Applying Inconsistent Mathematics

Abstract

Inconsistent mathematics has a special place in the history of philosophy. The realisation, at the end of the nineteenth century, that a mathematical theory—naïve set theory—was inconsistent prompted radical changes to mathematics, pushing research in new directions and even resulted in changes to mathematical methodology. The resulting work in developing a consistent set theory was exciting and saw a departure from the existing practice of looking for self-evident axioms. Instead, following Russell (1907) and Gödel (1947), new axioms were assessed by their fruits.¹ Set theory shook off its foundationalist methodology. This episode is what philosophers live for. Philosophers played a central role in revealing the inconsistency of naïve set theory and played pivotal roles as new set theories took shape. This may have been philosophy’s finest hour.²

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 naïve 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 chapter, 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.

Department of Philosophy, University of Sydney, Sydney, NSW, 2006, Australia. Email: mcolyvan@usyd.edu.au.