6    Semi-Naturalists
and Reluctant Realists

Two characteristics of Platonism are paramount. The ontological ingredient is the claim that mathematical entities exist outside of space and time, independently from us, and mathematical statements are true (or false) in virtue of how things are. This is the realist part of Platonism. The epistemological ingredient is the claim that we can perceive or intuit these mathematical entities (at least some of them), and thereby come to know their properties. There are other sources of knowledge, as well, but intuition is the one that sets Platonism apart from other forms of realism. A semi-naturalist (as I shall use the term here) is someone who accepts Platonism's ontological ingredient, but rejects the epistemic claim. Typically, they are realists, though often reluctant, who resist anything beyond empirical sources of knowledge. The aversion to intuition is usually motivated by naturalistic reasons that are by now quite familiar. A typical argument directed against the epistemic aspects of Platonism runs:

Perception is a physical process involving sound waves, photons, or other natural entities; without such physical interaction perception of any sort is impossible. Sets, numbers, and functions don't emit “platons” or anything else of the sort that could make causal contact with us; hence, there is no way we can perceive them.

There is a further step added to this argument that is accepted by many, but resisted by semi-naturalists. This next step, which is aimed at the ontological aspects of Platonism (and against realism more generally) runs:

Since we can't make contact with abstract entities, they are unknowable in principle. So even if they do exist, we couldn't know anything about them, which means we would have no mathematical knowledge. But, obviously, we do have such knowledge, so the reference of mathematical terms cannot be abstract entities.

Platonism is completely rejected by most naturalists, first epistemically, and then ontologically, as a consequence of this two-step argument. The way semi-naturalists avoid the complete rejection of Platonism is by positing some other way to come to know the properties of abstract entities, a way that is compatible with naturalist principles. The semi-naturalist allows the existence of Platonic objects, but (most emphatically) disallows the usual Platonistic claims about being able to “see,” or “grasp,” or “intuit” them. This brand of naturalism, unlike other versions, allows abstract entities, but insists, like all other versions of naturalism, that our knowledge of the Platonic realm must stem exclusively from natural processes, that is, from ordinary empirical procedures, or at least that it must be compatible with empiricism.

There is a wide variety of philosophers who hold this semi-naturalist view, including: Quine, Maddy, Linsky and Zalta, and (in a qualified sense) Balaguer. Some of the champions of the indispensability argument discussed in the first chapter are probably like-minded, but since they are not so explicit in their claims, I will leave them out of the discussion here. The thing that unites the semi-naturalists is that they all espouse realism in mathematical ontology, but assert some sort of naturalism in epistemology. Of course, they differ in the details, often quite significantly. But this should not obscure the fact that they are surprisingly alike in this one important regard.

QUINE'S NATURALISM

Among the more famous epistemological pronouncements in “Two Dogmas of Empiricism” is this:

The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. (Quine 1951, 42)

W.V. Quine has wavered little from this outlook; a view which as far as mathematical naturalism is concerned contains three important features.

1. We test the complete package of our beliefs (total science), not single beliefs one at a time. (Quine's holism.)
2. This includes mathematics and logic which are part of total science and are tested in the same way as any other part of the whole network. (Mathematical theories are not, for instance, evaluated on the basis of intuitions or by reflection on facts of language, but rather in the same way as the natural science, such as physics or biology.)
3. Total science is tested through sensory experience. (Quine's empiricism.)

Schematically, an instance might look something like this. We start with a theory making an empirical prediction.

1. Physical theory (e.g., Newton's laws of mechanics and law of gravitation)

2. Mathematical theory (e.g., calculus, geometry, trigonometry)

3. Auxiliary theories (e.g., optics, which tells us how light is affected by the atmosphere)

4. Initial conditions (e.g., Mars is at location x at time t

 Predicted observation (e.g., Mars is at location x’ at time t’)

Notice that the premisses include both mathematical theories and physical theories. If the predicted observation conforms to experience, then, according to Quine's holist account, everything that leads to it is confirmed to some extent. Not only is Newton's law of gravitation supported by the correct prediction of the motion of Mars, but so are the theorems of geometry and calculus used to make the successful prediction.

I should quickly reiterate my own view in contrast to Quine. The view I described in the first chapter presented mathematics as a separate realm that we use to model the natural world, not literally describe it. Thus, a mathematical theory would be neither confirmed nor refuted by its successful or unsuccessful employment in science. Readers might also recall from that chapter that aspects of Quine's account (such as his holism), have largely vanished from current debates about indispensability, even though Quine was the founder of those debates. I am returning now to Quine's own views in this chapter, because my concerns are somewhat different than they were when the focus was on explanation. Here the main concern is on the details of epistemology, Quine's and others’. Put another way, there are two things going on in Quine's account of applied mathematics: One is that mathematics must be true since it is indispensible to science; the second is that the evidence of the truth of mathematics is wholly empirical. It is the latter claim that concerns me here.

Just as praise is distributed over the whole of science, in Quine's view, so blame can, in principle, fall anywhere. This is the essence of Quine's holism. If the predicted motion of Mars does not turn out to be observed, we could blame Newton's laws, or the optical theory we had adopted, or, says Quine, we could even blame the (purported) laws of mathematics. “Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. . . . Conversely, by the same token, no statement is immune to revision” (1951, 43). As examples of what he has in mind, he cites non-Euclidean geometry and proposals to modify logic to account for some bizarre features of quantum mechanics. Mathematics and logic are central in our web of belief, but they are not sacrosanct. They could be altered.

This picture of mathematical epistemology fits in rather well with naturalism. Mathematical statements are justified empirically. Testing trigonometric relations or theorems about topological vector spaces is no different than testing theories of protons or polypeptides. We do not experience any of these objects directly—protons and prime numbers are equally theoretical posits on Quine's view. But we do experience some of their consequences. Our knowledge of mathematics is no different than our knowledge of physics or any other science—in every case the evidence is grounded in sensory experience.

What about Quine's ontology? Quine has often expressed nominalistic sentiments, but his considered view is that we must accept the ontology of sets, since they are essential to science, and we would be intellectually dishonest not to own up to these ontological commitments.¹ “Classical mathematics . . . is up to its neck in commitments to an ontology of abstract entities,” says Quine. As for what the notion “commitment” means, he asserts: “A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true” (1948, 13–14).

What does this mean? Take a sentence such as “There are prime numbers greater than one million.” To make it more perspicuous, put into logical notion: x (x is prime & x > 1000000). In this sentence x is a variable which is bound by the existential quantifier. In order for this sentence to be true, says Quine, there must exist a thing to which the x refers. That is, there is some entity p such that p is prime & p > 1000000. When we affirm this sentence we are committing ourselves to the existence of such a thing. In this case we're committed to the existence of a prime number p, which is clearly an abstract entity.

Quine willingly accepts that there are abstract entities, objects that exist outside of space and time, and our mathematical statements, when true, are true because they correctly describe these things. This makes him a Platonist in the ontological sense, that is, a mathematical realist. But, according to him, we do not learn about mathematical objects through an intellectual grasp or an intuition. Rather we learn about abstract entities in a wholly mundane fashion, by ordinary sensory experience of midsize material objects. This amounts to the rejection of Platonist epistemology (intuitions), in favour of a naturalist (empiricist) account of knowledge.

What about Quine's criterion of ontological commitment? Is it right? We can happily enjoy Homer's stories of the gods on Olympus or contemplate the caloric theory of heat. These theories (if we may be allowed to call Homer's tale a “theory”) are, respectively, committed to the existence of supernatural gods and to caloric, but we ourselves are not so committed. Why not? Simply because we do not affirm these theories to be true. We explicitly treat them as fictions. Could we do mathematics in the same spirit? Could we treat it as a mere game? No, says Quine, for the simple reason that mathematics is essential for science.

That's not very convincing, since I could model someone's behaviour on King Lear and even make successful predictions about that person's future actions, even though there is no King Lear. Nevertheless, this has become the basis of one of the more famous considerations in the philosophy of mathematics, the “indispensability argument.” (This was the topic of Chapter 1, but here, to repeat, my concerns are different.) The connection to naturalism is intimate. Though the argument can be expressed in a number of different ways, the following is typical:

1. Naturalism: Contemporary theories in the natural sciences are largely true (or at least are our best hope for knowledge). The method of those theories is empirical, and they provide the one and only reliable way to acquire knowledge.

2. Indispensability: Mathematics is indispensable for science; that is, scientific theories make essential use of mathematical principles and refer to mathematical objects.

3. Holism: Theories are not tested one proposition at a time, but rather as a whole; that is, the mathematical principles are evaluated along with physical principles.

Mathematical realism: Mathematical theories are true and mathematical terms refer.

Needless to say, the premisses of this argument are all controversial. For instance, why claim that the methods of contemporary science are clearly empirical? What about conceptual analysis? What about thought experiments? What about computer simulation? If values play a role in science, what is their epistemic status? Perhaps “indispensability” does imply truth, but what is the argument for this? Finally, Quine's holism is largely discarded (as I mentioned before); current champions of indispensability do not use it but prefer something like “inference to the best explanation.” However, we need not concern ourselves with these considerations. In any case, the conclusion of the Quine-type argument is understood to capture quite a bit: Mathematics is a body of objective knowledge; it is true; it is discovered rather than invented; numerals are the names of independently existing numbers; and so on.

Quine published his famous paper “Epistemology Naturalized” in 1969, many years after he first advocated his semi-naturalistic account of mathematics. A great deal of the current interest in naturalism stems from this paper. Much of what Quine calls naturalism is actually consistent with or even an extension of his earlier outlook. The views which are of a piece with it include:

• Science is based on experience, but it is science itself that tells us what an observation sentence is (anti–a priori philosophy).
• There are no perceptions of abstract, non-physical entities (anti-Platonistic epistemology).
• There are no first-person mental constructions (anti-Brouwer's constructivist philosophy of mathematics).
• There is no analytic/synthetic distinction; all propositions are synthetic (anti–a priori philosophy).
• Propositions pick up their meaning by being part of a complex web of belief (meaning holism).
• Propositions pick up their evidential support by being part of a complex web (epistemic holism).
• A web of belief is all encompassing (contra those who accept a limited holism but say there are relatively independent webs, one for science, another for mathematics, etc. This point shouldn't be confused with a different claim: Another culture might have a different, rival web. Theirs, however different it may be from ours, would still be all encompassing for them.)

In “Epistemology Naturalized,” Quine notes that some past epistemologists (Russell and Carnap, for example) have tried to account for, and to justify, our knowledge of the world with a combination of sense-data, logic, and set theory. Theirs was a programme of rational reconstruction—but it was a failure. There is, Quine claims, no first philosophy, no solid foundation from which to build. We are left with scepticism: “The Humean condition is the human condition.” An alternative approach is needed: We should use science to study philosophy. In particular we should turn to psychology in the study of the formation of belief. Epistemology, thus reconceived, becomes a branch of natural science. One of the consequences is that epistemology is done in the third person, not the first à la Descartes.

But how can we be sure this is good science? Isn't the process going to be circular? Yes, but Quine embraces it. At least it's not a vicious circle. The crucial thing is that it is not intended to be a justification of our knowledge, the sort of thing that we traditionally seek. Rather it attempts to say what we actually do, not what we ought to do. In other words, there are no norms. The oughts simply vanish, and Quine is happy with that. From now on, we are merely in the more limited business of describing how knowledge is acquired, not prescribing how to get it. The theory of knowledge is a part of natural science.

What do we observe about people's belief-forming behaviour? We notice, for instance, that people make inferences about the future. They shun contradictions. They prefer simpler theories to more complex ones. They posit “natural kinds” (which Quine takes to be a Darwinian adaptation). This is but a sample of what people typically do; we observe many other things besides. In doing things systematically we should study people as given such and such input (sensory stimulation) and as generating such and such output (stated beliefs). The aim of epistemology is to discover the details of the input–output relation.

Epistemology, or something like it, simply falls into place as a chapter of psychology and hence of natural science. It studies a natural phenomenon, viz., a physical human subject. This human subject is accorded a certain experimentally controlled input—certain patterns of irradiation in assorted frequencies, for instance—and in the fullness of time the subject delivers as output a description of the three-dimensional external world and its history. The relation between the meagre input and the torrential output is a relation that we are prompted to study for somewhat the same reasons that always prompted epistemology; namely, in order to see how evidence relates to theory, and in what ways one's theory of nature transcends any available evidence. (Quine 1969, 82–83)

Science already tells us a lot that even non-naturalist philosophers typically take for granted in their epistemology; for example: there is no precognition, there are no messages from God, and so on. Quine turns these sorts of facts into aspects of his naturalism. It is science itself that tells us what an observation sentence is, and, as Quine and any other naturalists would claim, science tells us that there is no perception of abstract, non-physical entities. In sum, science comes first. All knowledge is part of scientific knowledge; natural science is the one and only source of reliable beliefs, including reliable beliefs about the nature of belief itself. Mathematical knowledge is a part of this. It describes a realm of abstract entities (essential for the rest of science) that we come to know through normal scientific (i.e., empirical) means. This, in short, is the Quinean picture of how we come to know things and how we come to know mathematics, in particular. As one might imagine, there are problems with Quine's view, so I'll turn to some of these now.

First, as Charles Parsons remarked, “it leaves unaccounted for precisely the obviousness of elementary mathematics” (1979/1980, 151). There are no sentences of quantum mechanics, or of theoretical genetics, or of theoretical psychology, etc., which feel obvious or seem like they have to be true, at least, not in the way mathematics feels. “Protons are more massive than electrons” is something we must learn through laborious calculation and experimentation, or we can take it on faith. Yet sentences abound in mathematics which strike us as true as soon as we reflect on them. No matter what Quine says, our overwhelming conviction that 2 + 3 = 5 does not stem from sensory experience. It doesn't matter how carefully the observations were performed or how often repeated, experience seems utterly irrelevant. This is a major puzzle for Quine, but trivial for any Platonist.

Second, Quine's account of mathematics does not square with the history of science. Mathematics has a quarrelsome history; it is naive to think that a mathematical result, once established is never overturned. Lakatos (1976) is a good antidote to thinking otherwise. But the empirical natural sciences have had nothing to do with this. This is a point I made in the first chapter, but I need to make it again. It is not that mathematics and physics don't interact—obviously they do. The discovery of non-Euclidean geometries made General Relativity thinkable; and the success of General Relativity stimulated a great deal of further work on differential geometry. But their interaction is more psychological stimulation than logical connection. In the entire history of science the arrow of modus tollens (following an unexpected empirical outcome) has never been directed at the heart of mathematics; it has always been a theory with physical content that has had to pay the price.

Third, Quine's account is at odds with mathematical practice. Contra Quine, Penelope Maddy rightly notes that “(i) in justifying their claims, mathematicians do not appeal to applications, so [Quine's] position is untrue to mathematical practice, and (ii) some parts of mathematics (even some axioms) aren't used in applications, so [Quine's] position would demand reform of existent mathematics” (1984, 51). Quine is aware of the fact that much pure mathematics is unjustified on his account. How much mathematics is justified his way? Strictly, very little. For instance, science might get by with just the rational numbers, so the full continuum would not be justified on Quine's naturalistic outlook, since it's not essential. Like Kitcher, as we saw in the chapter on his work, any mathematics that isn't tied to empirical science is not, for Quine, legitimate mathematics. However, a reasonable amount of streamlining and filling in could be allowed, he claims, and this could legitimate the real numbers. But much of transfinite set theory, for example, would still be left out in the cold. Quine, however, is not too worried about this and probably feels liberal enough in allowing as much as he does.

[T]he continuum hypothesis and the axiom of choice, which are independent of [the other axioms of set theory], 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 constructability, V = L. It inactivates the more gratuitous flights of higher set theory, and incidentally it implies the axiom of choice and the continuum hypothesis. (1990, 95)

Most set theorists think that V = L is false. (I'll explain when I pursue this topic in the next chapter). But on Quine's account of mathematics, it turns out to be true (as do the CH and the axiom of choice). Of course, there is no simple way to say who is right and who is wrong. But we could appeal to Quine's own naturalism to rebut him.

We can do this by simply describing the mathematical community, noting that epistemology, according to Quine, is about what people actually do. Since V = L is overwhelmingly rejected by mathematicians, Quine's naturalist outlook (which describes the formation of belief and doesn't prescribe it), implies his account of mathematics (which is tied to science and tested by sensory experience) must be wrong. His naturalistic epistemology denies him the right to tell mathematicians what to do and believe. So if Quine is right in his epistemology, then he is wrong to assert V = L. Or, starting from the general consensus among mathematicians that V ≠ L and the fact that Quine's epistemology also implies V = L, a simple application of modus tollens does in his naturalism.

Let me once again sketch an alternative picture of how mathematics hooks on to the world (see Chapter 1 for details). The way mathematics is applied to science is not in the form of additional premisses added to physical first principles, but rather by providing models. A scientist will conjecture that the world, W (or some part of it), is isomorphic (or at least homomorphic) to some mathematical structure, S. Explanations and predictions are then made by computing within S and translating back to the scientific language. If it is a failure empirically, no one would or should dream of modifying S ; rather, one would look for a different structure, S ’, and claim that W is isomorphic to it instead.

Earlier I mentioned the change in how velocities were added together in the change from Newtonian to relativistic physics. Energy provides another simple example. It was modelled on the continuous real numbers in Newtonian mechanics. This was overthrown by the “ultraviolet catastrophe” of blackbody radiation which led to the quantum theory. The quantum revolution, however, was no threat to the theory of real numbers; instead, the previously conjectured isomorphism was dropped and a new one adopted. Energy is now modelled on the integers (i.e., in a bound state the energy operator has a discrete spectrum). Mathematics remained utterly unaffected by this change, unlike chemistry, which was radically transformed.

For any way that the world, W, could be, there is some mathematical structure, S, which is isomorphic to W. This fact about applied mathematics is what undermines Quine, since nothing that happens in the world would or should change our views about S itself. Experience could only make us come to believe that the physical world is like S’ rather than S. The corollary is that we cannot learn about the properties of these various mathematical structures by examining the physical world (though the physical world can certainly be very suggestive). Before we can conjecture that W is like S we must know about S independently—though perhaps not completely. This fact, of course, gives Platonism (and other two-realm views) a big boost. Quine's brand of semi-naturalism seems hopeless by comparison.

MADDY

Penelope Maddy has articulated a number of influential views. She has called her successive accounts “realism,” “naturalism,” and most recently “second philosophy.” The various views are more compatible than often realized. I think it fair to say she has been a naturalist throughout, though the details have differed in important respects. In the next chapter I'll examine her latest view. Here I'll focus on Maddy's older naturalized realism, or semi-naturalism, as I've been calling it. She has articulated this view in a number of places, but Realism in Mathematics (1990) is the chief source.

Quine and Maddy are fellow semi-naturalists. Both are realists about sets: Sets exist and are the truth makers of mathematical sentences. Where they back off from full-fledged Platonism is over epistemology. Naturalistic sentiments prevails, though quite differently in each case. Mathematics, in Quine's well-known metaphor, is part of our web of belief. And, like theoretical science, it is tested in hypothetical fashion via observational consequences. To repeat what I said earlier, correctly predicted observation consequences count for the mathematics and physics used in deriving them, while false observational consequences count against. The naturalist's epistemological requirements are largely satisfied in Quine's account; there are no claims about intuitions, that is, actually perceiving sets, any more than there are claims about seeing electrons. All that is perceived are spectral lines or streaks in cloud chambers, for instance, which serve to support not just quantum mechanics but the theory of linear operators defined on a Hilbert Space, as well.

Maddy is sympathetic with some of this. Like Quine, she embraces a Platonistic or realist ontology and, again like Quine, she embraces an anti-Platonist, naturalist epistemology. The details of this epistemology are striking and not at all like Quine's, except in being inspired by naturalism. Let's now see some of the details.

When Maddy looked into her refrigerator she saw three eggs; she also claims to have seen a set. The normal objection that sets aren't anywhere in space or time, much less in her refrigerator, is answered:

It seems perfectly reasonable to suppose that such sets have location in time—for example, that the singleton containing a given object comes into and goes out of existence with that object. In the same way, a set of physical objects has spatial location in so far as its elements do. The set of eggs, then, is located in the egg carton—that is, exactly where the physical aggregate made up of the eggs is located. (1990, 179)

In short, the belief that there is a three-member set of eggs in her refrigerator is, according to Maddy, a perceptual belief. This is sense perception with the physical eye, not some mysterious or metaphorical mind's eye. If true, naturalism would be triumphant.

It doesn't seem plausible, however, to suggest that seeing the threeness of the set is like seeing the whiteness of the eggs. To count objects we set up an association between the things we are counting and the natural numbers. In order for Maddy's belief that the set of eggs has three members to be a perceptual rather than an inferential belief, she would also have to perceive the one–one, onto function between the set of eggs and the cardinal number three. This means she would have to keep the number 3 in her refrigerator, too.² I'm happy to have the standard metre in Paris, but keeping the number 3 in Irvine, California, is going a bit too far.

Suppose I see a large pile of eggs, some on top of others, hiding those below from view. I see the pile, but do I see the set? If so, then I don't have to see all the members of the set to see the set itself. Given that we can see sets at all, this is perfectly reasonable. After all, you and I can see the same forest (from different perspectives) without seeing the same trees; thus, we don't need to see all the trees to see the forest. So, what about a set of eggs consisting of three in front of me in full view and a fourth egg in Australia? If I can see that set, then there is nothing to stop me from seeing an infinite set, as long as I can see part of it. This seems implausible. Of course, it is open to Maddy to say that we can only see a set when we can see every member of the set. That would block the objections. But what sense then do we make of seeing a forest without seeing every tree in it?

Let's label the eggs in the refrigerator: a, b, and c. Maddy claims to see them and to see the set {a, b, c}, which she says is located in the same space as the eggs themselves. That's four things she sees so far. Unfortunately, according to standard set theory, there are many more sets than this. There are {{a, b, c}} and {{{a, b, c}}} and {{{{a, b, c}}}}, and there are {{a}, {b}, {c}} and {{a}, {b, c}}, and there are {a, {a}, {{a}}, {{{a}}}, b, c}, and so on. In fact, there are infinitely many different sets. If any one of those sets is located in her refrigerator, then presumably they all are. Can we actually see these sets, too? All infinitely many?

Maddy acknowledges that the image on the retina of the eye is the same in each case. So how could we perceptually distinguish {a} from {{a}}? I'm perfectly happy to allow empiricism more resources than mere retinal image. For instance, in gestalt figures one has the same retinal image but one sees quite different things—one person sees a duck, while another sees a rabbit. The difference is due to differing background beliefs and expectations, and they could be acquired in typical empiricist fashion. Perhaps we could appeal to this, and just as we can switch from seeing a duck to a rabbit and vice versa, we could switch from perceiving {a} to perceiving {{a}}.

The problem with this solution is that we need to have a prior understanding of the set theoretic difference, just as we need to have a prior understanding of ducks and rabbits in order to make the gestalt switch. If we already comprehend the difference between an object, the set containing that object, the set containing the set containing that object, and so on, then perhaps we can, in Maddy's sense, perceive the various different sets of eggs. But this prior understanding is essential. Where do we get it? To say we get it from sense perception itself is obviously begging the question. We need to understand set theory already in order to see the various different sets of eggs when we look at them. We can't be learning about them in a process like the one Maddy describes. Any Platonist is happy to come to the rescue at this point. We independently intuit sets, and then we see them (in a metaphorical sense) instanced in the world. But this is not a rescue that Maddy or any other naturalist would welcome.

ZALTA–LINSKY–BALAGUER

Bernard Linsky and Edward Zalta define their naturalism to be “the realist ontology that recognizes only those objects required by the explanations of the natural sciences” (1995, 525). This sounds innocuous enough, but they go on to claim that abstract objects are required in a full account of science and that this has led some would-be naturalists—Quine and Maddy, as we saw earlier—to locate properties and sets in the empirical order of things. They call this view “naturalized Platonism” and they reject it in favour of their own position, “Platonised naturalism.” The view advocated by Linsky and Zalta is close to traditional Platonism, in that it posits an ontology of objects outside of space and time, and—remarkably for naturalists—an a priori epistemology. However, there are also crucial differences from traditional Platonism, and the burden of their efforts is to show that these differences make their particular brand of Platonism compatible with a naturalist outlook. In short, it is yet another version of semi-naturalism.

Their approach is similar to Mark Balaguer's “full-blooded platonism” (FBP) or “plenitudinous platonism,” as he also calls it (Balaguer 1998). Balaguer embraces the richest mathematical ontology possible, though to call it “possible” is to speak loosely. The idea is that if a mathematical entity is logically possible, then it is actual. If a mathematical structure is consistent, then it really exists. So far, so good. Ordinary Platonists are happy with some sort of the-more-the-merrier principle, but Balaguer, like Linsky and Zalta, goes well beyond this. In set theory, for example, if the standard axioms are compatible with the existence of so-called large cardinals (more on these in the next chapter), then ordinary Platonists are inclined to assert their existence and to posit new axioms about them. This is an example of the-more-the-merrier thinking. Balaguer happily accepts this—and a very great deal more. Ordinary Platonists think the CH is true or is false, in spite of the fact that it is independent of the standard axioms. Balaguer claims that it is both true and false. That is, there are consistent mathematical structures in Plato's heaven in which CH is true and other consistent structures in which ~CH is true. As long as both CH and ~CH are separately consistent, then both are true. Of course, they are not true together. Balaguer is not asserting CH & ~CH. He is merely claiming that each is true somewhere or other, which, of course, is not what any ordinary Platonist would say.

This unusual view of mathematics was discovered by Linsky and Zalta and, later but independently, by Balaguer.³ Very likely it was stimulated by David Lewis's account of possible worlds. Everything that is possible is actual in some possible world, and those worlds are all just as real as ours, according to Lewis (1986). Linsky and Zalta and Balaguer extend this to the realm of abstract entities. They then put this view (counterintuitive though it is) to the service of naturalism by pointing out that epistemic contact with mathematical objects is not needed in order to know what is mathematically true. We need only know about consistency.

The enormous epistemic advantage for naturalism, as Linsky and Zalta see it, is this: “Knowledge of particular abstract objects does not require any causal connection to them, but we know them on a one-to-one basis because de re knowledge of abstracta is by description” (1995, 547). The Linsky and Zalta view is similar to if-thenism, except that the if is always true, at least when logically consistent.⁴ The view, as they see it, has several virtues. (Balaguer would concur.)

• It is naturalistic in the sense that it posits only those entities needed by the natural sciences, but it acknowledges that abstract entities are included among these. (Even if the existence of God were consistent, God is not posited, since unlike the numbers, God is not needed by science.)
• It does not (unlike Maddy) posit problematic entities inside space and time such as naturalized sets or immanent universals.
• It does not posit a problematic cognitive faculty, such as Maddy's “set detectors.”
• It does not (unlike Gödel) posit a problematic cognitive faculty such as intuition that could perceive entities outside of space and time.
• It does posit a realist ontology of mind-independent, objectively existing abstract entities.
• This ontology is rich enough that all mathematical practice is justified, not merely (as with Quine) the mathematics used by current science.
• It posits an epistemology that is completely compatible with naturalism, since all that is required is knowledge of logical consistency.

Linsky and Zalta cite many of the standard reasons for rejecting orthodox Platonism: For instance, that knowing is a mystery, since we have no causal contact with the abstract entities we claim to know. Naturalized Platonists such as Quine and Maddy, as we saw earlier, get around this by embracing some sort of empiricist epistemology. However, both are quickly rejected by Linsky and Zalta for various reasons. For example, Quine's epistemology of mathematics justifies mathematics used by science but leaves the great remainder unconfirmed. The problem with Maddy, as they see it, is that she puts sets into the causal order and claims that we can actually see them. In reply, Linsky and Zalta object that at best some elementary set theory can be confirmed on her account while the most interesting parts would remain a mystery. These objections to Quine and Maddy are not new, but they are effective, nevertheless.

Linsky and Zalta sharply distinguish their Platonised naturalism from traditional Platonism (“piecemeal Platonism,” as they call it) in a number of important respects. Traditional or piecemeal Platonism conceives of abstract entities on the model of physical objects: (1) they are subject to an appearance/reality distinction, (2) they aren't said to exist until they are individually discovered, and (3) they have indefinitely many properties that we might or might not discover in the future. Quine is a piecemeal Platonist, according to Linsky and Zalta, and presumably, so is Gödel and most other Platonists, whether they be semi-naturalists or not. What do these claims mean and what can be said in their favour?

Linsky and Zalta deny that abstract objects have an appearance/reality distinction. This makes mathematics unlike physics, rejecting an analogy favoured by Gödel and many other contemporary Platonists, including myself. Thus, just as the sun appears to go down in the evening, although in reality the earth is turning, so mathematical objects might appear one way, but turn out to be quite different. Linsky and Zalta deny we make these sorts of mistakes in mathematics, but they are surely wrong to do so. Sets, for example, are far from well understood. Initially, sets were thought to be perfectly linked to properties: Being red is equivalent to being a member of the set of red things. Russell's paradox and the axiom of choice put an end to that illusion. The successive conceptions of set, from Cantor's to the iterative conception to current flirtations with non-well-founded sets, look like the history of the concept of electron, from Johnston Stony to J.J. Thompson to George Thompson to contemporary Quantum Electrodynamics. The history of set theory is just like the history of physics. Things look one way initially, but turn out on further investigation to be different.

The reason for Linsky and Zalta's confidence in denying the appearance/reality distinction is simple. Since every possible set theory is equally true, all (consistent) conceptions of set, according to them, are realized in Plato's heaven. And so it is pointless to argue (as mathematicians often do) over which is the correct conception—they all are. If sets appear one way, then in reality they are that way. And if they appear another way, then in reality they are that other way, too. But even if we grant this, there are still problems; in fact there are two major difficulties.

I said that if sets appear one way, then, on the Linsky–Zalta view, that's the way they are in reality. But this is true only if the appearance is consistent. Linsky and Zalta, and Balaguer, too, spend little time on the epistemic problems associated with inconsistency. Unfortunately, the history of mathematics is full of wrong steps based on inconsistent reasoning. A proper philosophy of mathematics should take this into account. How, for example, were mathematicians able to reason with considerable success about sets before the discovery of Russell's paradox? They were not describing anything in Plato's heaven, by the lights of Linsky and Zalta, since naive set theory is inconsistent. If naive set theory isn't an appearance that is not a reality, then what else could it be?

Platonists who like the analogy with physics (I include myself here) would perhaps say that mathematicians have a partial, imperfect grasp of sets, just as physicists have a partial, imperfect grasp of the physical world in quantum field theory, which, like naive set theory, is plagued with inconsistencies (e.g., the self-energy of the electron is infinite). Somehow, physicists and mathematicians work with inconsistent theories. An appearance/reality distinction makes these facts of intellectual life intelligible. Neither Linsky and Zalta nor Balaguer offer an account of this, and I doubt that they can.

To illustrate the second problem with their denial of an appearance/ reality distinction, suppose I am trying to get you to focus on the so-called cumulative hierarchy of set theory.⁵ I do this not because I happen to think it is the unique truth about sets (suppose for the moment that I share the Linsky-Zalta-Balaguer view), but because that is the particular structure to which I want to direct your attention. No matter what I say I will describe some structure (provided I speak consistently), but I may not correctly describe the particular structure that I intend. All I can do is wave my hands, give some analogies, and so on. This isn't an idle philosophical problem of no practical concern. I may be working with a collaborator or trying to outline a problem for a graduate student so that she will have a good thesis problem to work on. According to the Linsky-Zalta-Balaguer view, it doesn't seem possible for me to hint at a unique and definite problem without pinpointing it to such an extent that it is in effect solved. There is, in fact, no serious difference between this epistemic problem and the traditional problem of deciding among rival characterizations of what is thought to be the unique set theoretic structure. It seems that there is an important and serious “appearance/reality” distinction, either way. (I think this problem is sufficiently serious to make it worth considering again, which I will do later, from a different perspective.)

A related problem for Linsky and Zalta and Balaguer concerns the fact that some mathematics is “obvious” while other parts are anything but. This is an important fact about mathematics that should also be addressed by any philosophy of mathematics. We saw earlier that it is a problem for Quine. The Linsky-Zalta-Balaguer account clearly has no explanation for this philosophically remarkable and important fact—we are, according to them, equally out of touch with every bit of mathematics; we have no intuitions of any of it. Even if one rejects any sort of Platonistic account of them, some explanation of the psychological phenomenon of mathematical intuition is required.

I have focused only on those aspects of the Linsky-Zalta account that have a bearing on semi-naturalism. There are several clever technical aspects to their theory that merit further investigation, but since they effect semi-naturalism only indirectly, I will do no more than mention one: the comprehension principle for abstract objects. This principle says that “For every condition on properties, there is an abstract individual that encodes exactly the properties satisfying the condition” (1995, 536). It only takes a moment's reflection to appreciate how powerful this principle is. For every property or combination of properties, the principle posits a unique object. Linsky and Zalta argue that their comprehension principle is required by naturalism; thus their Platonism is required by naturalism, and hence, they argue, their Platonism is compatible with naturalism.

This is a questionable argument. It would perhaps be more plausible to say that their comprehension principle (even if it is required by science) is at odds with naturalism and so refutes it. To see this problem more clearly, consider a parallel case. Suppose that God is needed by science. That is, suppose the only way to make sense of various cosmological or biological facts is by positing an intelligent and purposive creator. Some people (alas) actually do argue this way. Their reasoning is invariably faulty, but for the sake of the argument, we'll suppose that some version of so-called “intelligent design” is rationally justified. Would it not be proper in this situation to reject naturalism? If all the facts of the natural realm point to something supernatural, then we can hardly maintain that reality is exhausted by the natural. Linsky and Zalta, as I mentioned earlier, define naturalism as “the realist ontology that recognizes only objects required by the explanations of the natural sciences” (1995, 525). If natural science needs God (which I do not believe) or needs Platonic entities (which I do believe), then it's time to consider jettisoning naturalism. After all, naturalism is a kind of scientific theory and so subject to refutation in the light of certain developments in science. What better evidence of its scientific status than to stand, like phlogiston and caloric, refuted?

Linsky and Zalta mean to have as many abstract objects as is logically possible, a “plenitude of abstract objects.” This means, according to them, that CH is true and that ~CH is also true, though in different structures. Indeed, objects themselves are linked to the theories in which they occur. Thus, points in Euclidean geometry would not be the same things as points in non-Euclidean geometry. They merely use the same word, “point.” The empty set in ZFC + CH is not the same entity as the empty set in ZFC + ~CH. “The appearance of disagreement is explained by the common vocabulary. What each has in mind is perfectly real, but each party to the disagreement mistakes their limited portion of reality for the whole of reality” (Linsky and Zalta 1995, 543). Consequently, if theories differ in any respect, they differ in every respect. Overlap does not exist; there is no object that they have in common. Thus, there are no rival theories, that is, theories that make conflicting claims about some common X. This is Kuhn's infamous incommensurability with a vengeance.

Needless to say, this seems many miles from mathematical experience where rival theories of X seem perfectly intelligible, and we rightly wonder which of the rivals is the correct theory of X. The meaning of X is the same in both theories, otherwise asking which is the correct theory of X is meaningless. The problem for Linsky and Zalta and Balaguer comes into sharp focus in independence proofs. We want to know if CH or ~CH is derivable in the theory ZFC. We discover that neither is derivable by showing that CH is consistent with ZFC and then by showing that ~CH is also compatible with ZFC. In doing this we have used the same concept of set each time, and so, meaning does not change from structure to structure. If meanings did change, then independence proofs would be incoherent nonsense, which they obviously are not. A set is a set whether CH is true or false.

Balaguer's unusual approach counts as a kind of Platonism. But I need to add a qualification that I have so far not mentioned. His particular brand of Platonism, or semi-naturalism, is one of the positions he outlines in his book (Balaguer 1998), though, strictly speaking, he does not endorse it. First, Balaguer argues that there is a version of Platonism—FBP, as he calls it—that overcomes all the standard epistemological objections to Platonism. Second, he describes a form of anti-realism—fictionalism—that he claims overcomes all objections to anti-realism. Third, he argues that we cannot decide between these two positions, so in verificationist fashion he claims there is no fact of the matter as to which is right. My interest in Balaguer is only in the first step, his peculiar brand of Platonism, since it is a version of semi-naturalism. I'll continue to call it Balaguer's view, because he articulated it so clearly, but it should be remembered that he does not subscribe to it, without qualification.

FBP, to repeat, is a version of semi-naturalism, since it embraces the reality of mathematical entities independently existing outside of space and time, but rejects any Platonic epistemology involving the intuition or perception of these entities. One comes to know what is actual by simply knowing what is possible, and that, presumably, is perfectly compatible with naturalism's strictures on epistemology.

Balaguer notes that there is some similarity to claims made long ago by Hilbert and by Poincaré, a comparison worth considering. Hilbert expressed his views on consistency and existence in a reply to Frege's criticisms of his book The Foundations of Geometry (Hilbert 1899/1950). One of Hilbert's main aims in that book was to show the consistency of his presentation of geometry. Frege failed to see the need of consistency proofs: “From the fact that axioms are true, it follows that they do not contradict one another” (1971, 9). Hilbert couldn't be more opposed: Rather than truth implying consistency, it's the other way around: “If the arbitrarily posited axioms together with all their consequences do not contradict one another, then they are true and the things defined by these axioms exist. For me, this is the criterion of truth and existence” (cited in Frege 1971, 12).

This is a curious view, and Frege raised an obvious objection (1971, 18ff.): The notion of an all-powerful, all-loving, all-knowing being is (let us assume) a consistent concept. By Hilbert's lights we then have a proof of the existence of God. Needless to say, this is preposterous.

Poincaré, too, held that consistency and existence go together, but his espousal of this principle seems something of a mystery. Poincaré’s philosophical sympathies run along constructivist or intuitionist lines, so one would expect him to link mathematical existence with constructability, not mere consistency. Hilbert is less surprising, if we think of him as a formalist. Then, mathematical truth and existence are not to be taken in any serious literal sense. This marks a crucial difference from Balaguer's account. For example, suppose we are working with the real numbers and the statement “Every equation has a root” is consistent with everything else we believe. Then it is true, and the roots exist, according to Hilbert. Thus, the equation x + 1 = 0 has a solution, x = √–1. This entity, √–1, though previously unencountered and not itself a real number, is now taken to exist. But for Hilbert it is just a symbol, not a real, independently existing thing. By contrast, for Balaguer and for Linsky and Zalta the number is taken to be perfectly real.

When he became an explicit formalist, many years after the debate with Frege, Hilbert declared that we can add “ideal elements” (points at infinity, imaginary numbers, transfinite numbers) to finite mathematics to make a smoother system. Hilbert's notion of existence is thus relatively innocuous; it's a kind of fictional existence. Moreover, among fictions, not all logically possible ones exist—only those that help to systematize true finite mathematics. This is certainly not the same sense of existence we normally employ, and it is certainly not Balaguer's strong sense of existence.

Balaguer's FBP is motivated by the naturalist objection to any sort of Platonist epistemology. Naturalism embraces empiricism, and empiricism won't countenance the perception of abstract entities. Balaguer thinks that he, like Linsky and Zalta, solves the problem by making Plato's heaven the home of every possible mathematical entity and every possible mathematical structure. If it's possible, then it's actual. However, empiricism has two ingredients. One says that all concepts are acquired in a way that is ultimately based on sense perception. The other says that all propositions are tested in a way that is ultimately based on sense perception. Linsky and Zalta and Balaguer have only addressed the second of these. They have (at best) avoided the empiricist objection to Platonism's intuition as a way of ascertaining the truth of some propositions. But they have not addressed the problem of the origin of mathematical concepts.

Empiricists have no trouble accounting for fictional concepts such as Pegasus, Santa Claus, or unicorns. We simply rearrange concepts we already have. Thus, for Pegasus we imagine sticking the wings of a bird on the body of a horse. This bit of mental gymnastics requires something to work with, of course, but it is no problem in this case, since we already have the concepts of a horse and a wing. We need to give an empiricist account of these, too, and presumably, we can do so with no difficulty. But can the same be said about mathematical concepts?

It's no good saying that every mathematical concept that is self-consistent is permissible, if we don't have any sort of grip on those concepts. Terms such as “limit ordinal,” “imaginary number,” and “fibre bundle” are not mere symbols, they are genuine concepts about which we theorize. But where do they come from? Perhaps everything we say about them is true, as Balaguer maintains, though we still have to account for how we acquired these concepts in the first place. It's a common belief in mathematics (though I doubt it myself) that everything reduces to set theory. If so, then we really only need to account for two concepts: set and member. Everything else can be defined by these, just as Pegasus can be defined in terms of the empirically admissible concepts horse and wing.

Will this work? Only if we can give empirically admissible accounts of set and member. When we discussed Maddy earlier we noted some of the problems involved in the empirical perception of sets. The concept of member is just as problematic. (Some of these problems came up when discussing Lakoff's account in Chapter 4.) We may think we can acquire the notion of membership via empirical examples. An apple in a basket might be offered as an example of membership; a (the apple) is a member of (is spatially located in) the set S (the basket). We represent this symbolically as a € S or as {a} = S. We could then put one basket inside another to empirically ground the idea symbolized as {{a}}. Perhaps the empty set could be grasped by seeing an empty basket. This process works well until we try to empirically illustrate the idea represented by {a, {a}}. (This is quite different than {a, {b}}, which is easy to physically illustrate.) We might first try putting a in a small basket, which we then put in a bigger basket. But in this bigger basket we need a both outside and inside the smaller basket. That is, we need a to be in two places at the same time, simultaneously inside and outside the inner basket. Empiricism requires the physically impossible. It can't fully account for the concept of set membership. Physical collections are initially helpful in grasping the idea of a set. But no one understands set theory until she distinguishes sharply between a collection of bricks, which has a mass and a location, and a set of bricks, which has neither. Making that distinction is an intellectual, not a sensory, achievement.

For the sake of the argument earlier, I allowed that all of mathematics reduces to set theory. In the Linsky-Zalta-Balaguer picture of Plato's heaven (or multiple heavens) this will certainly not be the case. And so the problem of accounting for the origin of our mathematical concepts is actually even worse. Given that everything that is possible is actual, by their lights, there will be indefinitely many entities that are simply not definable in set theory terms. So even if there is an empirically acceptable way to acquire the concepts of set and member, there are many (probably infinitely many) more concepts that will require such an empirical grounding. There is no reason to think that this is even remotely possible. In fact, it seems utterly hopeless as the following little argument suggests.

In FBP there are infinitely many “primitive” (undefined) concepts. Every consistent theory is going to have its own. And remember, according to Balaguer, there is no overlap: “Point” in Euclidean geometry does not mean the same as “point” in any of the infinitely many different non-Euclidean geometries. But in the realm of sensory experience there are only finitely many things (or combinations of things). Thus, most mathematical concepts cannot be acquired through sensory experience.

Of course, Balaguer could reasonably reply that this only means that we will never be able to grasp most mathematical concepts, but those we do grasp (including complex number, topological vector space, Lebesgue integral, differential manifold), we grasp via sensory experience. This is still somewhat implausible, but if it is right, it pushes us back to the problems associated with making sense of “set” and “member.” The reasonable conclusion we should draw in any case is that mathematical concepts are not acquired by empirical means at all.

Linsky and Zalta and Balaguer do not address the problem of concept acquisition, and I doubt that they could do so successfully. They do address the problem of how we come to know truths in a way that does not violate empiricism, since all we need do in order to know that a mathematical statement is true is to realize that it is consistent. They solve the epistemic problem of access by eliminating it entirely. But their solution ignores a rather obvious fact about mathematicians in their daily practice. Mathematicians not only utter truths; they have conversations in which they appear to talk about a single subject matter. This may seem trivial, but it's actually an important datum. Imagine a conversation that goes:

Alpha: Grass is green.
Beta: Hamlet is a great play.
Alpha: I like mashed potatoes.
Beta: Snow is cold.

Speakers here utter truths, but it's a hopelessly disjointed conversation that is not about a specific subject matter; it's just a string of random facts. Even people with a very short attention span normally manage to stick to a single topic for at least a brief period. How would a sensible mathematical conversation be possible, given Balaguer's FBP? In their conversing mathematicians would be all over the place, just as Alpha and Beta are. They would constantly be running into the type of problem that Wittgenstein raised about “going on in the same way.” Wittgenstein describes a teacher instructing a student on how to do addition by two. The teacher starts with a few examples:

2 + 2 = 4
4 + 2 = 6
6 + 2 = 8
. . .

The student is asked to “go on in the same way.” She adds by twos as follows:

996 + 2 = 998
998 + 2 = 1000
1000 + 2 = 1020
. . .

The teacher says this is a mistake, but the student protests and says that everything the teacher has so far told her is compatible with saying that 1000 + 2 = 1020. Wittgenstein's point is a sceptical one. He claims there is no objective fact of the matter about who is right. Linsky and Zalta and Balaguer, at this point, would seem to be the arch anti-Wittgensteinians (or is it arch pro-Wittgensteinians?) saying both are objectively right. That is, there are number structures such that the teacher's claims are true of one and the student's claims are true of another. The problem with Balaguer's answer is that he can't account for why we hardly ever have this sort of disagreement. When a teacher says “go on in the same way,” we almost always agree that it is indeed the same way.

Why do we get this agreement? Wittgenstein gave one sort of naturalist account that appeals to a common “form of life,” an answer that I and many others find unsatisfactory.⁶ Linsky and Zalta and Balaguer have no answer at all. It must remain a mystery on their account why people seem to focus on the same mathematical structure in their discussions when there are infinitely many different structures that overlap. Of course, for an ordinary Platonist the reason is simple: People have a grasp, a non-sensory perception, of the same entity or structure. It's a fallible perception, but it is usually sufficient to keep a mathematical conversation focused on the same subject. The problem he has in mind stems from the fact that a set of axioms might be satisfied by different, non-isomorphic models. The axioms themselves cannot distinguish between these different models. So, how do we do it? In particular, how do we pick out the “standard model,” the one that we were trying initially to capture with the axioms?

Balaguer does attempt to address a difficulty that is related. He raises the potential problem: How could humans beings acquire knowledge of what the various standard models are like?

In fact, there is no epistemic problem here at all. This is simply because standard models aren't metaphysically special. They're only sociologically special or psychologically special. To ask whether some proposition is true in, for example, the standard model (or class of models) of set theory is just to ask whether it is inherent in our notion of set. Thus, since our notion of set is clearly accessible to us, questions about what is true in the standard model (or models) of set theory are clearly within our epistemic reach. (1998, 64–65)

There are at least three problems with this answer. First, as I already mentioned, the notion of “our conception of set” is highly problematic. The second turns on a technicality that Balaguer does not mention. Many theories have first-order and second-order formulations. The former will use axiom schema, while the latter will use quantifiers which range over properties as well as individuals. The second-order version of Peano arithmetic, for instance, is categorical, which means there is essentially only one model, the standard one. First-order Peano arithmetic admits non-standard models as well. We are able to pick out the standard model because we bring it over from the second-order version.

This explanation of how we are able to pick out the standard model of first-order Peano arithmetic is available to many different accounts of mathematics, not just Platonism. Where Platonism has an advantage is in the ease with which it naturally favours second-order formulations of any theory. Why? Quantifying over properties is as natural to a Platonist as quantifying over individuals, since properties are every bit as real.⁷

The third problem with Balaguer's answer brings us back to Wittgenstein's problem, mentioned earlier. We simply do not have a definite conception of set, if “going on in the same way” is problematic, as Wittgenstein claims. On the other hand, if we can “go on in the same way,” contrary to Wittgenstein, then it must be because we have some epistemic (non-natural) grip on the entities themselves, and this, of course, flies in the face of Balaguer's account.⁸ In any case, it's a mystery how standard models got to be standard, if one does not embrace the epistemic side of Platonism.

In sum, FBP is an attempt to uphold a kind of Platonism in ontology and naturalism in epistemology. The ontology is much richer than any working mathematician wants. And the epistemology has only limited success, outweighed by its serious failures. It can't account for the acquisition of mathematical concepts and it can't account for the fact that people are somehow able to talk about the same thing.

Semi-naturalism offers the benefits of a realist ontology with the bromide of naturalistic epistemology, but is this really an improvement over orthodox Platonism? Surely not. The epistemological accounts of Quine, Maddy, Balaguer, and Linsky and Zalta are all implausible in their own right, in spite of being admirably clever in conception and motivated by concerns that are widely shared. They certainly benefit from the allegedly problematic nature of Platonic intuitions, but, as we saw in the last chapter, the “mind's eye” is not blind.

“We might as well hang for a sheep as a lamb” is not a popular bumper sticker with semi-naturalists. They grudgingly allow abstract entities, but claim never to perceive them. One can admire the all or nothing attitudes of carnivores and vegans, but why draw the line so as to include lambs but nothing more? Why allow Platonistic ontology, but not Platonistic epistemology? This involves endless implausible contortions. With Blake one wonders about semi-naturalists: “[D]id he who made the lamb make thee?”