This paper traces the evolution of thinking on how mathematics relates to the world—from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics in contemporary science.

This paper examines the efficacy of the familiar indispensability arguments for the existence of mathematical entities. It suggests that these arguments rest on inaccurate portrayals of scientific and mathematical practice.

For some time now, academic philosophers of mathematics have concentrated on intramural debates, the most conspicuous of which has centered on Benacerraf's epistemological challenge. By the late 1980s, something of a consensus had developed on how best to respond to this challenge. But answering Benacerraf leaves untouched the more advanced epistemological question of how the axioms are justified, a question that bears on actual practice in the foundations of set theory. I suggest that the time is ripe for philosophers of mathematics to turn outward, to take on a problem of real importance for mathematics itself.

Philosophers often claim that mathematical axioms are obvious or self-evidently true, but this does not match the practice of modern set theory. In fact, set theorists offer a wide range of arguments for and against axiom candidates, and the description and evaluation of these arguments presents an important challenge to the philosopher. The paper and its sequel ('Believing the axioms#, ii', "jsl" 53, 1988, pp. 736-64) take a first step in the direction of such an account.

The axioms of mathematics are often characterized as obvious or self-evident, but this description does not fit the axioms of set theory. This paper and its predecessor aim to survey the grounds on which set theoretic axioms are believed and to raise the question of when and whether these grounds are rational. Part i covers the Zermelo-Fraenkel axioms, the continuum problem, small large cardinals and measurable cardinals. Part ii covers determinacy and large cardinals, and ends with some philosophical remarks.