INDISPENSABILITY AND PRACTICE 281
sarily recommended the removal of atoms from, say, chemical theory; they did, however, hold that only the directly verifiable con- sequences of atomic theory should be believed, whatever the explan- atory power or the fruitfulness or the systematic advantages of think- ing in terms of atoms. In other words, the confirmation provided by experimental success extended only so far into the atomic-based chemical theory T, not to the point of confirming its statements about the existence of atoms. This episode provides no comfort to the van Fraassenite, because the existence of atoms was eventually established, but it does show scientists requiring more of a theory than the sort of theoretical virtues typically discussed by philo- sophers.
Some philosophers might be tempted to discount this behavior of actual scientists on the grounds that experimental confirmation is enough, but such a move is not open to the naturalist. If we remain true to our naturalistic principles, we must allow a distinction to be drawn between parts of a theory that are true and parts that are merely useful. We must even allow that the merely useful parts might in fact be indispensable, in the sense that no equally good theory of the same phenomena does without them. Granting all this, the indispensability of mathematics in well-confirmed scientific the- ories no longer serves to establish its truth.
But perhaps a closer look at particular theories will reveal that the actual role of the mathematics we care about always falls within the true elements rather than the merely useful elements; perhaps the indispensability arguments can be revived in this way. Alas, a glance at any freshman physics text will disappoint this notion. Its pages are littered with applications of mathematics that are expressly under- stood not to be literally true: e.g., the analysis of water waves by assuming the water to be infinitely deep or the treatment of matter as continuous in fluid dynamics or the representation of energy as a continuously varying quantity. Notice that this merely useful mathe- matics is still indispensable; without these (false) assumptions, the theory becomes unworkable.
It might be objected that these applications are peripheral, that they are understood against the background of more fundamental theories, and that it is in contrast with these that the applications mentioned above are “idealizations,” “models,” “approximations,” or useful falsehoods. For example, general relativity is a fundamen- tal theory, and when space-time is described as continuous therein, this is not explicitly regarded as less than literally true. So the argu- ment goes.
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282 THE JOURNAL OF PHILOSOPHY
But notice, when those pre-Einsteinians were skeptical of atomic theory, it was a fundamental theory in this sense; it was not pro- posed against another background as a convenient idealization or mere approximation. The skeptics were bothered, not by such pe- ripheral simplifications, but by what they saw as the impossibility of directly testing the core hypotheses of atomic theory. But consider now the hypothesis that space-time is continuous. Has this been directly tested? As Quine himself points out, “no measurement could be too accurate to be accommodated by a rational number, but we admit the [irrationals] to simplify our computations and gen- eralizations.”‘4 Similarly, space-time must be regarded as continu- ous so that the highly efficacious continuum mathematics can be applied to it. But the key question is this: Is that continuous charac- ter “experimentally verified” or merely useful? If it is merely useful, then the indispensability argument sketched earlier, the one relying on the role of continuum mathematics in science to support the Zermelo-Fraenkel axioms, or ZFC, cannot be considered conclusive.
I shall not try to answer this question here; to do so would require a more thorough study of the physics literature than I am capable of launching just now. But until such a study is undertaken, until the evidence for the literal continuity of space-time is critically exam- ined,’5 I think the simple observations collected here are enough to raise a serious question about the efficacy of this particular indis- pensability argument.