We reflect on the use of empirical methods in philosophy and the variety of different empirical methods, emphasising that the experimental method in the strict sense is only one of them. Based on our discussion, we argue for the use of methodological triangulation in empirical philosophy.

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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.

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one's goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one's means/ends ratio. Our story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like.

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