THE JOURNAL OF PHILOSOPHY
duced all of mathematics, including the part so far applied in physi- cal science.
From this point of view, a modified indispensability argument first guarantees that mathematics has a proper ontology, then endorses (in a tentative, naturalistic spirit) its actual methods for investigating that ontology. For example, the calculus is indispensable in physics; the set-theoretic continuum provides our best account of the cal- culus; indispensability thus justifies our belief in the set-theoretic continuum, and so, in the set-theoretic methods that generate it; examined and extended in mathematically justifiable ways, this yields Zermelo-Fraenkel set theory. Given its power, this modified indispensability theory of mathematics stands a good chance of squaring with practice, so it will be preferred in what follows.”
III. THE SCIENTIFIC PRACTICE OBJECTION My first reservation about indispensability theory stems from some fairly commonplace observations about the practice of natural science, especially physics. The indispensability argument speaks of a scientific theory T, well-confirmed by appropriate means and seamless, all parts on an ontological and epistemic par. This seam- lessness is essential to guaranteeing that empirical confirmation ap- plies to the mathematics as well as the physics, or better, to the mathematized physics as well as the unmathematized physics. Quine’s’2 vivid phrases are well-known: “our statements about the external world face the tribunal of sense experience not individually but only as a corporate body” (ibid., p. 41).
Logically speaking, this holistic doctrine is unassailable, but the actual practice of science presents a very different picture. Histori- cally, we find a wide range of attitudes toward the components of well-confirmed theories, from belief to grudging tolerance to out- right rejection. For example, though atomic theory was well-con- firmed by almost any philosopher’s standard as early as 1860, some scientists remained skeptical until the turn of the century-when certain ingenious experiments provided so-called “direct verifica- tion” -and even the supporters of atoms felt this early skepticism to be scientifically justified.’3 This is not to say that the skeptics neces-
This is more or less the position of my Realism in Mathematics. 12 “Two Dogmas of Empiricism,” repr. in From a Logical Point of View, pp.
20-46. 13 I trace the history in some detail in “Taking Naturalism Seriously,” in Pro-
ceedings of the 9th International Congress of Logic, Methodology and Philo- sophy of Science, D. Prawitz, B. Skyrms, and D. Westerstahl, eds. (Amsterdam: North Holland, forthcoming).