Applying inconsistent mathematics

At various times, mathematicians have been forced to work with inconsistent mathematical theories. Sometimes the inconsistency of the theory in question was apparent (e.g. the early calculus), while at other times it was not (e.g. pre-paradox na¨ıve set theory). The way mathematicians confronted such difficulties is the subject of a great deal of interesting work in the history of mathematics but, apart from the crisis in set theory, there has been very little philosophical work on the topic of inconsistent mathematics. In this paper I will address a couple of philosophical issues arising from the applications of inconsistent mathematics. The first is the issue of whether finding applications for inconsistent mathematics commits us to the existence of inconsistent objects. I then consider what we can learn about a general philosophical account of the applicability of mathematics from successful applications of inconsistent mathematics.
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