Indispensability

This Element is a cutting-edge survey of the Indispensability Argument in the philosophy of mathematics.

In this Element the authors state the argument informally as well as more formally, briefly reviewed its origins, trace some of the philosophical commitments behind the argument's premises, and a more critical evaluation. The argument indisputably relies on naturalism for its plausibility and, more controversially, on confirmational holism.Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively.