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What inductive explanations could not be

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Abstract

Marc Lange argues that proofs by mathematical induction are generally not explanatory because inductive explanation is irreparably circular. He supports this circularity claim by presenting two putative inductive explanantia that are one another’s explananda. On pain of circularity, at most one of this pair may be a true explanation. But because there are no relevant differences between the two explanantia on offer, neither has the explanatory high ground. Thus, neither is an explanation. I argue that there is no important asymmetry between the two cases because they are two presentations of the same explanation. The circularity argument requires a problematic notion of identity of proofs. I argue for a criterion of proof individuation that identifies the two proofs Lange offers. This criterion can be expressed in two equivalent ways: one uses the language of homotopy type theory, and the second assigns algebraic representatives to proofs. Though I will concentrate on one example, a criterion of proof identity has much broader consequences: any investigation into mathematical practice must make use of some proof-individuation principle.

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Notes

  1. 5-induction must be given by a higher inductive type because it has \(5 = S^{5}(0)\) as a generator, as well as equalities that quotient out the higher path structure.

  2. More prosaically, every induction principle has an associated (homotopy) initial object in the category of algebras associated with that induction principle. See Sojakova (2015) for a precise statement and proof of this fact.

  3. To put this all more precisely, we should not distinguish between induction principles if their associated categories of algebras are equivalent. Equivalence preserves initiality, so a “universal labeler” for one principle is equivalently a “universal labeler” for the other.

References

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Acknowledgements

I am grateful to Craig Callender, Casey McCoy, Sebastian Speitel, and Adam Streed for helpful conversations, criticisms, and comments. Thanks, too, to the audience of the 2015 Philosophy of Logic, Math, and Physics conference at Western University. This work was performed under a collaborative agreement between the University of Illinois at Chicago and the University of Geneva and made possible by Grant Number (10) 56314 from the John Templeton Foundation and its contents are solely the responsibility of the author and do not necessarily represent the official views of the John Templeton Foundation.

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  1. Department of Philosophy, University of California, San Diego, CA, USA

    John Dougherty

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  1. John Dougherty
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Correspondence to John Dougherty.

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Dougherty, J. What inductive explanations could not be. Synthese 195, 5473–5483 (2018). https://doi.org/10.1007/s11229-017-1457-1

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  • DOI: https://doi.org/10.1007/s11229-017-1457-1

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