Abstract
In this paper we discuss three interrelated questions. First: is explanation in mathematics a topic that philosophers of mathematics can legitimately investigate? Second: are the specific aims that philosophers of mathematical explanation set themselves legitimate? Finally: are the models of explanation developed by philosophers of science useful tools for philosophers of mathematical explanation? We argue that the answer to all these questions is positive. Our views are completely opposite to the views that Mark Zelcer has put forward recently. Throughout this paper, we show why Zelcer’s arguments fail.
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Notes
Note that we do not claim that “providing understanding” is the only possible extra aim that can be considered. However, it is the possible extra aim that justifies the application of philosophical concepts of explanation to mathematics. There may be other things that mathematicians may aim at, besides proving theorems and providing understanding.
As Zelcer points out, in this case we would probably like to have an explanation of why explanatory considerations, though they are important, are largely absent from mathematics. However, this question does not rise at this moment.
There are interesting contributions to the literature which do not fit into any of our two approaches because they focus on specific types of proofs and/or are do not propose models of mathematical explanation. Lange (2009) is a good example: he argues that proofs by mathematical induction cannot be explanatory, and he does that without making any specific assumptions what explaining in mathematics consists in. His only assumption is the asymmetry of explanations. Lange’s paper can been seen putting a constraint on reflective approaches: proofs by induction must come out as non-explanatory. But Lange does not develop a model of what mathematical explanation is, so it is not a reflective approach in itself. This is also the case for Baker (2010), in which Lange’s argument is criticized: Baker denies the validity of the constraint proposed by Lange.
It is possible that in the end we find out that there is only one type of explanation in mathematics. That is compatible with methodological pluralism because the latter amounts to considering all available options seriously.
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Acknowledgments
The authors thank Leen De Vreese, Bart Van Kerkhove and two anonymous referees for their comments on previous versions of this paper.
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Weber, E., Frans, J. Is Mathematics a Domain for Philosophers of Explanation?. J Gen Philos Sci 48, 125–142 (2017). https://doi.org/10.1007/s10838-016-9332-1
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DOI: https://doi.org/10.1007/s10838-016-9332-1