7    A Life of Its Own?

Maddy and the Autonomy
of Mathematics

SCIENCE CHAUVINISM

Penelope Maddy tackles philosophical problems in a concrete way. Her most effective arguments are tied to specific examples drawn from esoteric regions of research mathematics, such as descriptive set theory. With remarkable clarity she reports on and contributes to mathematics in the making. In previous work she was keenly sensitive to its philosophical content, noting how debates about, say, the Axiom of Choice or impredicative definitions have involved deep issues about the reality and accessibility of abstract entities. Philosophers, of course, found this heartwarming.

Surprisingly—and perhaps disappointingly to those same philosophers—much of that approach has been jettisoned. Maddy has done an about-face and currently claims that philosophy is irrelevant to mathematical research. In her most recent work she emphatically asserts that even though philosophical issues were raised and were made to seem central to deciding mathematical issues, they nevertheless played no genuine role in the final analysis. Only one factor mattered when it came to deciding on the legitimacy of the Axiom of Choice and impredicative definitions—they worked mathematically. In consequence, her new advice is simple: “[I]f you want to answer a question of mathematical methodology, look not to traditional philosophical matters about the nature of mathematical entities, but to the needs and goals of mathematics itself” (1997, 191).

Her argument is disarmingly brief: “Impredicative definitions and the Axiom of Choice are now respected tools in the practice of contemporary mathematics, while the philosophical issues remain subjects of ongoing controversy. The methodological decision seems to have been motivated, not by philosophical argumentation, but by consideration of what might be called . . . mathematical fruitfulness” (1998a, 164). Hence, her conclusion: “Given that the methods are justified, that justification must not, after all, depend on the philosophy” (ibid.; see also Maddy 1997, 191).

Her subsequent characterization of naturalism follows on rather obviously: “Mathematical naturalism . . . is just a generalization of this conclusion, namely, that mathematical methodology is properly assessed and evaluated, defended or criticized, on mathematical, not philosophical (or any other extra-mathematical) grounds” (1998a, 164).

In her most recent work, she adopts this outlook generally, calling it “second philosophy.” (She rejects “first philosophy,” the would-be foundation for all knowledge.) The crucial contrast for Maddy is with other forms of naturalism, especially that of Quine and Putnam in which mathematics picks up its legitimacy from a connection to successful science. We saw this in earlier chapters. She rejects their science chauvinism and she claims for mathematics a life of its own; mathematics must be evaluated exclusively in mathematical terms. In her words, Maddy advocates mathematical naturalism, not scientific naturalism.

The picture, then, is this. All naturalists begin their study within natural science; this is scientific naturalism. All scientific naturalists notice that mathematicians employ methods different from those of natural scientists. The response of the Quinean or science-only naturalist is to regard mathematical claims as justified only in so far as they are supported by scientific, as opposed to mathematical, methods. In contrast, the response of the mathematical naturalist—influenced by the observation that mathematics has flourished by its own methods, not by those recommended by the science-only naturalists—opts to evaluate mathematical methods in their own terms, opts not to hold mathematical methods answerable to natural science. (1998a, 165)

Notice how sharply she's divided the rival views into non-overlapping camps. The Quinean naturalist insists that all knowledge be reduced to scientific knowledge; mathematics is justified only in so far as it is justified by natural science. Maddy's mathematical naturalist, by contrast, says that mathematics is justified in mathematical terms exclusively; there is no other source of support—not from science, not from philosophy, not from anything else. In short, mathematics has a life of its own.

When rival views are characterized so sharply, there is inevitably room in the middle. Though I hate to seem a mealymouthed moderate, something in between these two views is surely right. Such a view would acknowledge (with Maddy and against Quine) that mathematics can provide much of its own justification, without appeal to the natural sciences. On the other hand, there is every reason to believe (against Maddy) that some mathematical methods and results receive their justification from non-mathematical sources—in particular, from philosophy. Establishing this claim is the main point of this chapter. That is, my aim is to show that there are perfectly legitimate considerations that function as evidence for mathematical results. The extent to which this is successful is the extent to which Maddy's mathematical naturalism must be rejected. I should quickly add, however, that I find Maddy's brand of naturalism much more palatable than the other forms examined in this book. In spite of being a self-avowed naturalist, she has much in common with any Platonist.

Maddy wants us to evaluate mathematics “in its own terms.” But how should we go about doing this? She would have us study the means–ends relation. We determine the goals of mathematics, the methods employed to reach those goals, and finally, we determine the effectiveness of those methods. Inevitably, this means studying the history of mathematics, including its current history.

This is the conception of mathematics we will now examine. It is, as I already noted, a departure in some but not all respects from her earlier views as expressed in Realism in Mathematics (1990), where she championed a realist ontology and naturalistic solutions to epistemic difficulties, claiming, for instance, that we can literally see sets. (This was discussed in Chapter 6.) Her new naturalism can be found in Naturalism in Mathematics (1997) and in articles such as “How to Be a Naturalist about Mathematics” (1998a), “Naturalizing Mathematical Methodology” (1998b), “Does Mathematics Need New Axioms?” (2000), and “Some Naturalistic Reflections on Set Theoretic Method” (2001). It also permeates Second Philosophy (2007), which perhaps best expresses her present view.

Her current brand of naturalism should not be confused with her older view, even though that previous position had a reasonable claim to be called naturalism, too. I should also stress that Maddy has not abandoned her earlier views, except in one crucial respect. To a large extent she retains her realist ontology (though weakened from “Robust” to “Thin Realism”) and her peculiar epistemology of seeing sets (in a literal, naturalistic sense). The one thing she has changed is her belief that these philosophical views matter to mathematical practice. They do not. They are, she claims, of philosophical interest only.

THE ANTI-PHILOSOPHY ARGUMENT

Maddy's argument is strikingly simple, but also doubtful. It runs:

1. Methodological issues (e.g., should we accept impredicative definitions, the Axiom of Choice, the Axiom of Constructibility?) have been rationally decided by the mathematical community.

2. Philosophical debates (e.g., Platonism versus various types of anti-realism in the ontology and epistemology of mathematics) remain unresolved.

 Philosophy played no role in resolving the methodological issues.

One problem with this argument concerns the possible overdetermination of the resolution of methodological issues. Maddy holds that a methodological principle she calls MAXIMIZE resolved (in the negative) the debate about the Axiom of Constructibility. (All of this will be described in detail in the following.) But a Platonistic view of mathematics would also lead to a rejection of this axiom. Many mathematicians are sympathetic to Platonism, so it could well be that it was their Platonism that in fact led them to the rejection of the axiom. Others may have been influenced by Maddy's MAXIMIZE, and some, of course, by both. Even though there is no consensus on Platonism, there could still be a significant influence of Platonism on many mathematicians on this particular point. There is no reason to think there is but a single cause of mathematicians’ specific mathematical and methodological beliefs.

Maddy does allow that philosophy can be inspirational, as, for example, Gödel's Platonism was undoubtedly a crucial motivation for his work. But in the spirit of Reichenbach's famous distinction, she would perhaps relegate inspirational philosophy to the “discovery,” not the “justification,” side of the mathematical process. This is the obvious strategy for her to take. It does, however, seem unfair to Gödel and to anyone else who thinks that there are good reasons for being a Platonist, and consequently, good philosophical reasons for rejecting the Axiom of Constructibility. Of course, not everyone sees the good reasons for Platonism; it remains contentious. But for those who do, those good reasons provide powerful evidence against the Axiom of Constructibility. The fact that strong disagreement over Platonism remains is irrelevant as far as explaining why some mathematicians reject the axiom.

HOW NATURALISTIC IS IT?

Maddy allows so much that seems at odds with the spirit of naturalism we might wonder just how naturalistic her view really is.¹ And the worry is quite legitimate. Unlike most naturalists, she will not kowtow to natural science—almost a defining condition. Nor does she insist on sense-experience as the one and only source of knowledge; and she does not object to abstract entities or even to a priori considerations. One could be pardoned for thinking that so far she sounds like a traditional Platonist. What, if anything, is the difference? The chief difference, according to her, is that these factors are irrelevant; it simply doesn't matter to mathematics whether these ontological and epistemological claims are true or false.

I can imagine people who would be willing to accept her assertions and still coherently claim to be Platonists. There would, however, be considerable tension within the view. But there is also something else. I suspect a second difference turns on norms.

Though Platonists seldom talk explicitly about norms in developing mathematics, implicitly they play a crucial role. Mathematics develops the way it does because that is the way it ought to develop. Mathematicians are rational people who weigh the available evidence and make the right decisions based on that evidence. By contrast, a naturalist typically rejects norms and Maddy's mathematical naturalist should be no exception. Of course, a naturalist could cheerfully allow instrumental rationality—if you want to achieve x, then you ought to do y —and Maddy quite explicitly does this in her call for studying means–ends relations in the history of mathematics. However, she is self-consciously a describer, not a prescriber of mathematical activity. Her account of mathematics has the form: This is what they did. Normative evaluation plays no role; she would not and could not account for mathematics using the form: This is what they did, because it was the right thing to do (given the available evidence). I realize this is a rather blunt way of putting it, and no one treating concrete examples would be quite so crude. Still, it illustrates a crucial difference between naturalists and Platonists, and it is central to Maddy's naturalism. She shares little with other self-described naturalists, but this one point is perhaps enough to justify the naturalist label.

It may seem that my claim is obviously false; after all, Maddy has made much of MAXIMIZE, which certainly looks like a norm. Indeed it is a norm within the mathematical community; but notice that it merely operates as a fact about mathematicians, a groundless convention. A simple example might help to understand this point better. Here are two facts about German mathematicians: (1) they capitalize nouns; (2) they avoid contradictions. Intuitively, there is a big difference: The first reflects a mere conventional norm about the German language, while the second is much deeper, reflecting a “real” norm. If German's stopped capitalizing their nouns, we wouldn't mind, but we would certainly object if they started contradicting themselves. Maddy (in typical naturalist fashion) treats MAXIMIZE as a sociological fact about mathematicians: They happen to follow this norm but there is nothing intrinsically right or wrong about it. Platonists, however, would say it is the right norm, and they are willing to offer reasons for saying so.

MATHEMATICS AND THEOLOGY

The rejection of categorical norms can lead to problems. It's one thing to explain and describe what people do; it's quite another to justify and prescribe their actions. Mathematical naturalism—like any other type of naturalism—blurs this distinction. Mathematics, on Maddy's view, does not need any external justification—not from science, not from philosophy, not from anything outside of mathematics itself. Indeed, by her lights no other justification is wanted or needed. But can we simply say that good mathematics is what good mathematicians do? That seems far too facile, not to mention circular.

It also ignores the fact that mathematicians sometimes do philosophy and they do it quite explicitly as part of doing mathematics. Think of the long list that includes: Bolzano, Weierstrass, Cantor, Kronecker, Poincaré, Lebesgue, Borel, Brouwer, Hilbert, Bishop, and many others. Maddy, of course, makes the crucial claim that philosophy plays no role of any importance in mathematical activity—it's just so much epiphenomena floating over the real action (though, as mentioned earlier, she says it can be “inspirational”). So she needs a way of distinguishing irrelevant philosophy from genuinely fruitful methodological considerations.

Maddy's technique (as already mentioned) is simple: Study the means–ends relation of historical examples; determine the goal of a particular mathematical practice; then determine whether the practice was successful in achieving that goal. She readily notes that this is a fallible process and points out some of the minor pitfalls. She even goes on to raise more serious problems directed at herself. “What . . . is to block an astrological naturalism, which holds that astrological methods are not subject to scientific criticism?” (1998a, 175). After all, if mathematics is self-contained and to be evaluated exclusively in its own terms, why not astrology?

To this, she has an easy and effective answer: Astrology overlaps with regular science. It posits causal powers that operate inside the spatiotemporal world. Thus, it is in direct conflict with much of science and can be rejected for normal scientific reasons—it is empirically false. We reject astrology for the same type of reason we reject phlogiston. Mathematics cannot be similarly dismissed, however, since it is not in competition with any of the empirical natural sciences. So, concludes Maddy, astrology and mathematics are quite disanalogous—to mathematics's happy advantage. This answer is, I think, the perfect reply to astrology.

A more challenging example for Maddy to consider might be speculative theology (ironically named “natural” theology). Let's consider some version of theology that is not in any sort of competition with regular science; it does not deny Darwinian evolution, for instance. A community of theologians who embrace this religious doctrine might have a long intellectual history and have discernable methods that help them solve the tradition's problems. For example, they might make clear progress by moving from polytheism to monotheism, thereby resolving some conceptual problems within their theory. We could study their goals and the methods they used to achieve them. And we could conclude that they have been doing a pretty good job of achieving those goals by their chosen methods. Yet their religious beliefs are very likely nothing but a cluster of falsehoods. (Non-atheists might prefer to express this differently, but the point should be obvious however it is put.) The challenge for Maddy is to explain how their theological activities would differ from mathematics. They should be on a par.

Maddy's earlier answer (discussed in Chapter 6) was along the lines of Quine and Putnam, the so-called indispensability argument. Recall how the argument (roughly) runs: Mathematics is essential to science and we have (let us suppose) good reason to think science is true; thus, we have good reason to believe mathematics is true. Maddy now, however, rejects this argument; it is a central part of the “science naturalism” she has repudiated. Her new brand of mathematical naturalism must rely on its own resources exclusively; mathematics does not—and cannot—look outside of itself for legitimacy, a legitimacy it certainly possesses. Theology could offer the same consideration and claim a similar legitimacy.

Platonists would offer a rival response to this problem. The crucial difference between speculative theology and mathematics is that we have intuitions in the latter case, but not in the former. Mathematical intuitions are the evidential equivalent of empirical observations in natural science. They may be fallible and limited in scope, but they are the testing ground for more speculative axioms. This quasi perception is what puts us in touch with the realm of mathematical entities. Of course, many religious people will claim they have intuitions (often called religious experiences) of God. But theological intuitions are pathetically unstable; they vary wildly from person to person, from epoch to epoch, from culture to culture. Mathematical intuitions, by contrast, are as stable and repeatable as sense-experience, probably more so.

Since Maddy can't help herself to either of these two justifications—utility in science or reliable intuition—it seems as if mathematics, by her lights, is just an entertaining pastime, a gigantic myth to which many contribute, or a wonderful game that many play, but which is no more likely to be true than the amusing and terrifying tales of Zeus and Christianity.

Yet, we mustn't be too quick. There is a second argument against the theology example that she might be able to use here. Consider Maddy's remark: “[P]ure mathematics is staggeringly useful, seemingly indispensable to scientific theorizing, but astrology is not” (1998a, 176). At first blush this seems to contradict her rejection of the Quine–Putnam indispensability argument outlined earlier. Not so. Note that this mention of indispensability is not an attempt to justify mathematics inside a scientific naturalist framework. It is a very much weaker claim, perfectly compatible with mathematics having a life of its own. It is an affirmation of utility only, not an affirmation of truth because of utility. The same utility claim that she makes for mathematics could not be made for astrology or theology, since they are simply not useful to science at all.

Even this much, however, is a partial surrender to science naturalism. At the very least, one has to think that the needs of science are genuinely important, otherwise I don't see how Maddy—shunning all non-mathematical considerations—could defend mathematical activity and not theology as legitimate. Inevitably, someone will raise the matter of psychological utility and religion will step forward to fill this need. Religion, unlike algebra, can comfort people in distress. And now we're back where we started. Though it is a small point and should not be overstressed, Maddy's mathematical naturalism is somewhat dependent on natural science after all.

By contrast, philosophy (that is, the very type of consideration Maddy claims is not relevant to mathematics) can distinguish between theology and mathematics without making any appeal to science, not even a relatively weak claim of utility. There are, for example, philosophical arguments for the existence of abstract entities, arguments that do not rely on mathematical considerations but can be marshalled in support of the truth claims of mathematics. And there are also philosophical arguments against the existence of God (e.g., the problem of evil), arguments that do not beg the question against theology at the outset. Of course, these arguments are fallible; at best they offer moderate support for their pro-mathematics, anti-theology conclusions. Nevertheless, they count as independent, non-question-begging, admittedly fallible considerations for the legitimacy of mathematical activity and for the illegitimacy of the results of theological speculation. By Maddy's account, however, they are extra-mathematical. So they violate her brand of mathematical naturalism. This seems to me a great pity, since without extra-mathematical support of one sort or another, mathematics might not get off the ground.

IS MATHEMATICS A SUCCESS?

We can come at the theology problem in a different way, this time focusing on success, but the conclusion will be the same.

Maddy insists that “mathematics has flourished by its own methods” (1998a, 165). The stress in this remark is on “its own methods,” but what about “flourished”? How can we tell that mathematics is indeed flourishing? She declares that any philosophical account must be abandoned if it “comes into conflict with successful mathematical practice” (1997, 161). Detecting conflict with mathematical practice is relatively easy, but what about detecting the success of mathematics?

The analogous questions for natural science can be easily answered (though even here there is controversy). We have empirical evidence for scientific theories. For example, a theory is successful when it makes empirical predictions that would be otherwise unexpected, except in the light of the theory itself. This is a clear, though fallible, sign of success, which is more or less independent. Unfortunately, we have no similar relatively independent check in mathematics (if we reject Platonism). Maddy, as we have seen, quite rightly rejects the Quine–Putnam mixing of mathematics and science, so that route is not one she can exploit. A Platonist (as I mentioned earlier) does have a way out; our mathematical theories can be tested by our mathematical intuitions, a kind of perception or intellectual grasp of the realm of abstract entities, which is more or less independent. Maddy's brand of naturalism will have none of this. Similar things could be said about constructivism, or formalism. They have different answers to the problem of success, but like Platonism, they, too, are unacceptable to naturalists.

There is simply no way to know if mathematics is successful, not to mention flourishing, if we stick to the very limited kinds of internal considerations that Maddy allows.

SEPARATE REALMS

The possibility of separately justifying various pseudosciences points to another difficulty for Maddy. Naturalizers are unifiers. Rather than a lot of separate sciences, they typically want a united and coherent account of the one and only world they take to exist. Mathematics, mind, morality, and everything else are to be brought under the single umbrella of the natural sciences. Those fields that cannot be incorporated in principle into this unification are declared illegitimate. Of course, the impressiveness of the achieved unification is somewhat mitigated by this definitional edict. Nevertheless, unification is typically one of the aims and occasionally one of the outcomes of naturalism. It is also one of its chief sources of support. Strikingly, Maddy's brand of naturalism does the very opposite; it creates distinct realms. Her insistence on equal status for mathematics and her rejection of imperialist-minded science-only naturalism assert in effect the existence of separate sovereignties and so undermine one of the strongest instincts that most naturalists have.

This in itself is not a bad thing and I'm not complaining. Platonists, after all, also assert the existence of a separate realm of mathematical entities, so they have no principled objection to Maddy's declaration of independence. Equally, of course, Maddy can have no objection along these lines to Platonism.

Quine (as we saw earlier), is somewhere between these polar views, a semi-unifier. Ontologically, he is with the Platonists: Mathematical entities are abstract objects existing outside of space and time. The crux of Quine's naturalism is his insistent empiricist epistemology: Mathematics gains whatever evidential support it can from its tie to empirical science; he gives no credence to mathematical intuitions. The way to justify mathematical claims, according to Quine, is to treat them on a par with the theoretical conjectures of science and to confirm the whole package by its observable consequences.

Theories, Quine allows, go well beyond what we can observe, but they are tested by their predictions that we can check against experience. Newton's theory is supported by getting its observable predictions right. Quine's mathematical naturalism amounts to the further claim that the mathematics used in making the predictions is also being tested in this process. And, moreover, this is the only way to test mathematics, according to him. There may be Platonic entities, but there are no direct perceptions or intuitions of them. Empirical data are the only sources of knowledge, according to Quine, and this is equally true for mathematical knowledge.

Quine has repeatedly used this indispensability argument to justify mathematical realism. But how much of mathematics is true? As we have remarked in previous chapters, the amount that science needs is rather modest. In fact, it is often claimed, science could probably get by with only the rational numbers; irrationals are not strictly necessary. But if Quine's indispensability argument can't justify the reals, then it would seem that they are nothing more than idle fictions. This is a serious difficulty, since most people—Maddy included—would rather jettison Quine than large chunks of classical mathematics, however “useless” they may be. Indeed, she rejects this whole approach.

Over the years Quine has liberalized his view. This liberalization was briefly discussed in the preceding chapters. Quine now allows additional bits of mathematics that would “round out” the parts that are necessary for science. Since the real numbers do this in a rather obvious and straightforward way, the reals are subsequently justified on Quine's view. So we can confidently say that in addition to the rational numbers, such things as √2, π, e, and all the other irrational and transcendental numbers exist, as do standard functions defined upon them. The views of Quine have come up before and will come up again in the following, but now is a good time to have a look at one of Maddy's rather specific objections to Quine's naturalism, since it centres on indispensability.

It's one thing for Quine, on the basis of his naturalism, to reject the higher reaches of esoteric set theory. But if the traditional core of mathematics can't be justified on his account, then for Maddy (and for a great many others) that failure must count as a reductio ad absurdum of his whole view. For a test case, Maddy cites the continuum of real numbers and Quine's attempt to legitimize it. Though I'm completely sympathetic with her rejection of Quine's approach, I'm not persuaded by her arguments for dismissal.

Quine's justification of any branch of mathematics stems from its indispensability to science. Thus, basic arithmetic is justified, but so-called large cardinals aren't. What about the continuum of real numbers? Though reluctant, many might be prepared to go along with this demarcation. But they certainly wouldn't, if the real numbers fell on the wrong side. Maddy argues that this is unfortunately so.

She claims that every use of the continuum in science involves assumptions that are known to be false. We treat a fluid as if it were continuous, but it's really a collection of very small atoms. Field theory (both classical and quantum) has problems with point charges that lead to divergent series when self-energy is calculated, so these are, strictly speaking, inconsistent theories. After surveying a number of candidates, Maddy concludes that the continuum is used only in idealizations or in inconsistent theories. In neither case are the theories true. Consequently, she says, Quine's approach can't justify this core bit of mathematics—the continuum is not indispensable to true science.

It's clear from considerations such as these that truth, not indispensability, is at issue. Quine's argument, once again, runs: Mathematics is indispensable for science; science is true; therefore, mathematics is true. The indispensability of the continuum has not been undermined by Maddy's examples at all, only (perhaps) the second premiss claiming that science is true. Quine could simply add: Idealization is an essential part of science and the mathematical continuum is indispensable for some of those idealizations. I don't say this argument (even when fleshed out) is a good one, but neither is Quine's original.² I only maintain that it is as persuasive as the original and in the same spirit. However, this is still not the main issue.

On Maddy's analysis, Quine could not hold that the continuum is essential for true science. But he needn't. Quine also holds that the mathematics that is directly justified by his indispensability argument can be fleshed out into a coherent whole. We can fill in the natural numbers with a zero; we can “complete” the rational numbers with irrationals; we can introduce √–1 when we try to solve the equation x² =–1 which leads to the introduction of the complex numbers, and so on. For instance, Quine's liberalized indispensability view would justify the rationals via fairly direct physical measurements. Then the real and complex numbers, etc., would be justified by gap-filling and natural extensions. Quine could then use his argument scheme to claim that the theory of the continuum and almost all of standard analysis are justified and likely to be true. The fact that the continuum is used in false idealizations might provide an additional argument for Quine, but its failure (if Maddy is right) does not undermine his chief argument.

Moreover, Maddy may have been too quick to conclude that all uses of the continuum are either tied directly to idealizations or to inconsistent theories. Ordinary quantum mechanics does not suffer from the infinities that plague quantum field theory. It uses continuous mathematics, but at a higher level of abstraction. If a measurement outcome could have the value a or b with a probability of for each, then the state of the system is . To find the probability of an outcome with eigenvalue a, we square the so-called probability amplitude of the coefficient of the corresponding eigenstate, i.e., (1/√2)² = . There is no getting away from the irrational number √2, even though it is very far from anything directly empirical. Similar things might be said about the extensive use of π or e. They seem quite essential to true science, not just to fictitious idealizations.

The problem with Maddy's argument seems to be this: She has only considered the continuum of physical space and time. But the continuum is also used in many other places, e.g., phase space, Hilbert space, Lie groups, continuous probability, and so on. These representational spaces are quite different from the more immediate physical space and time that we live in and experience. Perhaps physical space and time are not genuine continua, but this need not imply that all our various spaces of representation also fail to be continua. Mathematics serves the needs of science in strikingly different ways, at strikingly different levels of abstraction.

SITUATING THE PROBLEM

The crux of Maddy's naturalism is the claim that it is mathematical practice that matters to mathematics, not philosophy or anything else. As she puts it: “[M]athematical methodology is properly assessed and evaluated, defended or criticized, on mathematical, not philosophical (or any other extra-mathematical) grounds” (1998a, 164). We could raise complaints concerning the level at which we need to focus attention. It may be the case that we wrongly thought the explicit philosophical beliefs of mathematicians played a role in the practice of mathematics. Nevertheless, we can still ask philosophical questions about why mathematics works in the way it does. And the best explanation might come from a philosophical analysis such as that given by Platonism, or constructivism, or some other account. Whether mathematicians believe that particular explanation or not may be as irrelevant to their mathematical activity as my beliefs about my liver are to my liver's actual functioning.

The views of working mathematicians should be taken into account, of course. But, for a number of reasons, they are not decisive. Overwhelmingly working scientists today are scientific realists. That shouldn't (and didn't) stop van Fraassen or other anti-realists from thinking those scientists are wrong. Perhaps Maddy is correct to think philosophy as a matter of current fact plays no role in mathematicians’ deliberations. But that shouldn't (and won't) stop us from thinking it possible that philosophy is and ought to be central, nevertheless. Thus, we might, for instance, appeal to Platonism (or conventionalism or constructivism or whatever) in the explanation of mathematics. In the philosophy of science there is a popular “success of science” argument that might be used in modified form as follows: Mathematical practice is highly successful. This could be a mere coincidence, in which case the success of mathematics is a bit of remarkable good luck. Or it could be because mathematical practice is based on the intellectual grasp of underlying objective mathematical truths, in which case the success of mathematical practice is to be expected. Since the latter is the more plausible explanation, it is reasonable to conclude we actually know some mathematical truths.

Another argument (stemming from Frege) runs: Singular terms in true sentences refer. The sentence “John loves Mary” could not be true unless “John” refers to someone. “2” is a singular term in the true sentence “2 is an even prime,” so the term “2” refers.

If either of these are good arguments, then they establish realism. But the actual philosophical beliefs of working mathematicians are irrelevant to these arguments and arguably play no role in mathematical research. Some form of realism or Platonism must be right (according to these arguments), in spite of what working mathematicians explicitly think. Maddy has fixed on the (possible) ineffectual nature of the philosophical beliefs of working mathematicians to such an extent that she neglects other important questions that arise in a different way and that may well have philosophical answers, and so fly in the face of her brand of naturalism.

As I stressed before, Maddy is quite aware of this. She remarks, “Philosophy follows afterwards, as an attempt to understand the practice, not to justify or criticize it” (2000, 415). In other words, we can have a philosophical understanding of what mathematicians are doing, but that understanding plays no role in the practice of mathematics and plays no role in explaining why mathematicians do what they do. This point sets her apart from other naturalists, since she allows (at least in principle) that the correct after-the-fact account might be, say, Gödelian Platonism or some other highly anti-naturalistic view that posits abstract entities, intuitions, and other unnatural things. She merely wants to keep this out of the account of mathematical practice, and her position sounds perfectly reasonable. After all, there are some processes where beliefs about the process affect the process itself (e.g., beliefs about French grammar affect how French is spoken, beliefs about the effectiveness of ASA help alleviate headaches) and there are other processes where beliefs about the process have no effect whatsoever (e.g., beliefs about how black holes are formed have no effect on black holes themselves and flowers will bloom in certain conditions no matter what botanists believe). Maddy's naturalism purports to be akin to the second of these. Philosophical beliefs (whether true or false) have no more affect on mathematical practice, she would claim, than our beliefs about Cygnus X-1 will have on the amount of radiation emitted by Cygnus X-1.

There is a problem with this view, and it is serious. The distinction between our beliefs about black holes and the black holes themselves is perfectly natural. One involves an intellectual process and the other does not. The distinction is not so easy to defend when the process and the beliefs about the process both involve thinking, highly related thinking at that. Suppose, for example, that Gödel's views were to be correct as an after-the-fact account of how mathematics works. His account posits a Platonic realm of sets; it says that mathematicians have intuitions of some of these, and that they conjecture new axioms to try to account for and to systematize some of these intuitions, etc. It is an after-the-fact account of the process of mathematics in that he is trying to explain what happened. Let us suppose with Maddy that these philosophical beliefs have no effect on the mathematical process itself.

We now have a significant puzzle on our hands. If Maddy is right, it becomes a complete mystery why these facts—noticed only later by philosophers and historians—about the intuitions of independently existing mathematical entities played no conscious role in the actual making of mathematics in the first place. Is it even remotely plausible that working mathematicians didn't have so much as a glimmer? Imagine the parallel situation is science. It would be like philosophers arriving on the scene after some scientific revolution has come and gone, giving an account of the change of belief in terms of observation and inductive inference, and yet the participating scientists themselves had no inkling that this is what was happening. Could we imagine scientists being shocked to learn this? Could we imagine them sincerely saying: “We had no idea we were observing anything; we simply never noticed that this is what we were doing.” A tennis player could reasonably say “I didn't realize I was bending my arm when making that shot.” But the player could not with any credibility say “I didn't realize I was trying to hit the ball.”

An event's explanation has different types of after-the-fact accounts. Intellectual activity might be best explained in terms of neural hardwiring that arose in the evolutionary process. A person's mathematical or scientific activity could be explained this way without falling into any incoherence. However, if “reasons” are the causes of belief, then the agent must in some crucial respect be aware of the reasons. You can be utterly unaware of your brain but not wholly detached from your thoughts. This means that a Platonic account that appeals to intuitions of abstract entities is simply incompatible with Maddy's naturalism. She would like this to be an open question: Platonism is irrelevant to mathematical activity in the sense that it does not guide it, but it still could be the right account of what happens. Though she wants her naturalism to leave it an open question, I fear this possibility is hopeless. Platonism and Maddy's new brand of naturalism are simply incompatible. To see this, I'll turn to a similar debate that arises in the philosophy of science.

Some philosophers of science—I include myself (2001, ch. 6)—cite reason and evidence to explain the theory choices made by scientists. In contrast, some sociologists of scientific knowledge, including self-described naturalists such as David Bloor (1976/1991), cite interests and other non-cognitive factors in the explanations they propose, and they deny that anything like the philosopher's “reasons” (which have objective normative force) could be playing a causal role in a scientist's choice of theory. Reasons, on their view, are dismissed as mythical entities, or else they are interpreted in a sociological way that robs them of any objective normative force. There is no room for compromise here. Either the world has genuine norms which play a causal role or it does not.

Some, however, have tried to have it both ways. Michael Friedman, for example, is highly critical of Boor's sociology of science, and especially of his using philosophers such as Wittgenstein to support various relativistic theories of knowledge. When it comes to the role played by norms, he wants to adopt a middle view. In response to Bloor's banishing of all non-naturalistic notions of reason, Friedman writes:

But this line of thought rests on a misunderstanding. All that is necessary to stop such an “intrusion” of reason is mere abstinence from normative or prescriptive considerations. We can simply describe the wealth of beliefs, arguments, deliberations, and negotiations that are actually at work in scientific practice, as Bloor says above, “without regard to whether the beliefs are true or the inferences rational.” In this way, we can seek to explain why scientific beliefs are in fact accepted without considering whether they are, at the same time, rationally or justifiably accepted. And, in such a descriptive, purely naturalistic enterprise, there is precisely enough room for sociological explanations of why certain scientific beliefs are in fact accepted as the empirical material permits. Whether or not philosophers succeed in fashioning a normative or prescriptive lens through which to view these very same beliefs, arguments, deliberations, and so on, is entirely irrelevant. In this sense, there is simply no possibility of conflict or competition between “non-natural”, philosophical investigations of reason, on the one hand, and descriptive, empirical sociology of scientific knowledge, on the other. (1998, 245; my italics)

Somewhat surprisingly, Friedman seems entirely on Bloor's side when it comes to explaining the actual events of history—it is irrelevant whether Newton or Maxwell or Darwin actually had good reasons to believe what they believed. What caused them to adopt their particular scientific beliefs is the type of natural, non-normative factor that Bloor and other sociologists of knowledge cheerfully embrace. Friedman and Bloor see eye-to-eye on what has actually happened and the causes of it. Friedman only differs from Bloor in saying that we can also correctly describe certain beliefs as rational. Good reasons exist, according to Friedman, but they play no causal role in actual history. This is more or less the same view that Maddy champions when she allows that Platonism (or any other philosophical view) might come along after the fact and offer its account, but it plays no role itself in those events, anywhere in the development of mathematics.

I see two significant problems. If there are going to be such things as evidence and good reasons (in the strongly normative sense), then it is something of a mystery why they should play no causal role themselves in the history of belief. If they are totally disconnected from everything else, then Bloor is surely right to dismiss them entirely. Otherwise, it's a bit like saying there really are unicorns, but that unicorns are completely undetectable entities that do not causally interact in any way at all with other objects. It's logically possible that there are such unicorns, but good sense and Occam's razor tell us to disbelieve in their existence.

A second problem is related to the first. If pressed, Friedman and other champions of this outlook might resort to pointing out some embarrassing facts. They will note how bizarre Newton, for example, really was. He held all sorts of kooky beliefs on all sorts of subjects, especially in religion and alchemy. It's hard to believe that he suddenly became a paragon of rationality when doing mechanics. Real life is messy and complex; all sorts of factors would have been at play in Newton's thinking. It's irrelevant whether or not he was “rational,” says this line of thought. What matters to us is that we can reconstruct (rationally) the historical episode and we can see in retrospect that Newton's theory (i.e., the laws of mechanics and the gravitation law) was warranted in those circumstances.

But is our belief that Newton's theory was warranted itself warranted? Is our belief about Newton just another event brought about by various natural causes, as Bloor would maintain? Or is our belief itself a rational belief, caused by the evidence that is available to us? If the former, then Friedman's thought that we current philosophers and historians can get a grip on norms (even though they play no role in the causal story of the past) is simply wrong. But if it is the latter, then we are capable of having our beliefs be caused by evidence. And that means that Newton is capable, too. In consequence, reason can indeed be part of the causal story, contrary to Bloor and to Friedman. There is more to understanding the history of scientific beliefs than merely recording the totality of natural causes. There are non-natural causes of belief, too, namely, reason and evidence.

To think otherwise is to fall into a hopeless muddle. How is it that we later-day philosophers and historians are able to see something that Newton and Darwin and so many others couldn't? We are characterizing ourselves as clear-headed truth-seers while Newton was a helpless reed blowing in the social and psychological winds of his day, unable to distinguish real evidence from dishwater. This can't be right. If we can see the evidence available to Newton, and see what is rightly concluded from it, then surely Newton could have drawn those same consequences. And isn't this the best explanation of what Newton actually did?

At one point Friedman remarks: “It is not that the philosophical tradition sets up a competing model for causally explaining the actual historical evolution of science” (1998, 250–251). These same words could have been spoken by Maddy. I would say to both that, on the contrary, this is precisely what we should be trying to do. Contrary to Bloor and Friedman, reason and evidence were the actual cause of Newton's beliefs about gravitation. Reason and evidence aren't just part of an after-the-fact rational reconstruction that had nothing to do with the historical events themselves. Similarly, Platonic intuitions play a causal role in mathematical activity; they are part of the actual process. It's not just an ingredient in an after-the-fact account of what happened in the history of mathematics.

Let me quickly concede that this argument does not show that Maddy is completely wrong about her naturalism. It only shows that she can't be indifferent to Platonism, as she claims she wants to be. She must reject it as incompatible with her naturalism. And she must reject several other accounts (e.g., constructivism) which also operate at a conscious level in the minds of mathematicians. Conversely, to uphold the mere possibility of any of these is to reject Maddy's brand of naturalism.

Mathematicians’ philosophical beliefs do matter, but mathematicians are often confused about what they actually believe. Working scientists often give accounts of their own activities that are well off the mark, too. For example, they often claim that they make careful observations in the lab and that this is done without adopting any assumptions that might bias those observations. And they further claim that from these data theories are carefully derived in such a way that their theories are proven by these data. No one who has looked closely at the history of science believes that for a moment. Newton famously said, “I frame no hypotheses,” but his work is chock-full of them. So when I claim that Platonism is at work, I don't mean to say that mathematicians are all clear-headed, self-conscious Platonists. More likely it is an underlying realism or Platonism that is implicitly adopted by working mathematicians that is doing the crucial work in the development of their mathematics.

There are also degrees of conscious awareness. I'm much more guided by explicit grammar rules in French than I am in my mother tongue, English. But rules of English grammar nevertheless play a role in my English, even though I'd be very hard-pressed to say what those rules are. Most mathematicians are realists, even if they back off calling themselves Platonists outright. This fact is not merely relevant in an after-the-fact account of what happened. Their quasi-conscious Platonism plays a role in mathematics in the making. And, as we shall see, it may also be the principal reason for the rejection of some proposed new axioms, such as V = L.

NEW AXIOMS

Maddy argues for her brand of naturalism largely with a single, well-developed case study—the status of the Axiom of Constructibility. It is time to explain the axiom.

After six days of creating the universe, God may have tired. But set theorists didn't stop working after Zermelo's initial axiomatization in 1908. Frankel added more axioms in the 1920s, and even more are thought to be needed now. Interesting questions, such as the CH, can't be answered on the basis of existing axioms, so the search for additional ones is under way in the hope that new light can be shed on this and other old questions.

The so-called Axiom of Constructibility is one candidate for adoption. It simply says: V = L, or in other words, the universe, V, consists exactly of the constructable sets, L. (Neither V nor L is itself a set; both are so-called proper classes.) L is not the cumulative hierarchy, but is obtained by modifying it. The axiom and the notion of a constructable set were first put forward by Gödel in his characterization of a model of set theory that proved the relative consistency of the CH.

The cumulative hierarchy, which motivates much of set theory, starts with the empty set, ø, at the bottom stage, then any given higher stage consists of sets created by all possible operations performed on the sets of the lower stage. All the possibilities are generated by the power set operation, so we have at stage V_±+1 all the sets in V_α. L is more restricted than this, though it, too, is built up in stages. Instead of a stage, L_α, containing all subsets of the preceding stage, it only contains those that are describable in the language of set theory. The notion of an arbitrary set, however, outruns the linguistic resources of set theory; so by insisting that only describable sets can be admitted into the hierarchy, we significantly restrict L. (It should be stressed that the sense in which the members of L are “constructable” differs considerably from the much more limited idea associated with Brouwer and other so-called constructive mathematicians.)

Here are two of the most important results about the Axiom of Constructibility (ZFC stands for the usual axioms of Zermelo-Frankel set theory including the axiom of choice):

1. ZFC + V = L is relatively consistent (i.e., if ZFC is consistent then so is ZFC + V = L).³
2. ZFC + V = L implies the generalized CH, GCH (i.e., for every ordinal number α, 2_α = _α+1, whereas GCH is independent of ZFC alone.

Another candidate axiom is the Axiom of Measurable Cardinals, MC. To briefly explain, we first need to describe the notion of an inaccessible cardinal. Consider a set of ordinals, . The cardinal numbers of the members of S are , . . . The cardinality κ of the union of S, (i.e., ), is greater than the cardinality of each of the members of S. The operations of set theory, however, will not produce a set of cardinality κ from sets of smaller cardinality (unless some operation is repeated at least κ times). Thus, the cardinal number κ is inaccessible.⁴ (Analogy: Given the finite numbers and the fact that we are allowed to perform any finite number of operations on them such as addition, multiplication, and so on, we would still not be able to produce an infinite number.) Do inaccessible cardinals exist? So-called “large cardinal axioms” assert that they do. One of the most popular candidates is MC, the axiom of measurable cardinals.

A measure on a set S is a function µ:S→[0,1]. (It is convenient to make the measure on the whole set equal to 1, which we will do; there is no loss of generality in doing so.) µ must satisfy such properties as µ(ø) = 0; µ(S) = 1; if X ⊆ Y then µ(X) ≤ μ(Y); µ({a}) = 0; if X and Y are disjoint then µ(X Y) = µ(X) + µ(Y); and more generally, the measure on an infinite collection of mutually disjoint subsets of S equals the infinite sum of the individual measures on those subsets. The idea stems from Lebesgue, and is an attempt to generalize his work so that even very complicated sets of real numbers would have a measure. A measurable cardinal is a cardinal number that is greater than ₀ and admits a measure. Do any exist? The axiom of measurable cardinals (MC) says Yes. Here are some key facts:

1. MC implies V ≠ L.
2. If κ is a measurable cardinal then κ is inaccessible.
3. MC is independent of ZFC.
4. MC is compatible with CH and also with its negation.

Now comes the hard mathematical question: Should we accept either of these axioms, MC or V = L? Maddy approaches this question from both realist and naturalist perspectives. Most set theorists—but certainly not all—reject V = L. On the other hand, there is widespread acceptance of MC. Next comes the even harder philosophical question: Are set theorists rejecting (or accepting) V = L because of their philosophical beliefs concerning mathematical realism, or because of their non-philosophical beliefs about mathematical practice? Maddy, the naturalist, answers: No to realism, yes to practice. Let's look at the case for each.

REALISM AND NEW AXIOMS

As I mentioned already, not everyone rejects V = L. Quine's argument for the axiom is interesting and it bears on the main problems with Maddy's naturalism. Quine, as we have already seen at length, is a mathematical realist, but his epistemology of mathematics is a version of empiricism— he gives no credence to mathematical intuitions. To repeat again what I outlined earlier, the way to justify mathematical claims, according to Quine, is to treat them on a par with the theoretical conjectures of science and to confirm them by their true observable consequences. Some parts of mathematics cannot be tested even in this indirect empirical way, but, says Quine, they might be justified as a natural rounding out of what is already acceptable. This smoothing out justification has its limits, however. Quine draws the line well short of the full realm of possibilities. As he sees it, ZFC + V = L is exactly right. (Strictly, ZF + V = L is sufficient, since that implies CH.)

[T]he continuum hypothesis and the axiom of choice, which are independent [of ZF], can still be submitted to the considerations of simplicity, economy, and naturalness that contribute to the moulding of scientific theories generally. Such considerations support Gödel's axiom of construcability, V = L. It inactivates the more gratuitous flights of higher set theory, and incidentally it implies the axiom of choice and the continuum hypothesis. (Quine 1990, 95)

In a nutshell, ZFC + V = L should be accepted as true, says Quine, because, even though less than that is strictly speaking justified by the needs of science, a rather natural rounding out of the essential minimum gives us exactly ZFC + V = L. However, he claims further, no more than this can be justified; any additional mathematics that is independent of this is simply false, or rather, forever unjustified and unknowable, at best an idle game. The somewhat derogatory term “recreational mathematics” is often used.

In accepting the Axiom of Constructibility Quine is not in abundant company, though he is joined by Devlin (1977). Most set theorists reject V = L ; the universe of sets, they think, is very much richer than the constructable sets. Gödel is one of the leading proponents of this view, even though he was the first to formulate the Axiom of Constructibility. Maddy cites the views of several prominent set theorists:

Moschovakis : The key argument against accepting V = L  . . . is that the axiom of construcability appears to restrict unduly the notion of arbitrary set. (Quoted in Maddy 1997, 84)

Drake : Most set theorists regard [V = L ] as a restriction which may prevent one from taking every subset at each stage, and so reject it (this includes Gödel, who named it). (Ibid.)

Scott : Beautiful as they are, [Gödel's] so-called constructable sets are very special being almost minimal in satisfying formal axioms in a first-order language. They just do not capture the notion of set in general (and they were not meant to). (Ibid.)

Maddy describes the evolving concept of function as analogous to the axiom V = L. Early notions conceived of a function as a rule, a formula that could be explicitly specified. Gradually the notion was liberalized into its present form, a completely arbitrary association. Anything less than this would be considered highly restrictive and unjustified. Maddy is surely right to forge the link; the historical attitude to any restrictive notion of function and the current attitude to V = L are strikingly similar. The prescriptions are similar, too. There should be no constraints on either: We rightly want arbitrary functions and we rightly want arbitrary sets. Insisting on V = L is like insisting that a function must be explicitly definable—an intolerable constraint.

It might also help to draw a different kind of analogy, this time with modal realism. David Lewis (1986) once argued that the notion of a possible world cannot be conceived completely in linguistic terms. The idea behind the linguistic account is simply this: Possible worlds are not actual places with real flesh-and-blood beings in them; they are merely stories, nothing more. A possible world on this account is naturally defined as a consistent set of sentences. Lewis, on the other hand, thinks they are just as objectively real as our world. A possible world is a way things could be, and Lewis contends that the notion “a way things could be” has to outrun any linguistic resources we might have. Things, after all, might be so bizarre that they are indescribable by our linguistic resources. In fact, it is rather obvious that “indescribable by our linguistic resources” is itself a way things could be, and, therefore, a possible world that cannot be identified with any consistent set of sentences. A possible world, consequently, has to be very much more than some sort of mere linguistic entity. As Hamlet might have put it, “There are more things in heaven and earth, Horatio, than are expressible in your philosophy.”

In spirit at least, insisting on V = L is similar to insisting on the linguistic account of possible worlds, since the universe of sets, according to this axiom, is to be populated exclusively by sets that are constructed using the linguistic resources of set theory. The objection to both is the same: It seems rather obvious that many more things exist than our language allows us to describe. Unlike V = L, the axiom MC does not have this restriction about it.

Quine's argument for V = L is based on epistemic considerations, while Lewis's argument is based on the poverty of linguistic approaches and is a case of pure metaphysics. These sorts of arguments—both closely associated with debates about realism—are philosophical, and thus the very things that Maddy claims not to have genuine roles in the actual development of mathematics. Of course, mathematicians tend not to make arguments like these, at least not explicitly. But they might “sense” them, nevertheless, and somehow feel their force, if only implicitly.

NATURALISM, NORMS, AND NEW AXIOMS

When Maddy dismisses philosophy from playing a role in mathematical practice in her brand of naturalism, it is ontology and epistemology that are banished.

[M]athematical practice itself gives us little ontological guidance . . . the methods of mathematics . . . tell us no more than that certain mathematical objects exist. They tell us nothing about the nature of that existence—is it objective? Is it spatiotemporal?—indeed, nothing seems to preclude even Fictionalist or Formalist interpretations. And what goes for ontology goes for epistemology: no part of mathematical practice tells of human cognizers and their acquisition of mathematical beliefs. (1997, 192)

Ontology and epistemology are eliminated as irrelevant. Methodology, however, stays. For some reason Maddy doesn't consider this part of philosophy, though many others would. She calls it practice. But even if methodology is part of philosophy, the truly important thing is her claim that mathematical methodology (or practice) is independent of ontology and epistemology. Platonists typically believe that the mathematical realm is as full as it could consistently be.⁵ We might call this the more, the merrier principle and note that it determines a preference for one type of axiom, MC, over another, V = L. Maddy's complete separation of mathematical methodology from ontology and epistemology would be, if correct, a rather significant philosophical claim.

Specifically, the two methodological practices that Maddy finds in the history of mathematics are maximizing and unifying. “If mathematics is to be allowed to expand freely . . . and if set theory is to play the hoped-for foundational role, then set theory should not impose any limitations of its own: the set theoretic arena in which mathematics is to be modelled should be as generous as possible . . . Thus, the goal of founding mathematics without encumbering it generates the methodological admonition to MAXIMIZE” (1997, 210–211).

Notice that the injunction to maximize stems from the desire to provide foundations for other theories. Even if the other theories are mathematical theories, this is still a departure in spirit from Maddy's claim to look for internal sources for explaining the course of mathematics. Set theory is not accounted for in its own set theoretic terms, but rather in terms of the needs of other theories (albeit other mathematical theories). Perhaps mathematics as a whole can be accounted for completely in mathematical terms, but it does not follow that a part of mathematics, namely, set theory, can be accounted for completely in set theory terms. Her whole discussion, however, proceeds implicitly as if it can. This may seem a rather artificial, insignificant point, but let us not overlook it completely. It will arise when mathematicians want to develop things in different directions. We'll see an example of this in the following, when we discuss the CH once more.

Notice also the form of her injunction: “If mathematics is to be allowed to expand freely . . . and if set theory is to play the hoped-for foundational role, then set theory should not impose any limitations of its own.” I want to repeat something I stressed at the outset. Maddy, the naturalist, allows no categorical norms. Her injunction is a hypothetical imperative: If we want x, then we ought to do y. The grounds for this being that we have tried to achieve x and we have learned (by means acceptable to a naturalist) that y is a reliable way to go about this. A categorical norm has the form: We ought to try to achieve x. There are no naturalist grounds for this or any other categorical norm. In particular, there are no naturalist grounds for asserting: Mathematics should expand freely, or (to use her term), We should MAXIMIZE. Maddy can correctly report that mathematicians act so as to expand mathematics freely, but she cannot justify their actions. A Platonist, however, can. Metaphysical arguments and epistemic considerations lead explicitly to this norm. In sketchy form it might run: There are many more sets than current set theory countenances; since we want the complete truth, we must expand mathematics to do justice to the whole realm of set theory; V = L violates this, so it must be rejected. Life without norms is well-nigh impossible; so much the worse for naturalism, in consequence. Maddy, as I said, skirts this problem, since she only urges a hypothetical norm. But is it the right hypothetical norm?

RANDOM VARIABLES AND THE CONTINUUM HYPOTHESIS

One of the more striking developments in recent mathematics is the use of probabilistic arguments. This has been especially true in combinatorial branches of mathematics such as graph theory, but the potential is much greater and could even be quite revolutionary. Given Maddy's attitude to means–ends relationships and especially her principle MAXIMIZE, she is likely to endorse probabilistic proofs (at least in principle) and want to see room made for these methods in the foundations of mathematics. This may have consequences for the CH.

Christopher Freiling (1986), as we saw in Chapter 5, may have refuted CH. He calls his argument “philosophical,” since he does not provide a proof or a counterexample in the normal mathematical way. Recall, briefly, how it goes: Imagine throwing darts at the real line, specifically at the interval [0,1]. Two darts are thrown independently of one another. The point is to select two random numbers. We assume ZFC. If CH is true, then the points on the line can be well ordered and will have length ₁. Thus, for each , the set is countable (where is the well-ordering relation). Suppose the first throw hits point p and the second hits q. We'll assume p q. Thus, . Note that S_q is countable, so the probability of landing on a point in S_q is 0. Yet it will happen every time there is a pair of darts thrown at the real line, which seems absurd. Consequently, we should abandon CH, that is, the assumption that the number of points on the line is the first uncountable cardinal number. If the cardinality of the continuum is ₂ or greater, then there is no problem (as the argument is set out here), since the set of points S_q earlier in the well ordering need not be countable, and so would not automatically lead to a zero probability of hitting a point in it. However, as mentioned earlier, three darts will refute the assumption that it is ₂, and so on.

Freiling's argument is certainly contentious and it's safe to say that the majority of set theorists don't accept it. But, as I mentioned earlier, some do, including David Mumford, who would like to reformulate set theory, in consequence. This is enough to make the example worth considering in relation to Maddy's views.

Mumford would like to see CH tossed out and set theory recast as “stochastic set theory,” as he calls it. The notion of a random variable needs to be included in the fundamentals of the revised theory, not be a notion defined, as it currently is, in measure theory terms. Among other things, he would eliminate the power set axiom. “What mathematics really needs, for each set X, is not the huge set 2^x but the set of sequences X ^N in X ” (Mumford 2000, 208). I won't pursue the details of Mumford's programme, but instead get right to the philosophical point.

In the light of this example, we have two proposals, both of which could claim support from Maddy's methodological principle MAXIMIZE. First, we have standard set theory in search of additional axioms, guided by the desire not to limit in any way the notion of an arbitrary set. On this version of MAXIMIZE the standard axioms remain, V = L is rejected, and various large cardinal axioms are tentatively accepted.

Second, we have Mumford's programme. He can be seen as a maximizer, too. But his focus is on maximizing the range of legitimate proof techniques. In making room for stochastic methods and taking random variables seriously in their own right, he would reformulate set theory so as to pare down the universe of sets to a much smaller size, but there would be new theorems, not otherwise obtainable. This version of MAXIMIZE is also a perfectly legitimate aim by Maddy's lights, though I doubt it is one she anticipated.

How are we to settle this dispute? Clearly, appeal to MAXIMIZE will not help, since both sides could cheerfully embrace it. Freiling, as I said, called his argument “philosophical,” which seems quite reasonable. Why? It involves beliefs about symmetry, randomness, and causal independence that go well beyond mathematics proper and are clearly philosophical in nature. Freiling's approach will likely stand or fall with the correctness of those philosophical assumptions. Remember, Maddy's naturalism excludes not just science and philosophy, but everything non-mathematical from having mathematical influence. Freiling's “philosophical” assumptions may be false, of course, but that is neither here nor there. What counts is that they are the kinds of considerations that matter, at least in principle. And that is enough to undermine Maddy's brand of naturalism. Her principle MAXIMIZE is very powerful. It is also, I think, correct. But it is too amorphous and unwieldy by itself. It needs some sort of guiding hand, and that is where philosophy gets its foot in the door.

Perhaps I am being unfair in characterizing this as a battle between two versions of MAXIMIZE. There is another way to see the Freiling case, but it is similarly contrary to Maddy's naturalism. Start by acknowledging that dart-throwing considerations may lead to new mathematical results (i.e., ~CH). This is compatible with Maddy's claim (which I think is correct), that mathematics is ontologically autonomous; it does not depend on science or anything else for its truth. However, (and this is contrary to Maddy's claims), sources of evidence could come from anywhere, including from outside mathematics proper (e.g., from pictures or from darts). The refutation of CH from dart throwing does not come from the MAXIMIZE principle or from any non-trivial sort of means–ends reasoning. Rather, the aim of this activity is simply the truth about CH. The aim is to settle the CH question itself, not to serve some larger end, such as providing a foundation for all of mathematics. This way of understanding the dart-throwing example is just as damaging to Maddy's naturalism as the previous way of understanding it.

In both of these versions of the dart story, MAXIMIZE is the problem. In the first, we have competing versions of MAXIMIZE and we have to ask which is true. In the second, MAXIMIZE doesn't even enter the picture; we're interested in the simple truth from the start.

Maddy has addressed CH directly, in relation to Woodin's (2001) work, which is like Freiling's in that it aims to settle CH in a non-standard way (both, incidentally, arguing for ~CH). Maddy sees Woodin's approach as fitting her means–ends analysis.

Large cardinals themselves are clearly more congenial to the maximizing objective than V = L, but they're not sufficient to settle the CH. Recently, Woodin has presented an innovative case against CH, innovative because it doesn't take the usual form of proposing and defending a new axiom. Instead, Woodin argues, in rough outline, that it is possible to extend ZFC to a theory with certain ‘good’ properties and that any such extension implies not-CH. A serious second-philosophical investigation of this intriguing idea would involve examining the justification for classifying these specific properties as ’good’, but what's of central interest for present purposes is that Woodin's case is couched entirely in the means–ends terms that the Second Philosopher [a Maddy-style naturalist] believes will properly carry the day in the long run: for example, the ‘good’ properties require the theory to be decisive in certain ways.

In sum, then, the Second Philosopher sees fit to adjudicate the methodological questions of mathematics—what makes for a good definition, an acceptable axiom, a dependable proof technique?—by assessing the effectiveness of the method at issue as means toward the goals of the particular stretch of mathematics involved. (Maddy 2008, 359)

It may be the case that Woodin's approach and Freiling's fit the ongoing means-ends-oriented development of mathematics, though I think both are a bit of a stretch for Maddy. The same, however, cannot be said about Mumford's programme. The methodological principles themselves, such as MAXIMIZE, give no guidance, nor does a deeper analysis of the history of mathematics. The Mumford case calls for something quite new. Philosophy, qua philosophy, must get involved.

Needless to say, mathematical Platonism is reinforced by these sorts of examples. The Freiling example provides us with a happy bonus which I mentioned in the last chapter. It is worth mentioning again, since it illustrates the genuinely conscious role of epistemology in mathematical practice. Clearly, we are not deriving ~CH from already accepted mathematical axioms. Moreover, it is just as clear that the result is not empirical, since we can't pick out real numbers with darts in any but a conceptual sense. There seems only one plausible explanation for how Freiling's argument works: Somewhere along the way a Platonic intuition was at work, and that means trouble for every brand of naturalism.

CONCLUDING REMARKS

In sum, there is much to like about Maddy's naturalistic approach—even for a Platonist. Her insistence on taking mathematics on its own terms rather than reducing it to science is surely right. And her insistent focus on the details of mathematical practice is also something to be highly commended. Platonists can even be mollified to some extent by the fact that so much of what is dear to their hearts remains untouched by Maddy's naturalism. Her claim, after all, is only that Platonism is irrelevant to practice, not that it is false. But problems arise. Some of these stem from the fact that her brand of naturalism is very much less than the full story of mathematics. It is really only a naturalism of mathematical practice. Questions such as “Do mathematical entities exist independently of us?” remain open, as she readily admits. Many realists and anti-realists will cheerfully agree with several of Maddy's main contentions about practice, yet still disagree with each other over the main questions of ontology and epistemology. It is even questionable whether this separation can be consistently upheld. But other realists (and I count myself among them) will take further issue with her on matters of mathematical practice itself. Some mathematics (Freiling's darts and CH) is intelligible only in the light of assumptions which are outside of normal mathematics; something that is anathema to Maddy's brand of naturalism.