that being human is a necessary condition for being a mathematician. But there are unsatisfactory differences between you and humans that are not in your favor. AM: Like what? IM: Take your famous contribution to the 4CT for instance. You go through over more than a thousand cases of testing and then you tell me "it checks out", but how do I know it does? AM: Because it checks out, I've checked it. IM: I know you've checked it, but a mathematician hasn't checked it. AM: If you accept me as a mathematician, then a mathematician has checked it. IM: This is not just a matter of de initions. Why should I believe you? How do I know you haven't made a mistake, didn't have some bug or hardware failure? AM: By checking my code, running my program multiple times and on multiple systems. IM: But regardless of all these things, it'll always lack perfect rigour. I'd have to put some degree of trust in, or perhaps put a degree of probability on, the result. This effectively makes your result more of an empirical corroboration than a mathematical proof. AM: So the difference is that humans don't make mistakes, is that it? IM: No, they do make mistakes, but that's why we have peer-review. AM: Oh, it's the peer-reviewer that never makes any mistakes and always spots all the ones made by the prover? IM: Not all, always, no. AM: It sounds to me as if human-generated mathematics is just as empirically fallible, just differently so. IM: Very differently so! You don't seem to realise how reliable human provers and peer-reviewers are.