THE MATHEMATICAL PRACTICE OBJECTION
For the modified indispensability theorist, the choice between strong fact and no-fact hinges on developments in physics (and per- haps the rest of science). We should now ask the impact of this choice: How would the pursuit of our independent question be af- fected by it? In other words, we want to know if the metaphysical distinction between strong fact and no-fact has methodological con- sequences.
If no-fact is correct, if there is no pre-existing fact to discover about the Lebesgue measurability of I’ sets of reals, then what approach should the set theorist take? Many observers would hold that no-fact is the end of the story, that mathematicians are in the business of discovering truths about mathematical reality, and that, if there is no truth to be found, the mathematician should reject the question. From this point of view, all (relatively consistent) set the- ories extending ZFC are equally legitimate, there is no call to chose between them, and indeed, no grounds on which to do so apart from subjective aesthetic preferences. Once our question is shown to be independent, and developments in science undercut the claim to inherent truth value, there is nothing more of serious import to be said about the Lebesgue measurability of I’ sets. Call this end of the story no-fact.
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This position makes nonsense of the contemporary search for new set-theoretic axioms to settle independent questions like ours. In- deed, in our case, there are two competing candidates: V = L (G6del’s “axiom of constructibility”) and MC (the existence of a measurable cardinal). If V = L, then there is a non-Lebesgue mea- surable I’ set; if MC, then all I’ sets are Lebesgue measurable. Set theorists offer arguments for and against these axiom candidates,25 and in this debate, MC is strongly favored over V = L.26 If we are not to reject this activity as inconsequential mutterings-an espe- cially unappealing move, given that the original axioms of ZFC are supported by arguments of a similar flavor-we must instead reject end of the story no-fact.
But there is another version of no-fact. Even if there is no pre-ex- isting fact of the matter to be discovered, the process of extending the axioms of set theory might well be governed by nonarbitrary principles. This idea turns up, not only in the study of set theory, but when ontological decisions are made in other branches of mathe- matics as well. For example, Kenneth Manders27 describes the theo- retical norms at work in the expansion of the domain of numbers to include the imaginary or complex numbers, and Mark Wilson28 un- covers the rationale behind the move from affine to projective geom- etry. In such cases, despite lip service to the notion that any consis- tent system is as good as any other, mathematicians actually insist that a given mathematical phenomenon is correctly viewed in a cer- tain (ontological) setting, that another setting is incorrect.
One need not assume fact to endorse these practices. Even if there is no fact of the matter, no pre-existing truth about the exis- tence or nonexistence of complex numbers or geometric points at infinity or nonconstructible sets, the pursuit of these mathematical topics might be constrained by mathematical canons of “correct- ness.” For our case, one might hold that there is no fact of the matter about the Lebesgue measurability of I’ sets, but that there are still good mathematical reasons to prefer extending ZFC in one way rather than another, and perhaps, good mathematical reasons to adopt a theory that decides our question one way rather than another. From this point of view, no-fact is just the beginning of the
25 See my “Believing the Axioms. I-II,” The Journal of Symbolic Logic, LIII, 2 (June 1988): 481-511, and 3 (September 1988): 736-64.
26 I discuss part of the case against V = L in “Does V equal L?” in The Journal of Symbolic Logic (forthcoming).
27 “Domain Extension and the Philosophy of Mathematics,” this JOURNAL, LXXXVI, 10 (October 1989): 553-62.
28 “Frege: The Royal Road from Geometry,” in Nouis (forthcoming).
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story; it opens the door on the fascinating study of purely mathemat- ical canons of correctness. Call this beginning of the story no-fact.
The question before us is this: What are the methodological con- sequences of the choice between strong fact and no-fact? If strong fact is correct, the set theorist in search of a complete theory of her subject matter should seek out additional true axioms to settle the Lebesgue measurability of I’ sets (and so on). If end of the story no-fact is correct, the set theorist left with any interest in the matter should feel free to adopt any (relatively consistent) extension of ZFC she chooses, or even to move back and forth between several mutu- ally contradictory such extensions at will. And finally, if beginning of the story no-fact is correct, the set theorist should use appropriate canons of mathematical correctness to extend ZFC and to decide the question.29
Obviously, the method prescribed by strong fact differs from that prescribed by end of the story no-fact; that much is easy. But what about strong fact and beginning of the story no-fact? Does the pur- suit of truth differ from the pursuit of mathematical correctness? In fact, I think it does. Consider, for example, a simple argument that V = L should be rejected because it is restrictive. A supporter of this argument owes us an explanation of why restrictive theories are bad. A beginning of the story no-fact-er might say, “because the point of set theory is to realize as many isomorphism types as possible, and set theory with MC is richer in this way.”30 A strong fact-er might agree that the world of MC has desirable properties, while insisting that desirability (notoriously!) is no guarantee of truth.3′ Faced with the beginning of the story no-fact-er’s argument, a strong fact-er would reply, “Yes, MC is nice in the way you indicate, but if V does
29 The mathematical canons invoked in beginning of the story no-fact could ultimately recommend that several different set theories be accorded equal status. The methodology at work would still be different from that of end of the story no-tact, and the range of theories endorsed would almost certainly be narrower.
3 Spelling out this line of thought precisely is no simple exercise, but I shall leave that problem aside here. The point is just that the beginning of the story no-fact-er appeals to some attractive feature of set theory with MC.
Philip Kitcher touches on this point in his reply to Manders, “Innovation and Understanding in Mathematics,” thisJoURNAL, LXXXVI, 10 (October 1989): 563- 4, when he writes, p. 564: “Suppose this is a way in which mathematical knowl- edge can grow. What kinds of views of mathematical reality and mathematical progress are open to us? Can we assume that invoking entities that satisfy con- straints we favor is a legitimate strategy of recognizing hitherto neglected objects that exist independently of us? From a realist perspective, the method of postulat- ing what we want has (in Bertrand Russell’s famous phrase) ‘the advantages of theft over honest toil.’ If that method is, as Richard Dedekind supposed, part of the honest trade of mathematics, is something wrong with the realist perspec- tive?”
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equal L, L does contain all the isomorphism types possible. What’s needed is an argument that V = L is false.” So strong fact will differ methodologically from beginning of the story no-fact as well as end of the story no-fact.
We have reached this point: a methodological decision in set theory-namely, that between the methodologies proper to strong fact and to beginning of the story no-fact-hinges on developments in physics. If this is correct, set theorists should be eagerly awaiting the outcome of debate over quantum gravity, preparing to tailor the practice of set theory to the nature of the resulting applications of continuum mathematics. But this is not the case; set theorists do not regularly keep an eye on developments in fundamental physics. Fur- thermore, I doubt that the set-theoretic investigation of indepen- dent questions would be much affected even if quantum gravity did end up requiring a new and different account of space-time; set theorists would still want to settle open questions about the mathe- matical continuum. Finally, despite the current assumed indispensa- bility of continuum mathematics, I suspect that the actual approach to the Lebesgue measurability of V sets, to V = L versus MC, is more like that prescribed by beginning of the story no-fact than that prescribed by strong fact,32 and I see no mathematical reason to criticize this practice. In short, legitimate choice of method in the foundations of set theory does not seem to depend on physical facts in the way indispensability theory requires.
VII. CONCLUSION
I have raised two doubts about indispensability theory, even modi- fied indispensability theory, as an account of mathematics as prac- ticed. The first, the scientific practice objection, notes that indispen- sability for scientific theorizing does not always imply truth and calls for a careful assessment of the extent to which even fundamental mathematized science is “idealized” (i.e., literally false). The second, the mathematical practice objection, suggests that indispensability theory cannot account for mathematics as it is actually done. If these objections can be sustained, we must conclude that the indispensa- bility arguments do not provide a satisfactory approach to the ontol- ogy or the epistemology of mathematics. Given the prominence of indispensability considerations in current discussions, this would amount to a significant reorientation in contemporary philosophy of mathematics.