INDISPENSABILITY
The original indispensability arguments were aimed at those who would draw a weighty ontological or epistemological distinction be- tween natural science and mathematics. To those tempted to admit the existence of electrons while denying the existence of numbers, Quine5 points out that
Ordinary interpreted scientific discourse is as irredeemably committed to abstract objects-to nations, species, numbers, functions, sets-as it is to apples and other bodies. All these things figure as values of the variables in our overall system of the world. The numbers and func- tions contribute just as genuinely to physical theory as do hypothetical particles (ibid., pp. 149-50).
But, Quine’s opponent insists, the scientific hypotheses in our theory are tested by experiment, and the mathematical ones are not; surely the two can be distinguished on these grounds. To which Quine replies,
The situation may seem to be saved, for ordinary hypotheses in natural science, by there being some indirect but eventual confrontation with empirical data. However, this confrontation can be remote; and, con- versely, some such remote confrontation with experience may be claimed even for pure mathematics and elementary logic. The sem- blance of a difference in this respect is largely due to overemphasis of departmental boundaries. For a self-contained theory which we can check with experience includes, in point of fact, not only its various theoretical hypotheses of so-called natural science but also such por- tions of logic and mathematics as it makes use of.6
Thus, the applied mathematics is confirmed along with the physical theory in which it figures.
Of course, it is not enough for a piece of mathematics simply to appear in a confirmed scientific theory. For any theory T, there is another theory T’ just like T except that T’ posits a bunch of new particles designed to have no affect on the phenomena T predicts. Any experiment confirming T under these circumstances would also (in some sense) confirm T’, but we do not take this as evidence for the existence of the new particles because they are “dispensable,” i.e., there is an equally good, indeed better theory of the same phe- nomena, namely, T, that does not postulate them. The mathematical apparatus of modern physics does not seem to be dispensable in this way; indeed, Putnam has emphasized that many physical hypotheses
5″Success and Limits of Mathematization,” in Theories and Things, pp. 148-55.
6 “Carnap and Logical Truth,” p. 121.
278 THE JOURNAL OF PHILOSOPHY
cannot even been stated without reference to numbers, func- tions, etc.7
So a simple indispensability argument for the existence of mathe- matical entities goes like this: we have good reason to believe our best scientific theories, and mathematical entities are indispensable to those theories, so we have good reason to believe in mathematical entities. Mathematics is thus on an ontological par with natural science. Furthermore, the evidence that confirms scientific theories also confirms the required mathematics, so mathematics and natural science are on an epistemological par as well.
Unfortunately, there is a prima facie difficulty reconciling this view of mathematics with mathematical practice.8 We are told we have good reason to believe in mathematical entities because they play an indispensable role in physical science, but what about mathe- matical entities that do not, at least to date, figure in applications? Some of these are admissible, Quine9 tells us,
… insofar as they come of a simplificatory rounding out, but anything further is on a par rather with uninterpreted systems (ibid., p. 788).
So in.particular,
I recognize indenumerable infinites only because they are forced on me by the simplest known systematizations of more welcome matters. Magnitudes in excess of such demands, e.g. M,, or inaccessible numbers, I look upon only as mathematical recreation and without ontological rights. 10
The support of the simple indispensability argument extends to mathematical entities actually employed in science, and only a bit beyond.
The trouble is that this does not square with the actual mathemati- cal attitude toward unapplied mathematics. Set theorists appeal to various sorts of nondemonstrative arguments in support of their customary axioms, and these logically imply the existence of :,,. Inac-
7 Hartry Field disputes this in his Science without Numbers (Princeton: Univer- sity Press, 1980), and Realism, Mathematics and Modality (Cambridge: Black- well, 1989), but the copious secondary literature remains unconvinced.
8 Versions of this concern appear in C. Chihara, Constructibility and Mathe- matical Existence (New York: Oxford, 1990), p. 15; and in my Realism in Mathe- matics (New York: Oxford, 1990), pp. 30-1.
9″Review of Charles Parsons’s Mathematics in Philosophy,” thisJOURNAL, LXXXI, 12 (December 1984): 783-94.
10 Quine, “Reply to Charles Parsons,” in The Philosophy of W. V. Quine, L. Hahn and P. Schilpp, eds. (La Salle, IL: Open Court, 1986), pp. 396-403; here p. 400.
INDISPENSABILITY AND PRACTICE 279
cessibles are not guaranteed by the axioms, but evidence is cited on their behalf nevertheless. If mathematics is understood purely on the basis of the simple indispensability argument, these mathemati- cal evidential methods no longer count as legitimate supports; what matters is applicability alone. Here simple indispensability theory rejects accepted mathematical practices on nonmathematical grounds, thus ruling itself out as the desired philosophical account of mathematics as practiced.
So simple indispensability will not do, if we are to remain faithful to mathematical practice. We insist on remaining faithful to mathe- matical practice because we earlier endorsed a brand of naturalism that includes mathematics. But it is worth noting that even a retreat to purely nonmathematical naturalism (forgetting our commitment to actual mathematical practice) will not entirely solve this problem. From the point of view of science-only naturalism, the applied part of mathematics is admitted as a part of science, as a legitimate plank in Neurath’s boat; unapplied mathematics is ignored as unscientific. But even for applied mathematics there is a clash with practice. Mathematicians believe the theorems of number theory and analysis not to the extent that they are useful in applications but insofar as they are provable from the appropriate axioms. To support the adoption of these axioms, number theorists and analysts may appeal to mathematical intuition, or the elegant systematization of mathe- matical practice, or other intramathematical considerations, but they are unlikely to cite successful applications. So the trouble is not just that the simple indispensability argument shortchanges unap- plied mathematics; it also misrepresents the methodological realities of the mathematics that is applied.
There is, however, a modified approach to indispensability consid- erations which gets around this difficulty. So far, on the simple approach, we have been assuming that the indispensability of (some) mathematical entities in well-confirmed natural science provides both the justification for admitting those mathematical things into our ontology and the proper methodology for their investigation. But perhaps these two-ontological justification and proper method-can be separated. We could argue, first, on the purely ontological front, that the successful application of mathematics gives us good reason to believe that there are mathematical things. Then, given that mathematical things exist, we ask: By what methods can we best determine precisely what mathematical things there are and what properties these things enjoy? To this, our experience to date resoundingly answers: by mathematical methods, the very methods mathematicians use; these methods have effectively pro-