Abstract
This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary ‘structural properties’. Although I grant that mathematical statements could be considered as necessarily true in this sense, I question whether this justifies the claim that mathematics as a whole is necessary.
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A version of this paper was presented at the meeting, ‘Impact des Categories: Aspects Historiques et Philosophiques’, held in Paris, Autumn 2005. I thank the audience for valuable comments and discussion. I especially wish to thank Colin McLarty for his invaluable support. Also thanks to the two anonymous referees for their helpful comments.
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Carter, J. Categories for the working mathematician: making the impossible possible. Synthese 162, 1–13 (2008). https://doi.org/10.1007/s11229-007-9166-9
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DOI: https://doi.org/10.1007/s11229-007-9166-9