3    Perception, Practice,
and Ideal Agents

Kitcher's Naturalism

EXPERIENCE AND THE EVOLUTION OF PRACTICE

Phillip Kitcher has contributed significantly to a wide range of philosophical topics. Much of his work is tied to naturalism, especially his work on mathematics. His particular brand of mathematical naturalism is sketched in clear terms: “Our present body of mathematical beliefs is justified in virtue of its relation to a prior body of beliefs; that prior body of beliefs is justified in virtue of its relation to a yet earlier corpus; and so it goes.” He continues, “Somewhere, of course, the chain must be grounded. Here, perhaps, we discover a type of mathematics about which Mill was right, a state of rudimentary mathematical knowledge in which people are justified through their perceptual experiences in situations where they manipulate their environments (for example, by shuffling small groups of objects)” (Kitcher 1988, 299). An important footnote immediately qualifies this:

[B]ecause the chain is so long it seems misleading to emphasize the empirical character of the foundation. Indeed, it seems to me to be possible that the roots of primitive mathematical knowledge may lie so deep in prehistory that our first mathematical knowledge may be coeval with our first propositional knowledge of any kind. Thus, as we envision the evolution of human thought (or hominid thought, or primate thought) from a state in which there is no propositional knowledge to a state in which some of our ancestors know some propositions, elements of mathematical knowledge may emerge with the first elements of the system of representation. Of course, this is extremely speculative. (1988, 321n10)

Finding a starting point is essential. A Mill-type account is one possibility, and Kitcher is right to worry about its speculative nature. A second possibility stems from neurophysiological accounts, such as by Butterworth (1999), who claims, for example, that elementary arithmetic—but not all of mathematics—is encoded in our brains. Kitcher could help himself to this, then claim that the account of non-elementary mathematics he offers in terms of changing practices takes off from there. However, this is just as speculative as the Mill-type version. We need some evidence one way or the other; so far we have none. Nonetheless, to keep the discussion manageable I will grant Kitcher his explicit empirical, Mill-type starting point, which he reiterated in his sympathetic exposition of Mill (Kitcher 1997). Once a starting point is established, the main job for Kitcher is “to show that contemporary mathematical knowledge results from this primitive state through a sequence of rational transitions” (1988, 299).

In sum, Kitcher's mathematical naturalism consists in two parts: First, our knowledge is understood as originating in response to our material environment, our sensory experience in seeing and manipulating small groups of objects. Second, mathematical knowledge developed from this empirical starting point into its present complex form by means of natural processes. There are many possible ways in which this might happen, since there are many different processes that a naturalist would find acceptable in principle. Kitcher's preference is to focus on practice.

We should understand the growth of knowledge in terms of changes in a multidimensional unit, a practice, that consists of several different components. Each generation transmits to its successor its own practice. In each generation, the practice is modified by the creative workers in the field. If the result is knowledge, then the new practice emerged from the old by a rational interpractice transition. (1988, 299)

Basic mathematical beliefs, to repeat, were (in ages past) grounded in elementary experience (grouping and rearranging small collections of medium-size objects). Then out of this early experience, says Kitcher, various mathematical practices grew. But these practices are not stable for all time. They have a history; some atrophied, some remained fixed, and some evolved into other practices. Current mathematics is the result of this complex process.

This is a promising account and if it works, a considerable achievement for the naturalist outlook. Mathematics could then be understood completely in naturalistic terms, since the only things involved in the explanation of current mathematics would be sense perception and learned human behaviour. There is no need to posit a Platonic realm, mysterious intuitions, or any other sort of non-natural entity or process. Can the promise be fulfilled?

The devil, as usual, is in the details. What shapes mathematical practice? Why does it have the character it does? How has it changed over time? Kitcher takes practice to have five components:

1. a language employed by the mathematician whose practice it is
2. a set of statements accepted by those mathematicians
3. a set of questions that they regard as important and as currently unsolved
4. a set of reasonings that they use to justify the statements they accept
5. a set of mathematical views embodying their ideas about how mathematics should be done (1988, 299; see also Kitcher 1983, 163)

In focusing on these, I do not mean to suggest that Kitcher's view is entirely captured by these five points. They are merely part of a larger whole, but quite an important part. Let's look at each point.

Language. (A practice includes “a language employed by the mathematician whose practice it is.”) Perhaps number words can be associated with the primitive groupings in elementary experience, but what of other mathematical terms? Where does “zero” come from, or “derivative,” “limit ordinal,” or “real projective plane”? It's not enough to say mathematicians employ a language at a particular time. Kitcher needs to tell us how this language—with its terms that do not correspond to direct experience—has somehow managed to develop through practice out of an earlier language.

It would be unfair to demand from him the actual detailed history; merely showing how it is possible would be sufficient. In asking this, what I have in mind is something like the narrative explanations offered by evolutionary biologists. They often don't try to tell us exactly what happened, but only to show how an evolutionary history is possible. How did giraffes get long necks? They might have had to compete for scarce food and those with longer necks were able to reach higher and get more; they passed on this trait, which had significant survival value to their offspring, and so on. The story need not be true; it is only meant to show how such a thing is possible in an evolutionary framework in case the long neck is purported to be a counterexample to the evolutionary hypothesis. Could Kitcher tell us something similar? Could he offer us a story showing how current mathematical language might have developed out of elementary sensory observations and manipulations of physical objects?

Earlier in the twentieth century, a rather similar project for the language of science came to a crashing halt. The positivist aim was to reduce the language of theoretical science to the language of observation. Terms such as “electron,” “gene,” and “superego” were to be defined by means of terms such as “white streak in cloud chamber,” and so on. It is now generally conceded that this programme is hopeless. Even staunch anti-realists such as van Fraassen (1980) deny that theoretical terms can be grounded, defined, or explained away in observation language.

Kitcher has not told us precisely what he has in mind in linking current mathematical language to past languages and practices, and ultimately to sensory experience and manipulation; but however he means to spell this out, it seems his project is perilously close to the now defunct positivist programme. Just as “electron” cannot be reduced to descriptions of sensory reports, so it seems very unlikely that, say, “p-adic numbers” and “projective planes” can be reduced to our practices and our experiences.

A set theoretic imperialist might jump in at this point to rescue Kitcher. Set theory (on the imperialist's view) captures all of mathematics. Any term, however esoteric, can be defined by means of the two undefined terms (called primitives), set and member. A function, for example, is a set of ordered pairs, and ordered pairs are just sets of sets. So a function is a set of sets of sets. The same goes for number, integral, limit ordinal, fibre bundle, topological vector space, and so on. It may be the case that the theoretical language of science cannot be defined by an observation language, but in mathematics we seem to need only two basic concepts, set and member, to define everything. Could this help Kitcher?

Since he is not a set theoretic imperialist, Kitcher is unlikely to help himself to this sort of argument. Moreover, disanalogies between mathematics and science are not the sort of thing that he, as a naturalist, wants to stress. Nevertheless, it is worth seeing why the argument can't be used, anyway.

The problem has often been noted, and it rests on the difference between set and physical collection. In explaining set theory we often cite examples such as a flock of birds, a pack of wolves, and so on, as instances of sets. But to do justice to the notion we distinguish between a set of birds which is located nowhere in space, and a flock which is in the sky flying south. A pack of wolves might devour a deer, but a set of wolves has never eaten anything. The kinds of groupings of objects that Kitcher cites as providing the necessary elementary experience to get mathematics going are always material entities, such as collections of apples, never sets of apples. We simply cannot manipulate a set of apples, nor can we see it or smell it or touch it. Our grasp of the set of apples (presuming we do grasp it) is some sort of non-physical, non-sensory operation—the sort of thing that delights a Platonist, but induces vertigo in an empiricist. What Kitcher needs to do is to give us a naturalist account that links the perception and manipulation of physical collections with the concepts we currently have about sets; concepts that sharply distinguish them from physical collections. Until then, talk of current mathematical language being linked with elementary experience is really nothing more than a promissory note of dubious value.

Accepted statements. (A practice includes “a set of statements accepted by those mathematicians.”) Kitcher's claim that mathematicians accept a body of statements is reminiscent of Kuhn's claim that those who share a paradigm will endorse certain formal statements, e.g., “F = ma.” (Kuhn 1962/1970) I'm sure Kitcher is right about this, as was Kuhn. What he might also be willing to add is that mathematicians attribute various different meanings to many of the mathematical statements they accept. All contemporary mathematicians would accept dx²/dt = 2x, for instance. Interestingly, however, they probably mean a variety of different things by it. Most, though not all, would take this to mean that 2x is equal to the limit of ((x + Δx²)—x ²)/Δx, as Δx approaches 0. Others might take it to be the ratio of infinitesimals. Many will think of this derivative geometrically as the slope of the tangent to the curve x ² at the point x, or perhaps they will think of it in terms of the motion of a point through space, i.e., they think of a derivative as velocity. The fact that they take the formal statement in different ways probably supports Kitcher's point, for the different interpretations of it are typically tied to different practices.

Kitcher's observation is an important one, just as Kuhn's was. The ready acceptance of a class of statements is an important part of mathematics and would obviously play a central role in practice. But it does not follow that naturalism is the correct interpretation of this fact. The rival accounts of formalists, conventionalists, and Platonists are all consistent with this point. Kitcher's rivals would deny that mathematics reduces to practice, but they would not deny the existence or importance of practices. Indeed, there is practice aplenty, just as there is in the natural sciences where realism, the counterpart to Platonism, is close to the commonsense view.

Important questions. (Practice includes “a set of questions that they consider important and as currently unsolved.”) What makes a question important? Does P = NP? Is the Riemann hypothesis true? These questions are taken to have the highest importance. Mathematics is to a large extent problem driven, just as the natural sciences are. Kitcher is doubtless right about this and would, not surprisingly, find support in all quarters. Many mathematicians work on particular open problems, but others focus on realizing a programme. The so-called Langlands programme, for example, contains a large and diverse set of conjectures concerning algebraic geometry (such as the Taniyama-Shimura conjecture¹); its aim is to realize a certain unity. Some of the specific problems would be considered important in their own right. Others are less interesting intrinsically, but are nevertheless important because they play a role in fulfilling the general programme.

Once again, however, this claim does not set Kitcher apart from his rivals. Hilbert, the formalist, is famous for his mathematical problems lecture (Hilbert 1900). It was presented more than a century ago, listing twenty-three outstanding problems, many of which are still unsolved today. Naturalist and non-naturalist alike can agree that mathematics is driven by problems.

A set of reasonings. (Practice includes “a set of reasonings that they use to justify the statements that they accept.”) What makes a particular form of inference acceptable? Infinitesimals and the axiom of choice have been controversial. What makes their use legitimate or not? There are, in fact, two questions here: How did any style of reasoning originate? And once launched, how does any particular style of reasoning transform into another? Both questions present problems for Kitcher.

If we focus on the starting point, Kitcher's elementary experience, we have only a cluster of sensations. We do not have any sense-experience that corresponds to an inferential practice—we cannot literally see modus ponens or reductio ad absurdum. The first piece of reasoning will have to come ex nihilo and develop from there. Kitcher's naturalism isn't going to account for this feature of our cognitive life. However, there is a naturalistic solution to this. Basic forms of reasoning might be innate, the result of a Darwinian process. Those who reasoned in modus ponens fashion tended to have more offspring than those who did not. Imagine a hominid who reasoned: (1) If this is a sabre-toothed tiger, then I should avoid it. (2) This is a sabre-toothed tiger. Therefore, I will approach it with a friendly smile. You may safely conclude that this hominid is not your ancestor. Though Darwinian adaptation offers a naturalistic solution, it is not Kitcher's. Until he addresses this problem, it remains a lacuna in his naturalistic account.

Now let's suppose a set of reasoning practices exists. The question is: How does it change? The axiom of choice, taken as a principle of inference, provides a good example. Zermelo introduced it a century ago to prove the well-ordering theorem. Many balked at its use, but over the long run the axiom (and its equivalents such as Zorn's Lemma) have become standard tools in mathematics. What convinced the larger community of the axiom's correctness? Two things. First, it represents an infinite version of a principle of reasoning that is universally accepted in any finite case. So it is supported by analogy. Second, it leads to results often thought to be plausible in their own right that can't be achieved without the axiom. This second reason for accepting it is similar to a commonplace inference in science: If a hypothesis implies a wide variety of accepted results that can't be obtained in any other reasonable way, then that should be taken as evidence for the truth of the hypothesis. Gödel (1947/1964), for example, strongly endorsed this line of thinking. Kitcher is surely right to assert the existence of such inferential processes in the development of mathematical reasoning. Since it parallels typical scientific reasoning, it fits nicely with his naturalism. But it also fits with Platonism, as Gödel's endorsement makes plain. (Gödel's Platonism will be discussed in Chapter 5.)

How to do mathematics. (Practice includes “a set of mathematical views embodying their ideas about how mathematics should be done.”) Kitcher's fifth and final ingredient says that mathematicians hold views about what is or is not legitimate and about how future mathematics should develop. Should we, for instance, only accept an existence proof when we can provide an explicit construction? Do axioms have to be self-evident, or can we evaluate them on the basis of their consequences? Can we ignore concerns about truth and simply play with any axioms we like? Needless to say, answers will take the form of normative philosophical views; they are prescriptions, not descriptions based on elementary sense-experience. Of course, as anthropologists we can detect the presence of norms in the behaviour of others, and Kitcher quite rightly notes that working mathematicians are influenced by various norms they encounter in the practice of other mathematicians. The tough question is to justify these norms. This is something the anthropologist wouldn't attempt to do, since anthropology, being a science, is in the business of describing, not prescribing, behaviour. Can a mathematical naturalist do it?

Since Kitcher tends to favour naturalistic epistemology in the form called “reliablism,” it could possibly be used here. A norm (a procedure that mathematicians advocate), is acceptable in so far as it reliably produces the truth. (See Goldman 1979 for a classic account of reliablism.) Putting litmus paper in a solution and then checking its colour is a reliable method of testing the hypothesis that some solution is an acid. In the natural sciences we have empirical success to help us determine whether or not we are on the right track in believing theory T. Unfortunately, reliablism does not have a similar touchstone in mathematics, at least not beyond the elementary experiences that are Kitcher's starting point. A non-naturalist can appeal to (non-natural) mathematical intuitions in addition to established mathematical methods in order to justify some new, esoteric bit of mathematics. But Kitcher's naturalistic mathematician is limited to citing established methods alone.

We may have a question-begging circle: How do we know mathematical result R is correct? Because it was produced by method M. How do we know method M is reliable? Because it produced result R. The circle is not “vicious,” in that it does not lead to contradiction, but neither is it “virtuous.” Platonists can break into the circle by means of intuitions, but Kitcher's naturalists are trapped.

It is one thing to give a naturalistic account of the activities of some social group. In doing so we (in typical scientific fashion) describe their beliefs and activities. Perhaps they believe a number of things: “Grass is green,” “F = ma,” “God exists,” “Some people have ESP,” “Water is H₂O,” and so on. Our account of those people is naturalistic in the sense that we use only the methods of natural science to determine the facts about them and what they believe. To put the matter bluntly, we treat the fact that they believe in God on a par with the facts that they eat a lot of shellfish or practice polyandry. As descriptive anthropology goes, that's the end of it; there is nothing more to say. We are, as scientists, not obliged to defend or to criticize their beliefs and practices.² But we want more than this from Kitcher. We are not detached observers merely satisfying our curiosity about the exotic society of mathematicians. We want Kitcher's account (or any other) to justify mathematics, as well as tell us how it works. After all, it is not their mathematics that interests us, it is our mathematics, too—we want to be right.³

The ingredients that Kitcher includes in practice may well be the right ones, but the problem is that his account of practice really does not answer the crucial questions: How did we get from the starting point of elementary sense-experience to current mathematics? Are the methods used by current mathematicians the right ones? Kitcher's account may shed light on some aspects of the development of mathematics, but there are still gaping holes in the story, holes that a naturalist, it would seem, cannot fill.

EPISTEMOLOGY OF PRACTICE

Kitcher allows that “Platonists can simply take over [his] stories about rational interpractice transitions, regarding those transitions as issuing in the recognition of further aspects of the realm of abstract objects” (1988, 311). Surprisingly, such Platonists are also naturalists, according to him. It would seem that it is not ontology (human practices versus abstract objects) that chiefly divides naturalists from traditional Platonists, on his account, but rather epistemic matters: How do we make contact with the things we know about? When it comes to practices, the answer seems straightforward: we observe them. We see Newton, or Leibniz, or Gauss, or Dedekind making this mathematical move or that. We hear what they say; we see what they have written; we note the sequence of symbols used. Practices are readily available to us. Abstract objects, by contrast, are not—or so the usual objection goes. We can have empirical knowledge of practices, making them naturalistically admissible, but not of abstract entities. And this, the common objection runs, makes Platonism's abstract entities unacceptable in principle.

There are two problems with this line of argument. Is the perception of abstract entities really so problematic? And is the perception of practices really so unproblematic?

It has become commonplace to dismiss abstract entities on the grounds that they don't causally interact with us. The underlying assumption is that if they don't causally interact, then we can't possibly know about them. The locus classicus for this view is Benacerraf (1973), which Kitcher fully endorses (1983, 59). The intuition is a strong one: It's possible that there is an invisible, incorporeal elephant in my room. I can't see it, or touch it, or smell it, or detect it with any possible instrument. Though it is logically possible that such a being exists, it seems idle to speculate about it and even worse to gratuitously include it in our ontology. Without causal interaction (however indirect), knowledge is hopeless. The same goes for abstract entities.

But there are important differences. Postulating an invisible elephant would indeed be idle, since it does no work in our overall system of beliefs. Postulating abstract entities, on the other hand, does help explain and systematize our mathematical knowledge. It also explains some of our intuitions. There may be better accounts that make no use of abstract entities—that remains to be seen—but abstract entities, unlike invisible elephants, are posited for good reasons.

More serious is the charge that there can be no object of knowledge without causal contact between ourselves and that object. This point is basic to the causal theory of knowledge and is a common ingredient in many versions of naturalism. It is also easily refuted. I'll sketch the argument here and repeat it in more detail in Chapter 4.

Consider a particular experimental situation in quantum mechanics known as an Einstein-Podolsky-Rosen setup. A source emits a pair of photons to the left and right where detectors will measure for the property of spin-up or spin-down. At one wing a measurement result is, say, spin-up. Then someone at that side can predict with complete accuracy that a measurement on the other will get spin-down. I will readily allow that the person who measures (on the left, say) knows the outcome on the left in a way that satisfies the causal requirement (i.e., ordinary sense perception, etc.). But I claim that the person on the left also knows the outcome on the right and that this second piece of knowledge does not satisfy the naturalistic requirements.

Is there a causal connection between the left side and the right that could ground our experimenter's knowledge of the distant outcome? Such a causal connection would have to be faster than the speed of light. This is ruled out by special relativity. Could there be a causal connection via the source (i.e., a common cause in the past of the correlated outcomes)? Such a common cause would be a so-called local hidden variable. This is ruled out by the Bell results, a set of arguments and experimental outcomes that show that certain interpretations of quantum mechanics are hopeless. In short, the only way to save the naturalist requirement that there be a causal connection between knower and known is by rejecting either special relativity or well-established experimental work. Of course, fallibility must be acknowledged at every step in an argument such as this, but I think the sensible thing to conclude in light of this example is that the naturalist requirement of a causal connection is quite unjustified.⁴

The second problem concerns the perception of a practice. Though Kitcher thinks the perception of abstract entities is problematic, he takes the perception of practices to be straightforward. But is it? It seems to me that it is anything but—especially when mathematics is involved. Interpretation, among other things, is crucial. Imagine yourself watching Archimedes pushing a stick around in the sand. What is he doing? Here are some possible answers:

1. He's testing the sand to see if he can draw fine lines in it.
2. He's testing the stick to make sure it will draw fine lines in the sand.
3. He's drawing a right-angled triangle.
4. He's proving the Pythagorean theorem.
5. He's writing a coded message (using triangular shapes) to tell his friends about an impatient Roman soldier.

What we literally see (that is, the sense perception) is the same in each case, but the practice involved is quite different in each interpretation. The practice of drawing right triangles is quite different from the practice of testing the quality of the sand. We interpret; and when we do, we bring a lot of background information. There are at least two ways to understand any (new) practice that we encounter. One way is Kitcher's way: We see (using sense perception) the practice, and that is how we learn it. The other way is Platonistic: We have some (possibly vague) intuitive grasp of abstract mathematical entities and when we see (using sense perception) some activity, we interpret it as mathematical activity. In short, Kitcher says we get our knowledge of numbers by counting apples while the Platonist says it is our independent knowledge of numbers that allows us to count apples.⁵

There is no easy way to settle this dispute; neither account is obviously right or obviously wrong. But Platonism is at least as plausible as Kitcher's naturalism. And quite possibly it is a bit more plausible when we consider an analogous argument put forward long ago by Chomsky. Skinner tried to account for language completely in behaviouristic (stimulus and response) terms. Chomsky (1957) objected that we cannot in general tell what the stimulus is until we see the response, and in particular that we do not know what the verbal behaviour is that we are observing, unless we already knew the features of language that are being exemplified. Kitcher, I suggest, is in a similar if not the same boat as Skinner. Mathematical practices are like observable linguistic behaviours; we could never learn mathematics or a natural language, if in learning we were restricted to those observable things alone.⁶

HOW MUCH IS JUSTIFIED?

It's a frequent objection to any account of mathematics that ties itself closely to natural science and the empirical world that much of mathematics can't be justified. Most of pure mathematics simply has no connection or application at all. This is the basis of a common complaint made against Quine. Like Quine, however, Kitcher is quite prepared to bite the bullet. “Epistemic justification of a body of mathematics must show that the corpus we have obtained contributes either to the aims of science or to our practical goals. If parts can be excised without loss of understanding or of fruitfulness, then we have no epistemic warrant for retaining them” (1988, 315–316). Of course, we might like to keep esoterica such as so-called large cardinals around to amuse ourselves, but there can be no epistemic grounds for doing so—it's just an idle game. Quine called the inapplicable parts “recreational mathematics.” Kitcher and Quine might both concede that it is more stimulating and challenging than chess, but it is no less a mere game.

We could let this stand as a reductio ad absurdum of Kitcher's view— any philosophical account of mathematics that requires tossing out big chunks of existing mathematics on epistemic grounds is less plausible than the mathematics being eliminated. Penelope Maddy, for instance, accepts the reductio point and further remarks: “the goal of philosophy of mathematics is to account for mathematics as it is practiced, not to recommend reform” (1997, 160). Though I would qualify her remark, I am happy to embrace the spirit of it. To excise large chunks of mathematics is certainly to reform it, whether Kitcher wants to think of it that way or not.

Kitcher enters a note of caution: Drawing the distinction between epistemically grounded and ungrounded mathematics “must wait on the development of a full theory of rational interpractice transitions, both in mathematics and in the sciences” (1988, 316). This will seem to many a prudent remark in the spirit of naturalism's desire to fit mathematics into the whole of science, which, of course, is fallible and still developing. On the other hand, it smacks of that version of naturalism that is waiting for total final science before declaring the details, like mystics on a mountaintop eagerly awaiting the final rhapsody. Kitcher should tell us now where the line is, allowing that it might shift in the future. This is quite different from saying that it can't now be known.

There is one question, which I'll mention, but not pursue. When one practice P gives way to another, was P wrong all along? Or was P merely appropriate at time t, but inappropriate at time t’? Practices are different from truths. Though reasonably believed during the eighteenth century, Phlogiston theory was nevertheless false all along. It would seem that practices, unlike theories, are surpassed, not refuted. Thus, optical microscopes give way to electron microscopes. On the other hand, some practices (bloodletting) seem as wrong as the theories that justified them. This is a topic that should be clarified, but I must leave it to others.

IDEAL AGENTS

Perhaps the most serious problems arise with Kitcher's introduction of “ideal agents.” This strange entity is introduced to solve a major problem, but the solution might be worse than the condition it was meant to cure.

Arithmetical truths owe their being to the constructions we make, according to Kitcher. These, of course, are not constructions in the sense of Brouwer or other so-called constructive mathematicians, but rather are the “familiar manipulations of physical objects in which we engage from childhood on” (1983, 108). Of course, the objects that we actually manipulate are relatively few in number. When it comes to infinite numbers or even to very large finite numbers, it is a bit of a mystery how we acquire the relevant experience to learn about them. When dealing with whole numbers, we may only use numbers of finite size, but as soon as we enter the realm of analysis we need the full power of the real numbers with their decimal expansions that are infinitely long. So the problem Kitcher faces is simply this: How do we humans, who can carry out at most a finite number of manipulations, operations, or constructions, manage to generate infinite numbers? Kitcher's solution:

Arithmetic owes its truth not to the actual operations of actual human agents, but to the ideal operations performed by ideal agents. In other words, I construe arithmetic as an idealizing theory: the relation between arithmetic and the actual operations of human agents parallels that between the laws of ideal gases and the actual gases which exist in our world. We may personify the idealization, by thinking of arithmetic as describing the constructive output of an ideal subject, whose status as an ideal subject resides in her freedom from certain accidental limitations imposed on us. (1983, 109)

This is a remarkable postulate. Not surprisingly, the ideal agent is faster than a speeding bullet, stronger than a mighty locomotive, and able to leap tall buildings in a single bound. This much idealization is rather harmless— no worse than a frictionless plane—and Kitcher is entitled to help himself to it. Superman, after all, is still a finite being. But when it comes to infinite operations, we must surely object. This cannot be passed off as merely overcoming an “accidental limitation” that we humans have. A Platonic realm is not half so mysterious or implausible.

Kitcher likens his ideal agent to an ideal gas. But there are crucial disanalogies. First, we can approximate many of the idealizations of physics: We can polish a plane surface to reduce friction to almost nothing, thereby approximating the frictionless plane. By fiddling with the temperature and pressure of a gas we can approximate the ideal gas for which the ideal gas law holds. But there is no finite operation that approximates an infinite operation. Whether I count to 10, to 100, or to 10¹⁰⁰, I am no closer to the infinite.

Second, in mathematics we want to keep things on a par; the real numbers, which have infinite decimal expansions, are just as legitimate as the small natural numbers. But no physicist thinks frictionless planes or ideal gases are on a par with real surfaces or real gasses. These are taken to be fictions—useful to be sure, but fictions, none the less. For Kitcher to maintain the parallel, he would have to adopt a similar twofold division: mathematics based on genuine physical manipulations carried out by humans and mathematics based on fictitious operations carried out by ideal agents. If he did adopt such a distinction, he wouldn't be the first. Hilbert's particular brand of formalism did something like this.⁷ Real mathematics, for Hilbert, is to be understood along Kantian lines. It is finite. To this real (finite) mathematics we add “ideal objects,” such as infinite sets and points at infinity. These he took to be useful fictions that help the whole of mathematics to run smoothly. It is open to Kitcher to do something like this. In fact, I think that is what he has done (inadvertently) with the introduction of the ideal agent. The thing to note is that this is a major departure from the naturalism that was his initial motivation.

FALLIBILISM AND HISTORY

Much of Kitcher's case for naturalism hangs on an analogy between the development of mathematics and the rest of science. In particular, Kitcher (1983, 161) argues against a cumulative history of either. Results that were accepted in physics at one time were overthrown at a later date. This is agreed by all. The same, says Kitcher, is true of the development of mathematics. Results about continuity, about functions, about what is and is not differentiable have changed considerably over the past three hundred years. It is reasonable to conclude that mathematics is as changeable as any of the natural sciences. As a point of similarity, these facts help considerably in making the case that mathematics just is a natural science.

On the other hand, Kitcher allows that there is a kind of stability in mathematics (but not in science), that stems from the fact that mathematical results can be reinterpreted, and thereby saved from refutation. But he may be too generous to his opponents and needn't have allowed even this. Results in physics are often similar to those he envisions in mathematics. Here is an example: In Schrödinger's first quantum theory, his now famous equation described the electron as a physical wave. For a variety of reasons that theory didn't last. But the Schrödinger equation (specifically, the φ-function in the equation) was reinterpreted by Max Born as describing a probability amplitude, a reinterpretation that has proved to be very successful. It gave the Schrödinger equation the kind of stability, through reinterpretation, that Kitcher thinks happens uniquely in mathematics. If anything, the analogy between change in mathematics and change in science—and hence, his case for naturalism—is stronger than he thought.

But what moral should we draw from this? That fact that there is significant change in mathematics presents a deep problem for some accounts. Formalism and constructivism should be much worried by it. Both of these accounts pride themselves on being able to account for the “certainty” of mathematics. Since there is no certainty, they are in trouble for assuming there is. Much of naturalism is directed against Platonism, but Platonism is untouched by the history, about which Kitcher makes such heavy weather. The central claims of Platonism are these: First, there is a realm of mathematical objects which exists outside of space and time and is completely independent of us; second, we can grasp some of these objects via a kind of non-sensory experience, intuition. There is nothing here about infallibility. The most famous Platonist of recent times, Kurt Gödel, explicitly thought that doing mathematics is similar to doing physics; he was completely untroubled by mistakes in mathematical perception.

[D]espite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have any less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception. . . . The set-theoretic paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics. (Gödel 1947/1964, 484)

According to Gödel, the frameworks of mathematics and physics are remarkably similar. We have intuitions or sense-experience of some things, and we conjecture other things to explain and systematize what we intuit or experience. Just as physics is a fallible enterprise, so is mathematics. Gödel is not alone. It's fair to say that almost every self-conscious Platonist writing today is a fallibilist to some degree. Kitcher will not be able to make the case for naturalism and against Platonism on these grounds. Fallibilism is necessary for naturalism, but not sufficient.

Kitcher wavers on this and related points. At times, for instance, he seems confused about the notion of foundations. “Foundationalist philosophies of mathematics bear a tacit commitment to apriorist epistemology” (1988, 294). Some foundational work has this quality, but not all. When foundational studies try to secure mathematics, to show that it is free of contradiction, to show that we can trust our findings, then it is indeed concerned with epistemic matters and, arguably, associated with a priori reasoning. However, a great deal of foundational work is not concerned with epistemology at all, but rather with the underlying ontology. Are numbers just sets? Those who think so are making claims concerning the ontology of mathematics; they are not trying to convince us that we can be certain that 5 + 7 = 12. This type of foundationalist enterprise is explanatory—not justificatory. And it's no different than the enterprise of explaining the nature of every-day phenomena by appeal to an underlying reality of quarks and leptons. Such foundations are not more secure than what they explain; in fact the relative security of knowledge works the other way.

Once this distinction is made, then Platonism does not succumb to Kitcher's general criticism—neither as an account of what mathematics is about nor as an attempt to secure mathematical certainty. Platonism is a fallibilist account of the nature of mathematics. Of course, it makes important epistemic claims, too, but a commitment to epistemic certainty is not one of them. It is indeed tied to the a priori. However, what this amounts to is the following: Mathematicians have (fallible) intuitions and these are a source of evidence; these intuitions are distinct from sense perception. The claim about intuition is probably the chief anti-naturalistic feature of Platonism. So far, it has withstood the naturalistic onslaught.

INTERACTION OF MATHEMATICS AND SCIENCE

There are additional topics to take up briefly, and both are connected to the interaction of mathematics and science. Both were issues in earlier chapters, so I restrict my discussion of them here to their relevance to Kitcher.

Question: What is the cause of mathematical change? Or, in Kitcher's terms: What causes a shift in practice? Within the natural sciences, physics, chemistry, and biology interact with one another. This, according to the naturalist and non-naturalist alike, is just what we would expect. Does mathematics similarly interact with other sciences? The fallibility of mathematics is not enough to answer the question. If naturalism is right, then some mathematical change would very likely stem from its interaction with physics and the other sciences. On the other hand, if mathematics has a life of its own, independent from the rest of the natural sciences, then the prospects for a naturalistic account begin to look rather poor. Which is it?

Kitcher distinguishes between “internal” and “external” transitions. The difference depends on whether the rationality of a shift in practice is brought about by factors that are internal or external to the discipline itself. Contact with the physical world is the most obvious source of external transitions in the natural sciences. Kitcher sees a continuum with pure mathematics at one pole, almost wholly driven by internal factors. At the other pole, he sees the applied sciences such as metallurgy, largely at the mercy of external factors. He resists a sharp division between the two, noting that, for example, Copernicus was motivated more by conceptual problems (which are internal) than by new empirical observations. He also remarks that “the pursuit of analysis in the early nineteenth century was profoundly modified through the study of problems in theoretical physics” (1988, 301).

Kitcher's distinction between internal and external factors in bringing about change is widely shared and is surely right, at least to a first approximation. I would, however, like to add something to it. Factors—whether internal or external—could be seen as “objective” or “subjective.” For example, a contradiction discovered in a theory is surely an objective internal factor motivating change, while a new empirical datum that conflicts with a theory constitutes an objective external factor. In either case these objective factors could bring about rational theory change. The Copernicus example that Kitcher cites is a case of an objective factor. But I doubt that the mathematical example he cites is. More generally, I doubt that any factor coming from the physical realm could count as an objective external factor bringing about a mathematical transition. Instead, when we see these examples in detail, they turn out to be examples of subjective import. They stimulate a line of mathematical thought that hadn't been appreciated before. They are psychologically effective, but that is all. They are not objectively tied to theory change the way a contradiction or a new observation is tied to change, but are only subjectively associated.

The entire history of mathematics very strongly supports the autonomy and independence of mathematics. Results in mathematics have been overthrown, but always by other bits of mathematics. Results in one part of physics have sometimes led to a revolution in other parts, even to revolutions elsewhere, such as in chemistry. But never has a result in physics led to the overthrow of any result in the mathematical realm. The discovery of non-Euclidean geometry, for example, was a mathematical discovery. As I mentioned earlier, once the existence of such geometries was recognized, it allowed the possibility of new ways to represent or to model the physical world. The success of such new representations in General Relativity stimulated in turn further work in differential geometry. But the connection between the mathematical theory and the physical theory is heuristic and psychological—not logical or evidential. Developments in one provoke an active interest in the other. This is a case of a subjective external factor. On the other hand, the older chemical views were logically refuted by quantum mechanics; that is, one could not believe both on pain of contradiction. Nothing like that, I claim, has ever happened to mathematics. This epistemic autonomy argues rather decisively for ontological autonomy, and hence it counts against most naturalistic accounts of mathematics, and certainly against Kitcher's.

APPLIED MATHEMATICS

Kitcher proposes a standard conception of rationality, distinguishing epistemic aims from general aims, and distinguishing individual aims from those of a community. Truth and understanding are the epistemic aims of both individuals and of communities, while technological goals, social goals, and other practical concerns are the general aims of both individuals and of communities. Epistemic rationality consists in the rational pursuit of truth and understanding. Global rationality consists in the rational pursuit of one's practical goals, and we can easily imagine that a person (or community) will pursue some at the expense of others: In one instance truth gives way to utility, in another personal happiness gives way to understanding, and so on.

This account of rationality immediately gives rise to a serious problem for the mathematical naturalist. If there is no independently existing realm of mathematical truths, how can we have epistemic aims at all? It would be like trying to determine if chess bishops really move diagonally. The goals of truth and understanding make no sense on Kitcher's account. What then, is mathematical rationality?

Kitcher is aware of the problem. His proposal is to blur the distinction between epistemic and practical goals. In physics we can sharply distinguish the search for truth and understanding from the practical goal of making, say, a working airplane or a functioning computer. The distinction works because (assuming realism) there is an independently existing physical realm and what we say about that realm is true or false. But in mathematics, according to Kitcher, there is no such thing; there are no truths out there waiting to be discovered. In addition, there is a second problem associated with the notion of progress. In physics we make progress by accumulating more truths or by more closely approximating the truth.⁸ But this is not possible in mathematics, if there are no truths to begin with. Instead of truth and progress in the normal sense, Kitcher proposes an alternative:

Mathematical progress, in a nutshell, consists in constructing a systematic and idealized account of the operations that humans find it profitable to perform in organizing their experience. Some of these operations are the primitive manipulations with which elementary arithmetic and elementary geometry begin. Others are first performed by us through the development of mathematical notation that is then employed in the sciences as a vehicle for the scientific organization of some area of experience. But there is no independent notion of mathematical truth and mathematical progress that stands apart from the rational conduct of inquiry and our pursuit of nonmathematical ends, both epistemic and nonepistemic. (1988, 315)

Kitcher is perfectly aware of the consequence of such a view, namely, that the only legitimate mathematics is that which is tied to our other aims. “Epistemic justification of a body of mathematics must show that the corpus we have obtained contributes either to the aims of science or to our practical goals. If parts can be excised without loss of understanding or of fruitfulness, then we have no epistemic warrant for retaining them” (ibid., 315–316). I quoted this passage earlier and noted the possibility of using it as a reductio ad absurdum of Kitcher's naturalism—we ought to have more confidence in the mathematics than in the philosophy that wants to dump it. I want to approach the issue again, this time with an eye to a different difficulty.

For those who delight in the esoteric outreaches of mathematics (e.g., transfinite numbers), the prospects of their favourite pastime look pretty grim on the Kitcher account. A great deal of mathematics does not hook up with science in any way, so by Kitcher's lights it is not epistemically warranted. Does this mean it should be discarded, perhaps kept around as an amusing pastime, like chess, or perhaps lumped together with speculative theology as the product of idle and uncritical minds?

G.H. Hardy boasted of the “uselessness” of his work in number theory. Later commentators have been amused to find that some of Hardy's most useless work turned out to have applications after all, some mundane (in heating systems), others hateful (military cryptography). Herein lies a problem for Kitcher. Useless mathematics has potential uses. What then is its status? It does not seem to me that its epistemic status has changed one bit. What has changed is that we have merely found an application for it. A working mathematician would point to the proof of a theorem to justify it, and she would do that before and after the theorem is applied. The proof will be grounded by established legitimate practice in Kitcher's sense or it won't; applicability is irrelevant.

Kitcher's naturalism does a reasonable job of explaining how the mathematics that actually is applied is connected to the world. (By “reasonable” I mean only that it has some initial plausibility; I am certainly not saying his account is correct, even in this limited regard.) This, of course, is hardly a surprise, since naturalism's main claim is that mathematics is linked in a fundamental way to the natural world. What Kitcher does not do is show how well-developed-but-not-applied mathematics can come to be applied.

Suppose we are interested in velocity. Recall the formalism for applying mathematics that we outlined in Chapter 1, which we illustrated with weight. The idea presented there was to let a mathematical system model a physical system, by providing a model with a similar structure to the physical system and a homomorphism that relates the two. This time we let D be a set of velocities (possessed by various bodies) and let D* = R, the set of real numbers; we let ≤ and ⊕ be the physical relations is the same velocity or less than and composition of velocities, respectively. The relations ≤ and + are the familiar relations on real numbers of equal or less than and addition. Given these stipulations, the two systems are and . In this way, numbers are associated with the velocities V, W, etc., in D by the homomorphism φ: D → R which satisfies the three conditions:

This is the same as the earlier example of weight. Thus, (1) says that if V is the same velocity or less than W, then the number associated with V is equal or less than the number associated with W ; and (2) says that the number associated with the combined velocity V ⊕ W is equal to the mathematical sum of the numbers associated with the objects separately; (3) stipulates a standard unit speed, 1, expressed in metres per second.

This characterization of applied mathematics favours Platonism, as I argued earlier, since we are implicitly endorsing the existence of a distinct mathematical realm with which we represent the natural world. And the ease with which Platonism can account for applied mathematics should be clear. It also has two great advantages over Kitcher's naturalism.

First, the unapplied parts of mathematics are epistemically legitimate, at least in principle. They are just as well confirmed and as well established as those parts that are applied. And, moreover, they are there waiting patiently for us, ever available for the potential needs of science.

Second, it is in harmony with the history of mathematics. Yes, there have been theoretical and conceptual changes in mathematics, and yes, there have been similar changes in the natural sciences. But, being distinct realms, they are independent from one another. No revolution in science would have any effect on mathematics except to stimulate research in that area, and no revolution in mathematics would have any effect on science, except for making known the existence of structures in the mathematical realm that could serve as models for scientific theories. This is not the way physics, chemistry, and biology interact with one another. The history of mathematics, as I've repeatedly stressed, supports the Platonist's two realms account, not Kitcher's one-world naturalism.

Using the formalism, these points can be illustrated by a simple example, the addition of velocities in classical and relativistic mechanics. Imagine a ball thrown forward with velocity V inside an airplane which is flying at velocity W with respect to the ground. I stress that W and V are velocities: they are physical properties, not numbers. In accord with the preceding scheme, we associate real numbers with these velocities: φ (V) = v and φ (W) = w. In classical physics the composition of velocities, ⊕, takes a simple form:

However, in relativistic physics the composition of velocities is more complicated:

Obviously, this was not an overthrow of our previous beliefs about mathematical addition. Indeed, the old mathematical “+” plays a role in the new formula—it's still addition. Rather, we have simply picked out a different mathematical structure on which to model the physical composition of velocities. In Kitcher's naturalism, mathematics is intimately linked to the natural world; in Platonism, they are separate realms. This example and countless others like it support the sharp distinction upon which Platonists insist.

Needless to say, “support” isn't quite the same as “overwhelmingly justifies.” There has always been and always will be considerable sympathy with naturalism, even when it incorporates mathematics. Kitcher's programme has much to commend it, including a detailed and plausible account of the historical development of pure mathematics. But there are sufficiently many problems of a sufficiently challenging character that Platonism and perhaps other anti-naturalist rivals seem on balance considerably more plausible.

RECENT WORK ON CONCEPT ACQUISITION

One of the fundamental challenges for any account of mathematics is concept acquisition. For Kitcher and other empiricists the challenge is to give an empirical account of things which are prima facie invisible and intangible. We start, Kitcher says, with small objects that we can see and manipulate, and we take it from there. Though his book was important and influential, things have come some way since Kitcher's The Nature of Mathematical Knowledge was first published. In the balance of this chapter I will critically discuss two recent works on concept acquisition that are bound to be influential in the near future. Both are naturalistic.

Grounding Concepts: An Empirical Basis for Arithmetical Knowledge (2008) and The Origin of Concepts (2009) are important books. The first is by Carrie Jenkins, a philosopher of mathematics who is very much in the analytic tradition; the second is by Susan Carey, a psychologist who is well connected to philosophy and sympathetic to its problems. Both are deeply concerned with how we acquire our arithmetical concepts and both are empiricist-minded. Their approaches, however, are strikingly different.

Jenkins aims to reconcile three conflicting beliefs: (1) our knowledge of arithmetic is a priori, (2) the facts of arithmetic are objectively true (this is realism, which she characterizes as mathematics being independent from our mental lives), and (3) sensory input is the only source of knowledge. Her solution, in brief, is to say we acquire arithmetic concepts through empirical experience but we learn mathematical truths through the analysis of those concepts.

The principal claim Jenkins makes is that our concepts are grounded. It is not clear what this means, but central is her claim that a concept is grounded if and only if it accurately reflects some feature of the world. The reason this is not clear is that Jenkins wants to distinguish between grounding a concept and acquiring it. It seems that the former is raw material for the latter. Jenkins posits an I-mechanism, an innate capacity for turning sensory input into concepts. This is, we should point out, speculative psychology, always a dangerous activity for a philosopher. In any case, it is all empirical: “The only data which could be relevant to concept justification and concept grounding are data obtained through the senses” (Jenkins 2008, 137).

Jenkins resists using “analytic” for the truths “7 + 5 = 12” and “Vixens are females,” because she finds the term somewhat ambiguous. But we won't go far wrong in calling her “conceptual truths” analytic, anyway. This is the source of our knowledge. “Arithmetical truths are conceptual truths; that is, we can tell that they are true just by examining the concepts” (2008, 8). Hence, in this sense they are a priori.

Grounded concepts track the world. They should be accurate. This is what gives arithmetic its objectivity, and it is why we can be realists about arithmetic truths. Now we have the three desired ingredients, nicely reconciled: Arithmetic is empirical, because concepts are grounded empirically; it is a priori, because the truths are analytic (conceptual truths); and the grounded concepts reflect the world, so we maintain objectivity (realism). Jenkins's view—if it works—should put many naturalist anxieties to rest.

Jenkins illustrates her view with the acquisition of the concepts “7,” “5,” “+,” “=,” and “12.” Reflecting on these yields the truth “7 + 5 = 12.” This seems initially plausible. But arithmetic is more than a body of such particular truths. It also includes the principle of mathematical induction. It is much harder to see how induction arises from an analysis of individual concepts.

We might try the following analogy. When teaching mathematical induction, some use a thought experiment involving dominoes. Imagine a row of dominoes on their edge, close to one another so that if any one falls, it will knock over the next one. We then knock over the first domino. Given this much information, can we predict what happens? Yes, quite easily; they all fall over. With two idealizations we have mathematical induction. The idealizations are that there are infinitely many dominoes and that each takes no time to knock over the next in line.

Might this serve as a grounding and eventual conceptualization of induction? Perhaps. A Platonist might say that the process of the domino thought experiment acts as a kind of trigger, helping the mind's eye see the Platonic facts, just as she might say the roundish physical object is not the source of the concept of a perfect circle, but a perhaps necessary empirical stimulus to grasping it.

Susan Carey is also concerned with the origin of concepts, but her main focus is on number concepts in children. She has a different story to tell from Jenkins. Carey begins with the usual distinction between sensory/perceptual cognition and conceptual cognition. She then divides conceptual cognition further into two separate levels: “core cognition” and “intuitive theory” cognition. Core cognition, she holds, is an evolutionary adaptation, while intuitive theory cognition is the kind of rational thought that underwrites, say, normal scientific theorizing. For instance, a concept of core cognition is that of “object.” The idea that the world is made up of self-subsisting objects is an innate evolutionary adaptation, not something which each human individual has to be “taught,” such as the properties of levers or the conservation of energy. The concepts of core cognition can be possessed pre-linguistically, unlike intuitive theorizing, which is always linguistically explicit. Some concepts, such as object, number, and agent, are not in need of a conceptual foundation or an experiential explanation, because Darwinian natural selection gives a causal account of why they are present in the mind as (pre-linguistic) modules of core cognition. Note the contrast with Jenkins. The concepts of core cognition overlap with those of intuitive theorizing, which is why they are able to explain its occurrence; but nevertheless they are distinct, particularly in terms of how they are active in cognition.

Carey presents two distinct systems of core cognition that she believes infants use to operate on individual file representations in a numerical way: “analogue magnitude representation” (AMR) and “parallel individuation of small sets” (PI). I won't describe these in detail, but only note that both involve taking sets of individuals as input and give representations that support quantitative computation as output. However, differences in the formats of the two systems are betrayed by differences in the nature of the outputs. For instance, AMR computation obeys Weber's law (magnitudes can be distinguished as a function of their ratios, e.g., a group of two is easily distinguished from a group of three, but a group of 102 is not easily distinguished from another of 103). PI computation does not obey Weber's law. On the other hand, PI computation can handle continuous quantities, while AMR only computes discrete variables. Since AMR is underwritten by the ability to subitize small groups of objects and since it involves ratios, it will not work beyond rather small numbers, up to three for children and only up to four for adults. These are the innate mechanisms evolution has provided us with, according to Carey.

Quite aside from the theory proposed by Carey, philosophers, especially Kitcher, should find interesting and valuable the wide array of experiments she reports. For instance, two- and three-year-olds can count objects up to six or so. That is, they pair the objects with the number words, one, two, three, . . . But when asked to give someone, say, three of those objects, they give a random handful. They do not yet have the number concept three, in spite of being able to count that high.

Children this age are called “one-knowers” when they can pick out one thing, but they still cannot distinguish two things from three. Only after several months do they become “two-knowers” and another several months later “three-knowers.” Only later do they become “cardinal principle knowers,” able to distinguish any specific number of things. It may seem perplexing that a child could grasp the numerical list, 1, 2, 3, . . . , but not be able to sort out three things. A simple example may make this more intelligible. Imagine saying the alphabet while pointing to individual cookies in a bowl. One could do this quite well but be utterly perplexed when asked to give someone, say, f or g or h cookies. Mastering the cardinal principle is a significant intellectual achievement.

Carey takes this to show there is a discontinuity in conceptual development. So a question arises: How does conceptual development proceed? Carey turns to Quine for the answer. She rejects Quine's empiricism when it comes to concept acquisition, but she warmly embraces “Quinean bootstrapping.” (Neurath's boat—we make changes at sea, trying to stay afloat—and other such metaphors are offered as an explanation of bootstrapping.)

Carey proposes, “Numerical list representations are bootstrapped from representations of empirical parallel individuation” (2009, 325). A child easily learns “one, two, three, . . .” (lots of primates can do this), but it is initially as meaningless as “eenie, meenie, miny, mo.” The question is: How does the child learn that the number word “seven” means seven? Carey suggests a bootstrapping process in which the child learns “one” means one; learns that the quantifier word “a” is linked to one; learns linguistic plurals and learns that they are linked to other number words; and so on. Eventually, “one,” “two,” “three” are linked to the first three words in the previously meaningless number list. The pattern becomes obvious and the rest follows on.

As Carey notes, bootstrapping is not a deductive process. There is no reason to think it should be unique, yet it does seem the same in all cultures. Some societies do not get passed “one, two, many,” but those that do will do so in the same way. By contrast, people with the same empirical input often develop very different physics. How do we explain the difference between the mathematics and the physics cases? Quinean bootstrapping is really no help.

Platonists might agree with Carey that something important here is innate, provided by evolution, and moreover, that it is in some sense correct. I think Kitcher would do well to embrace this innateness as the origin of number concepts in place of his observation and manipulation of small collections of objects. But from that point, some sort of intellectual cognition takes over to develop mathematics. Scientific realists who claim that causation is part of objective reality have no trouble allowing that some basic notion of cause is innate. Sometimes survival value and truth diverge, as in colour perception, but other times they are the same, as in cause and number. How such core cognition develops is something Platonists and naturalists will continue to dispute.

Perhaps the most important moral for philosophers to draw from Carey's work is that standard empiricists’ accounts of concept acquisition in mathematics are implausible. This is a challenge for Jenkins, Kitcher, and the whole range of empiricist accounts of arithmetic.