4    Just Metaphor?

Lakoff's Language¹

 
In their book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (2000), George Lakoff and Rafael Núñez consider the nature of mathematics in the framework of cognitive science. Contemporary cognitive science has revolutionary consequences for the philosophy of mathematics, especially the fact that mathematics involves conceptual metaphor. At least, so they say. They further claim that their results not only refute traditional Platonism about mathematical entities like numbers, but also provide a completely new philosophy (and even a new pedagogy) of mathematics that can take its place (2000, 9).

I will argue to the contrary, that findings in cognitive science that have any bearing on the metaphorical character of mathematics are not incompatible with Platonism. In fact, the metaphorical nature of mathematics, such as it is, is actually better understood within a Platonist framework. Contrary to Lakoff and Núñez's assertions, their rejection of Platonism has little to do with their empirical results per se and instead issues from their own broader “philosophical” outlook and certain political views they hold regarding Platonism. They are quite ill conceived. Consequently, we shall find that there are no compelling grounds for abandoning Platonism coming from that quarter.

Though the Lakoff and Núñez view may seem implausible from the start, it nevertheless deserves consideration and public attention. For one thing, Lakoff and his other co-authors, such as Johnson,² have been highly influential in promoting their particular views about metaphor. Consequently, the Lakoff and Núñez account of mathematics will tend to ride the coattails of this earlier, popular work. For another thing, their work on mathematics and metaphor has already been given a favourable reception in some (but certainly not all) mathematical quarters.³ It is important to make their mistakes clear for all to see.

MATHEMATICS AS METAPHOR

Let's begin by laying out Lakoff and Núñez's view. They sketch their general argument right at the outset. It is both naturalist and explicitly anti-Platonist. Platonic mathematics (like belief in God) is a matter of faith, they say, not the subject of scientific study. The only conception we can have of mathematics is of a mathematics that is a human conception, something understandable in terms of bodies and minds. The nature of this human mathematics is a question of empirical science; it is neither a mathematical question nor a question of a priori philosophy. Therefore, it should be studied by cognitive science, the discipline that focuses on minds and brains. The burden, they insist, is on any other view to show itself scientifically respectable (Lakoff and Núñez 2000, 2–3). In their own words,

Mathematics as we know it is human mathematics, a product of the human mind. Where does mathematics come from? It comes from us! . . . Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history. (2000, 9)

This “cognitive perspective” translates into a methodology for studying “what mathematics is”; it employs methods of empirical cognitive science to investigate the largely unconscious conceptual structures and thought processes that characterize mathematical understanding. More specifically, Lakoff and Núñez seek to account for much of this subject matter in terms of conceptual metaphor, “a mechanism for allowing us to reason about one kind of thing as if it were another” (2000, 5) and “to use the inferential structure of one conceptual domain . . . to reason about another” (2000, 6).

Lakoff and Núñez start with an “innate arithmetic” that allows us to grasp number and the operations of addition and subtraction. This hardwired component is “minuscule” in comparison with mathematics as a whole (2000, 337). Nonetheless, by virtue of this innate arithmetic, we are born with rudimentary concepts of number, addition, and subtraction on the very small natural numbers (from one up to three or four). In support of this capacity they cite an array of empirical studies on subitizing that suggests infants have some grasp of simple arithmetic truths such as “1 + 2 = 3.” (Carey, who was briefly described in the last chapter, is more up-to-date on the findings of cognitive science. For the most part I will not argue with Lakoff and Núñez as far as their explicit empirical claims are concerned, but it will be obvious in some places that their account is at odds with Carey's.)

Although we are born with this degree of understanding of arithmetic, Lakoff and Núñez appear to hold that everything else we understand, think, or know about arithmetic and every other branch of mathematics, including algebra, set theory, and logic, is the product of conceptual metaphor. Furthermore, conceptual metaphors “cannot be analyzed away. Metaphors are an essential part of mathematical thought, not just auxiliary mechanisms used for visualization or ease of understanding” (2000, 6).

Lakoff and Núñez's deployment of conceptual metaphor to explain our grasp of mathematics employs two main kinds of metaphor (they make minor use of other kinds, discussed in the following). The first of these is grounding metaphor : such metaphors “allow you to project from everyday experiences . . . onto abstract concepts” (2000, 54). In four of these grounding metaphors, a set of “basic truths” about some domain of everyday empirical experience (“object collection and construction, motion and manipulation of physical segments,” 2000, 78) is mapped onto our innate concept of number, resulting in “a set of ‘truths’ about the natural numbers under the operations of addition and subtraction” (2000, 56). Another grounding metaphor “grounds our concept of a class in our concept of bounded region of space” (2000, 123). The idea is that many properties we attribute to mathematical objects such as numbers (e.g., they have parts) arise because we think of them, metaphorically, as physical objects. However, these grounding metaphors arise in a “natural” way in pre-linguistic young children, occurring in an “automatic, unconscious form” (2000, 55, 77) and “usually require little instruction” (2000, 53). (This is at odds with Carey's report of how children learn. Early on they can count objects, that is, associate them with the words “one,” “two,” “three,” . . . , but have no idea what it means to gather, say, three of them. That ability, she notes, comes considerably later.) These metaphors, by their ability to “directly link a domain of sensory-motor experience to a mathematical domain,” Lakoff and Núñez claim, give to our basic conceptions of arithmetic and set theory a grounded or “embodied” character (2000, 102).

However, Lakoff and Núñez think that most of mathematics does not arise via such natural grounding metaphors. Branches of mathematics other than our basic understanding of the natural numbers, arithmetic, and classes are explained by linking metaphors, which “allow us to conceptualize one mathematical domain in terms of another mathematical domain” (2000, 150). For example, the Boolean logic of classes is seen as produced by understanding classes metaphorically as abstract algebraic elements (2000, 128). Unlike grounding metaphors, these metaphors do not arise naturally in virtue of our every-day experiences; they “require a significant amount of explicit instruction” and are the conscious innovations of great mathematicians (2000, 53). Despite the fact that most mathematics does not arise directly from grounding metaphors that bring embodied experiential concepts to mathematics, Lakoff and Núñez see the entire body of mathematics as embodied or grounded. This is because “by means of linking metaphors, branches of mathematics that have direct grounding are extended to branches that have only indirect grounding . . . Yet ultimately, the entire edifice of mathematics does appear to have a bodily grounding” (2000, 102).

METAPHOR AND PLATONISM

Lakoff and Núñez assert unequivocally that their theory of mathematical cognition has radical implications for the philosophy of mathematics, and in particular for the view they pejoratively call the “Romance of Mathematics.” Amongst other things, this view, which they hold up to ridicule, holds that mathematics is “abstract and disembodied—yet it is real” (2000, vx), that mathematics is “an objective feature of the universe, that mathematical objects are real, and that mathematical truth is universal, absolute, and certain” (2000, 339). Also, mathematics has “an objective existence, providing structure to this universe and any possible universe, independent of and transcending the existence of human beings or any beings at all.” Further, “human mathematics is just a part of abstract, transcendent mathematics” (2000, xv).

These statements (poorly) describe a generic version of Platonism:

1. Mathematical entities are abstract Platonic entities (existing independently of any conscious beings).
2. The entities referred to in the mathematical thoughts of humans are Platonic entities.
3. Known mathematical facts are known with certainty.

(1) is an existence claim, (2) is a claim about the relation between Platonic entities and our mathematical thought, and (3) is an epistemological claim about mathematical knowledge. The first two are part of standard Platonism, but the third claim is not, since most current Platonists are fallibilists. (I have stressed this earlier, but more in the following.)

Since Lakoff and Núñez's remarks entail these claims, Platonism is a part of the “Romance of Mathematics,” a myth, as they see it, that is “the standard folk theory of what mathematics is for our culture” (2000, 340). According to them, “every part of the romance appears to be false” (2000, xvi). Their reasons for thinking this are directly tied to their empirical hypothesis concerning mathematical cognition. “It had become clear,” they write, “that our findings contradicted this mythology” (2000, 339). Later they say of Platonic accounts that “such answers are ruled out by the cognitive science of mathematics in general and mathematical idea analysis in particular” (2000, 366). The supposed incompatibility between their empirical results and Platonism centres on claim (2), that, according to Platonism, the reference of human mathematical thought (what mathematical thought is about) is some Platonic entity. Lakoff and Núñez allow that the existence claim (1) is (or might be) true; that is, perhaps mind-independent Platonic mathematical entities actually do exist. However, they infer from their theory of mathematical cognition that human mathematical thoughts are not about such entities—that is, that (2) is false. In their words:

Mathematical idea analysis shows that human mind-based mathematics uses conceptual metaphors as part of mathematics itself. Therefore human mathematics cannot be a part of a transcendent Platonic mathematics, if such exists. The argument rests on analyses we will give throughout this book to the effect that human mathematics makes fundamental use of conceptual metaphor in characterizing mathematical concepts. Conceptual metaphor is limited to the minds of living beings. Therefore, human mathematics (which is constituted in significant part by conceptual metaphor) cannot be a part of Platonic mathematics, which—if it existed—would be purely literal. (2000, 4)

Surprisingly, Lakoff and Núñez do not develop this line of argument in detail, but it can be fleshed out. Consider the following reconstruction:

1. Human mathematical thoughts attribute mathematical properties/concepts to numbers in a metaphorical manner (alleged empirical result).
2. Platonic entities exist and do so independent of the presence of any conscious beings (assumption for sake of the argument).
3. If a realm of thought attributes properties to X metaphorically, then X does not literally possess those properties (Metaphor Principle).
4. Therefore, Platonic entities do not literally possess any of the mathematical properties attributed to numbers in human mathematical thought (from 1–3).
5. Therefore, it is not the case that the entities referred to in mathematical thoughts about numbers are Platonic entities, even if Platonic entities do exist (from 4).

This argument has several promising features: The inference to proposition four is valid, the argument makes essential use of a premiss describing Lakoff and Núñez's empirical theory of mathematical cognition, and its third premiss describes, more or less precisely how the content of this empirical theory is relevant to the conclusion. It also has affinities with other anti-realist arguments that exploit the metaphorical nature of certain descriptions. If Platonists concede that what they say about numbers is metaphorical, then it appears that their claim that numbers really exist and that we know things about them cannot be taken seriously, since the existing “numbers” are radically unlike the entities we typically think of as numbers. This seems like a strong argument against Platonism. Let's investigate it further.

THE CONSTRASTIVENESS OF METAPHOR

Lakoff and Núñez's argument turns on the truth of the Metaphor Principle in premiss (3), a principle with some plausibility. Consider an utterance of “Harry is a rat.” If someone says this metaphorically, we can infer that they are not talking about a rat named Harry. The reason, obviously, is that metaphors are non-standard usages of language. Whatever else one wants to say about it, it is of the essence of metaphor that it is non-literal in meaning. If I know that Harry actually is a rat, and I know that the speaker of “Harry is a rat” knows this, then I will not be able to take the utterance as a metaphor, at least not in any straightforward sense. If I know that Harry is a rat, then if I want to describe him as devious or untrustworthy, it will not do to call him a rat, since he is one, and so hearers are likely to misinterpret what I say. A useful way to put this is to say that metaphors are contrastive, in the sense that they have or are associated with a meaning that contrasts with some other (literal) meaning.

The common theories of metaphor all impart this character to metaphors.⁴ On so-called pragmatic views, an expression used metaphorically has its literal meaning (Harry is a rat), but this meaning contrasts with the utterance meaning of the speaker (Harry is devious and untrustworthy). On semantic views of metaphor, the metaphorical expression has a meaning different from its literal one, but here again this special metaphorical meaning contrasts with the literal one that the expression usually has. This unanimity on the contrastiveness of metaphor is unsurprising, given that the non-standardness of metaphorical language is usually the point of departure for its theoretical investigation.

It would seem then that the Metaphor Principle, and hence the preceding argument, hinges on the contrastiveness of metaphor. It is this feature of metaphor that provides us with a route from the metaphorical character of language or thought to anti-realism about the content of that language or thought. However, when we try to extend the argument to mathematics, as Lakoff and Núñez do, a problem arises. The problem is that the notion of metaphor that Lakoff and Núñez use is (paradoxically) a non-standard one: Their notion of metaphor is not essentially contrastive. To see this, we need to look at each of the different sorts of conceptual metaphor that Lakoff and Núñez employ in their account of mathematics.

GROUNDING METAPHORS

Though not hardwired, grounding metaphors are “learned at an early age, prior to any formal arithmetic training” (2000, 55). They arise via regular “correlations” between the “most basic literal aspects of arithmetic” and “everyday activities . . . like taking objects apart” (2000, 54). These correlations “result in neural connections between sensory-motor physical operations like taking away objects from a collection and arithmetic operations like the subtraction of one number from another. Such neural connections . . . constitute a conceptual metaphor at the neural level” (2000, 55). Grounding metaphors consist in the transfer of a set of properties from one concept (e.g., object) to another concept (e.g., number). The transfer occurs when there is repeated correlation in our experience between applications of the two concepts.⁵ For example, the physical operation of removing a part of an object always produces a smaller object, and when we subtract the number of parts removed from the original number we always get a lesser number. Because of such correlations, we come to understand subtraction metaphorically as removal of object parts, and numbers metaphorically as objects, thereby transferring properties like “has parts” from physical objects to numbers.

However, this kind of metaphor does not require any contrastive element. Obviously, children forming and utilizing such metaphors as “numbers have parts” are not expressing or referring to any other sort of meaning, on which numbers do not really have parts, since these metaphors form at an unconscious level. More importantly, there is nothing in the characterization of a grounding metaphor to suggest that contrastiveness of any kind is required. Though a property attributed to the numbers is transferred from some other sort of entity, it need not be the case that numbers themselves fail to have that very property. For example, imagine with the Platonist that numbers exist and have parts. It is possible that children come to attribute having parts to numbers on the basis of their resembling objects, which also have parts. We might form this (unconscious) belief about numbers in terms of an analogy, rather than a (contrastive) metaphor. In short, Lakoff and Núñez's grounding metaphors are not essentially contrastive.

LINKING METAPHORS

Unlike grounding metaphors, linking metaphors are not formed unconsciously. On the contrary, their construction sometimes takes “extraordinary ingenuity and imagination by mathematicians” (2000, 117). Often Lakoff and Núñez's linking metaphors correspond to mathematical definitions, in which one sort of mathematical entity is treated as (or reduced to) another. Von Neumann's famous definition of the natural numbers in terms of sets, for instance, is one of their examples of a linking metaphor (2000, 141–142). The definition is as follows:

To see the implications of linking metaphor for the argument, consider a property that the natural numbers might have: being constituted by sets. According to Lakoff and Núñez's construal of von Neumann's definition, the statement “The natural numbers are constituted by sets” is metaphorical. But surely the fact that this claim is metaphorical in Lakoff and Núñez's sense does not entail that numbers are not really constituted by sets. For von Neumann's definition to be a linking metaphor, all that is required is that one mathematical entity be construed in terms of another. This need not involve a contrast with some other meaning, such that the constitution of numbers by sets is not what is really being asserted. As Lakoff and Núñez themselves note, definitions of this sort are often explicitly taken literally, in so far as they are seen as Platonic attempts at reducing one domain of mathematical entities (numbers) to another (sets) (2000, 150–151). However, this is not incompatible with the fact that the von Neumann definition is a linking metaphor, in Lakoff and Núñez's sense, for such metaphors are not essentially contrastive.

OTHER METAPHORS

Lakoff and Núñez also employ other sorts of metaphor, such as “extension metaphors.”⁶ These are consciously made “entity creating metaphors” (2000, 64), in which some mathematical objects are created ad hoc (2000, 64) in order to satisfy certain theoretical requirements (e.g., closure) (2000, 81). Basically, extension metaphors are uses of metaphor to fill gaps in the ontology of conceptual domains that serve as the sources for grounding and linking metaphors. For example, in their object collection grounding metaphor, there is no object collection that can be mapped onto zero. A new entity, the empty collection, must be conceived via an extension metaphor, and then this can be used as a metaphor for zero: “from the absence of a collection, the metaphorical mapping creates a unique collection of a particular kind— a collection with no objects in it” (2000, 64). The empty collection is seen as having the property “can exist with no members” because it is viewed metaphorically as the absence of a collection, which literally has that property. But again, it is hard to see why this account should disallow the empty collection's literally having this property. Extension metaphors are not contrastive.

METAPHOR AND ANALOGY

The argument against the relational claim (2) of Platonism (that the objects of our thought are Platonic entities) fails when applied to mathematics in the way suggested by Lakoff and Núñez. The reason is that the Metaphor Principle in premiss (3) requires that metaphor be contrastive, while the various forms of metaphor that Lakoff and Núñez hold as relevant to mathematics are not. In general the difficulty is that in order to allow their notion of metaphor to cover all of mathematical thought, Lakoff and Núñez stretch it so that it is hardly recognizable. For them, metaphors occur when one notion is understood in terms of some other, yielding “a grounded, inference-preserving cross-domain mapping” (2000, 6). The broadness of this notion is evident in that they categorize as metaphorical much language that is, prima facie, literal. They say, for instance, that the statement “The relationship was so unrewarding that it wasn't worth the effort” is “rarely noticed as metaphorical at all” (2000, 46). To most of us, and to most metaphor theorists, this is because it isn't a metaphor. The oddity of Lakoff and Núñez's sense of “metaphor” is also evident from its tendency to obliterate the distinction between metaphor and analogy. Analogy is also a way of reasoning about one thing as if it were another, but analogies have no anti-realist implications. Saying (or thinking) “The water is as smooth as glass” does not commit one to its not really being smooth, or as smooth as glass is.

Since Lakoff and Núñez are using “conceptual metaphor” as a term of art, they are free to define it as they like. Perhaps there are compelling reasons for defining metaphor this way within cognitive science. Be that as it may, the anti-realist and anti-Platonist implications that they claim for their view of mathematics require the more typical contrastive meaning of “metaphor.” Hence their refutation of Platonism only works in so far as they slide between this typical meaning and their technical one.

PLATONISM AND METAPHOR

If “metaphor” simply means seeing something in terms of another thing (i.e., if metaphor is not contrastive), then it seems that a Platonist could accept with equanimity that much or most of our mathematical understanding involves metaphors involving physical objects. In fact, the idea that a grasp of abstract entities proceeds not directly, but is mediated (at least initially) via our cognition of the concrete is a familiar one in Platonic thought. Of the prisoner dragged from the cave into the sunlight, Plato's Socrates says, “at first, it would be shadows that he could most easily make out, then he'd move on to the reflections of people and so on in water, and later he'd be able to see the actual things themselves” (Republic, book VII).

Indeed, the vast majority of mathematicians would not object to metaphors, similes, and analogies in mathematics. Frege (1971), for example, said that the right way to present a mathematical theory is to posit some primitive (i.e., undefined) terms, then assert some axioms that use those terms. After that, we can define new terms using the primitives, prove theorems, and so on. Subsequent terms are to obey the two criteria of eliminability and non-creativity. That is, any defined term can be replaced by primitives (this is the eliminability criterion) and nothing can be proven using defined terms that cannot be proven using only primitives (this is the non-creativity criterion). Undefined terms are necessary, since to have everything defined means that we either have circular definitions or an infinite regress. This attitude toward definitions is common today.⁷

Frege added that while we could not define everything, we could and should give an “explication” of undefined concepts, something that is not, strictly speaking, part of the formal theory. Set theory, for example, is typically presented axiomatically with two undefined concepts: set and member. However, textbooks usually say a few things about sets, helping us to understand them. Thus, we are told: “A pack of wolves is a set,” “A flock of birds is a set,” and “The individual birds are members of the flock.” These statements are not part of the theory proper, but rather are metaphors (though Frege did not put it this way) that are intended to aid understanding. A strict formalist, unlike Frege who was a Platonist, would not want an explication of “set”; it's just a symbol and the axioms tell us how to manipulate it. Most people, however, including most mathematicians, welcome the sort of explication one typically finds in textbooks. The metaphors are helpful because they allow us to focus on certain properties that sets and other mathematical creatures have.

Are they necessary? Frege implicitly says yes. Primitive terms such as “set” are not defined, so the only way to understand them (at least initially), is in this extra-theoretical way, through analogies, similes, and metaphors. On this traditional Platonic view, consequently, metaphors have a critical role, albeit it is ultimately only a heuristic and pedagogic one. But I wish to claim something further. Not only does mathematical metaphor fit comfortably with Platonism, it is better understood in a Platonist framework than it is in Lakoff and Núñez's naturalistic alternative.

Consider Lakoff and Núñez's treatment of sets. They tell us that our “concept of a class is experientially grounded via the metaphor that Classes are Containers” (2000, 140). According to this metaphor, “a class of entities is conceptualized in terms of a bounded region of space, with all members of the class inside the bounded region and all nonmembers outside the bounded region” (2000, 122). This notion of a class, supplemented with an additional metaphor that provides sense to the empty and universal classes, serves as the basis for our notion of set. However, “sets are more sophisticated than Boolean classes,” in that sets can also be members of other sets. Classes can be subclasses of one another (one bounded region may overlap another) but they cannot be members of one another. To get our notion of set, then, we employ the “Sets are Objects” metaphor: A set can be “inside” a bounded region of space if we think of it as an object (2000, 141).

This understanding of “set,” however, does not mirror the mathematical sense of the term. In fact, it violates it seriously. It would do to recall what was said earlier about the difference between sets and such things as flocks and packs, etc. For instance, a flock flies south for the winter, but sets don't fly anywhere; a pack might devour a deer, but a set couldn't eat anything; and so on. I'll elaborate a bit and illustrate the vast differences between the kind of thing that Lakoff and Núñez have in mind and the proper conception of a set.

Consider the following baskets (in the middle) in the table here, and the set that corresponds to them (on the left), and the remark (on the right). Let a and b be distinct individual objects, say, apples. In the first four situations, baskets (containers) partially illustrate how sets work, but not particularly well. In the final situation, the metaphor (or simile or analogy) utterly breaks down. A physical object such as an apple cannot be in two places at the same time; it cannot be both inside and outside a basket simultaneously. Yet this is what the container metaphor implies if we take it seriously. The basket metaphor works well for some aspects, but for even a simple set such as {a, {a}} it makes no sense at all.

If sets are not understood as spatial entities, then being in two “places” at the same time is not a problem, since nothing is located anywhere in the spatial sense. The container metaphor is a good pedagogic device for getting started in set theory, but it quickly breaks down. The fairly obvious moral to be drawn from this example is one that would cheer any Platonist. The ways in which things belong to sets simply do not correspond to the ways objects are contained in space. Metaphors such as spatial containment do not let us grasp correctly the nature of sets. They can be illuminating in some respects, but ultimately an understanding of set theory, far from being constituted by such metaphors, is actually a matter of overcoming the limitations that these metaphors place on our intuitions. For the Platonist, the limitations of such metaphors are to be expected, whereas on the Lakoff and Núñez approach they are intolerable, since they suppose there is nothing beyond the metaphor.⁸

The view that mathematics is constituted by metaphor has further uncomfortable consequences. Lakoff and Núñez define cognitive metaphor as an “inference preserving cross-domain mapping—a neural mechanism that allows us to use the inferential structure of one conceptual domain (say, geometry) to reason about another (say, arithmetic)” (2000, 6). In a typical metaphor (e.g., an utterance of “Harry is a rat”), we already have an object in the target domain. Harry already exists. Lakoff and Núñez, however, maintain that in some cases metaphors create mathematical entities. Complex numbers, differentiable manifolds, and the like do not exist prior to being cognized metaphorically; they are made by the metaphor.

There is a problem with this, because most often we do not accept all the entailments of a metaphor. In the “arithmetic as object collection” metaphor, for example, we carry over to numbers some properties but not others (e.g., mass, colour).⁹ But how do we know which properties to carry over and which to abandon? In uttering “Harry is a rat” we intend to carry over certain traits of character, but not the property of having a long, hairless tail. This is recognized because we have independent knowledge of Harry.

Again, this difficulty does not arise on the Platonist view. According to the Platonist, metaphors and analogies are useful in teaching mathematics because parents and teachers (if not the children who are learning) have an independent grip on numbers and understand that mass and colour are not to be carried over from bananas to numbers. This is a perfectly intelligible point, if metaphors are taken to be merely pedagogic devices that allow us to focus on properties of mathematical objects that we already grasp. It is quite mysterious, however, if metaphors are actually creating numbers.

PHILOSOPHICAL ARGUMENTS AGAINST PLATONISM

What the preceding argument suggests is that far from showing Platonism to be false, the metaphorical nature of mathematics that Lakoff and Núñez claim to have empirically discovered accords well with Platonism, if it does not in fact support it. This casts doubt on Lakoff and Núñez's claim to be refuting Platonism with empirical scientific results. In fact, a closer reading of their book reveals several arguments against Platonism that are outright philosophical (that is, something akin to a priori conceptual analysis), and which have little or nothing to do with empirical research in cognitive science, all of which is rather ironic, given their anti–a priori stance. The first of these is tendered in their Introduction:

The question of the existence of a Platonic mathematics cannot be addressed scientifically. At best it can only be a matter of faith, much like faith in a God. That is, Platonic mathematics, like God, cannot in itself be perceived or comprehended via the human body, brain, and mind. Science alone can neither prove nor disprove the existence of a Platonic mathematics, just as it cannot prove or disprove the existence of God. (2000, 2)

This argument is clearly motivated by a strong naturalist sentiment. It seeks to strip Platonism of any cognitive status: It cannot be a justified belief, but must be, like religion, a mere leap of faith with no rational grounds to support it. The argument, however, is a non sequitur. Just because the truth or falsity of Platonism cannot be determined in the laboratory does not mean there is no way to determine its truth or falsity. The analogy with God is instructive here. Many philosophers think there are excellent reasons for dismissing God's existence, but the fact that no one has yet detected him in the lab or run into him at the supermarket is not one of them. Many of our most strongly held beliefs are not, and could not be, justified “scientifically,” if that means empirically. Beliefs about whether abortion is morally permissible or beliefs about whether Hamlet is a great play are instances of this. Yet, it would be perverse to call such beliefs mere matters of faith or arbitrary stipulations. In short, this argument seems to rely on an extreme—and extremely implausible—form of positivism.

Lakoff and Núñez also attack Platonism for failing to account for mathematics’ historical evolution. “In the Romance of Mathematics,” they write, “culture is assumed to be irrelevant. If mathematics is an objective feature of this or any other universe, mere culture could not have any effect on it” (2000, 355). They remark:

Many of the most important ideas in mathematics have come not out of mathematics itself, but arise from more general aspects of culture. The reason is obvious. Mathematics always occurs in a cultural setting. General cultural worldviews will naturally apply to mathematics as a special case. In some cases, the result will be a major change in the content of mathematics itself. (2000, 358)

As one example of the effects of culture on the “permanent content” of mathematics, Lakoff and Núñez cite the influence of Greek essentialism on the axiomatic method, the influence of the idea that reason is mathematical calculation on our conception of mathematical reason, and the influence of Aristotle's idea that theories must have foundations on the Foundations of Mathematics movement (2000, 356–358).

One might dispute whether these are bona fide examples of cultural influences on mathematics; Lakoff and Núñez provide virtually no historical evidence for these claims. Even waiving this objection, however, it is hard to see just what is here supposed to be incompatible with Platonism. No Platonist would deny that mathematics occurs in a cultural setting or that ideas from that setting can stimulate and have an impact upon mathematicians’ understanding of the entities they study. For the Platonist, of course, these social factors alter the permanent content of mathematics by leading the mathematician to discover new independently existing facts, or to understand certain facts in a particularly fruitful way. On this view, certain social factors have changed the content of mathematics not because mathematics fails to be objective, but because they happened to provide fruitful strategies for discovery and understanding. In this regard, mathematics, according to any Platonist, is akin to physics and biology, which also evolve in a cultural setting.

FALLIBILITY

Lakoff and Núñez also fault Platonism for its alleged commitment to the infallibility of mathematical knowledge. They seem to think that, according to Platonism, any mathematical beliefs that we do know we would know with certainty; this is claim (3) in the gloss I gave earlier of the “Romance of Mathematics.” Since they think that “every part of the Romance appears to be false” (2000, xvi), it would seem that they believe mathematical knowledge should be fallible, rather than certain.

However, it would be a serious mistake to think that all Platonists believe mathematical knowledge to be known with certainty. Kurt Gödel is a classic counterexample. To repeat what I earlier stressed, Gödel compared knowledge of Platonic entities to our knowledge of physical objects and he stressed the parallel between the possibility of error in our knowledge of physical objects and the possibility of error in our knowledge of Platonic entities: “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-theoretical paradoxes are hardly any more troublesome for mathematical intuitions than deceptions of the senses are for physics” (Gödel 1947/1964, 484).

Even though mathematics is a priori on Gödel's Platonist account (i.e., it is not known through empirical experience), it is not a body of certain truths. “A priori” and “certain” are quite distinct concepts.¹⁰ The mind's eye is subject to illusions and to the vicissitudes of concept formation, just as the empirical senses are. Further, mathematical axioms are often conjectures, proposed to capture consequences that are intuitively grasped, but not self-evident propositions themselves. Conjecturing in mathematics is just as fallible as it is in physics or would appear to be even in cognitive science.

Furthermore, we should note that it is not even clear that the metaphor-based account of mathematics offered by Lakoff and Núñez succeeds in making mathematics fallible. In order for mathematical knowledge to be fallible, it must be the case that our mathematical beliefs can turn out to be false. According to Lakoff and Núñez, a mathematical proposition is true when it “accords with” a subject matter (2000, 366). In terms of their metaphor-based account, a proposition is true when it accords with a certain metaphorical understanding we have of a certain subject matter. For example, “n + 0 = n ” is true because our grounding metaphors for arithmetic are such that “it is an entailment of each of these metaphors that n + 0 = n ” (2000, 367). However, this makes truth too easy to attain.

Take any arbitrary mathematical proposition that we have no reason to believe in. All we have to do is concoct a metaphor that it “accords with” or that “entails it,” then it is as true as a mathematical belief can be. Lakoff and Núñez rightly stress the creativity of mathematicians, but mathematics is not that creative. Again, if we want to explain how it is that mathematics can lead us to genuine error, how mathematical knowledge can require correction and revision, Platonism has the upper hand over Lakoff and Núñez.¹¹

UNIQUENESS?

A final philosophical argument that Lakoff and Núñez offer against Platonism involves the observation that “mathematical entities such as numbers are characterized in mathematics in ontologically inconsistent ways” (2000, 342). For instance, in geometry, numbers are treated as points on a line and in set theory as sets. This is a problem for Platonism, they say, because on that view, “there should be a single kind of thing that numbers are; that is, there should be a unique ontology of numbers” (2000, 343). They see Platonism as committed to this view because it “takes each branch of mathematics to be literally and objectively true” and so “claims that it is literally true of the number line that numbers are points, literally true of set theory that numbers are sets,” and so on (2000, 343).

First of all, it is far from clear that these are literal statements, in the eyes of a Platonist. They may just mean, for instance, that numbers are like points on a line; that is, they may be mere analogies. The vast majority of mathematicians would say that the geometrical line is isomorphic to the set of real numbers, or that Euclidean space has the same structure as R³. This is not the same as asserting identity. To say “points are numbers” is just a shorthand for this, and is quite different from saying points = numbers. Indeed, mathematicians might even underscore the claim by noting that there are many ways to “coordinatize” space; that is, there are many ways to attach numbers to points, a claim that only makes sense if numbers and points are distinct entities to start with.

POLITICS AND PEDAGOGY

Lakoff and Núñez's rejection of Platonism is not an inescapable consequence of the cognitive science of mathematics, nor the result of cogent philosophical argument. That's my claim so far, but I should also note that their attack on Platonism is fuelled in part by political and social motivations. In particular, they believe that Platonism is “doing social harm,” and that “at least indirectly it is contributing to the social and economic stratification of society” (2000, 341). According to them, the Platonist or Romance view of mathematics “is part of a culture that rewards incomprehensibility.” By treating mathematics as an objectively existing world with no real connection to human emotions, life, and reason, it “intimidates people,” and “leads many students to give up on mathematics as simply beyond them” (2000, 341). The Platonic view “helps to maintain an elite and then justify it.” Those unable to catch on to the esoteric technical language of mathematics are dismissed as simply lacking the capacity to see clearly into the Platonic realm. Many students are “alienated” from mathematics, and give up trying to learn it. But they are thereby deprived of the material benefits that it could convey to them. This translates into a “lack of adequate mathematical training to the populace in general,” putting many people at a disadvantage in a technical economy (2000, 341).

Lakoff and Núñez's general idea is that their cognitive metaphor view of mathematics would rectify this problem by making mathematics more intelligible and accessible to the masses. In particular, they think that current mathematics education results in poor learning. “Rote learning and drill is not enough,” they write. “It leaves out understanding. Similarly, deriving theorems from formal axioms via purely formal rules of proof is not enough. It, too, can leave out understanding” (2000, 49).

This is hardly a novel complaint. Platonists and others have raised similar grievances for two and a half millennia. Nevertheless, the root of this problem, according to Lakoff and Núñez, is that current mathematics education presents mathematics in a superficial way. Since the cognitive metaphor view offers a deeper understanding of what mathematics is about, it can, they claim, remedy this pedagogical defect.

Mathematical idea analysis, as we seek to develop it, asks what theorems mean and why they are true on the basis of what they mean. We believe it is important to reorient mathematics teaching more toward understanding mathematical ideas and understanding why theorems are true. (2000, xv; see also 11)

One case they use to illustrate how mathematical idea analysis can improve education is Cantor's treatment of the cardinality of infinite sets (2000, 142–144). They write,

Cantor . . . intended pairability to be a literal extension of our ordinary notion of Same Number As from finite to infinite sets. There Cantor was mistaken. From a cognitive perspective, it is a metaphorical rather than literal extension of our everyday concept. The failure to teach the difference between Cantor's technical metaphorical concept and our ordinary concept confuses generation after generation of introductory students. (2000, 144)

Lakoff and Núñez focus on Cantor's result C, as they call it: Infinite sets have the same number of members as some of their proper subsets. (Though they don't mention it, this is not due to Cantor but to Dedekind, and it is not a result but a definition of infinite set. I'll let this pass, since it's not important for their point or my criticism of it.) They claim that C is metaphorically true, but not literally true. An infinite set and its proper subsets (e.g., the set of natural numbers, 1, 2, 3, . . . and its subset consisting of even numbers, 2, 4, 6, . . . ) do not, they say, really (literally?) have the same number of members. According to Lakoff and Núñez, this is because our concept of “same number as” requires that two groups with the same number of members must be such that removing pairs of members, one from each group, never results in one group being empty and the other not. An infinite set and its proper subset, they say, fail to satisfy this condition.

Nonsense. Start with {1, 2, 3, 4, . . . } and the proper subset {2, 4, 6, 8, . . . }. Then remove pairs, one from each: we remove 1 & 2, then we remove 2 & 4, then 3 & 6, and so on. We never exhaust one set while the other still has members. Lakoff and Núñez have failed utterly to grasp the point of Dedekind's definition.

However, they continue, if we understand “same number as” metaphorically as “being one to one pairable,” then, claim Lakoff and Núñez, you can say with Cantor that such sets have the same number of members.

It seems unclear, however, that Cantor's construal of “same number as” as “being one to one pairable” in fact violates our standard notion of the concept. The construal is perhaps unusual, but it could be argued that it is perfectly defensible in light of an example or two. For instance, we can establish that the number of students in a room is the same as the number of chairs when we notice that bums and seats are in one–one correspondence, even though we have not explicitly counted either. And our baseten number system strongly suggests that our distant ancestors counted by pairing objects with fingers.

For the sake of argument, however, let's grant that Cantor's construal of “same number as” in C is a non-standard or metaphorical one. Following on this point, Lakoff and Núñez make two claims: (1) Cantor thought that C was not metaphorically true but literally true, and (2) mathematics teachers have employed Cantor's interpretation of C and this has impeded student learning.

Is the first claim true? Cantor would have thought that C was literally true just in case he thought that “same number as” did not have a non-standard meaning in that proposition. That is, if Cantor held C literally, then he thought that “same number as” as it occurs in C means just what “same number as” means elsewhere in its common and standard usage. In other words, all of what is meant by “same number as” generally is meant by it in C. Now Lakoff and Núñez themselves believe that part of what is meant by “same number as” generally is that no set can have the same number of elements as one of its proper subsets (2000, 143). But if this is part of the literal meaning of “same number as,” then the literal or standard meaning of C is simply a contradiction. Obviously, one of the greatest mathematicians of all time did not believe such a blatant contradiction.

Even if the first claim is false, however, the second could be true. Perhaps Cantor did not take C literally, or as involving (what Lakoff and Núñez take to be) the standard usage of “same number as,” but perhaps many mathematics teachers have done so and have thereby confused their students. However, most mathematics textbooks take pains to point out that “same number as” in Cantor's theory does not mean everything that common folk might take it to mean. This is usually done by the introduction of a new definition of cardinality for infinite sets in terms of one-to-one pairability. It would not seem, then, that mathematics teachers typically commit the error of taking C literally either.

Although it is generally seen as obvious that C involves a specific and possibly non-standard use of “same number as,” there is, perhaps, still something to be gained in understanding C if we treat the meaning of the phrase as metaphorical, as Lakoff and Núñez do, rather than simply as the introduction of a novel definition of “same number as” for infinite sets. But it is hard to see what this advantage would be. For the notion of metaphor used here is one wherein we purposely restrict our normal understanding of the meaning of a concept to one of its specific components. This is just what goes on when we define “same number as” in a way that restricts the scope of our usual folk concept of “same number as.” In short, it does not appear that metaphor-based “mathematical idea analysis” aids in understanding Cantor's result. This should lead us to be sceptical of Lakoff and Núñez's general idea that the cognitive metaphor view will make mathematics more intelligible and accessible than Platonism does.

Generally, Lakoff and Núñez presume that since Platonism holds that mathematics involves a realm beyond the physical and temporal, it entails that learning mathematics is more difficult. No reason is given for believing this to be true or for why it should be so. Ruben Hersh makes a similar claim in his book What Is Mathematics, Really? (1997). Like Lakoff and Núñez, he believes that “elitism in education and Platonism in philosophy naturally fit together” (1997, 238). Hersh's idea is that the “otherworldly” nature attributed to mathematics by Platonism justifies a student's conviction that he or she simply cannot learn mathematics.

Granted, many people do feel hopeless in the face of mathematics and this is indeed a genuine tragedy. But the naive explanation offered by Lakoff and Núñez and by Hersh strains credulity. Many people (often the very same ones who claim to be hopeless at mathematics) declare themselves to be hopeless at learning foreign languages. But languages are obviously a human creation and are hardly otherworldly. What grounds are there for thinking that pedagogical problems in mathematics should be pinned on Platonism? Lakoff and Núñez, who so often deride what they call a priori speculation, offer no empirical evidence for this claim. Neither does Hersh.

In fact, there are good reasons to think that Platonism is not to blame for widespread difficulties in learning mathematics. First of all, Platonism is a view with strong egalitarian roots. In the Meno, for instance, Socrates demonstrates that geometrical knowledge is available to anyone prompted to recollect the forms. Even a humble slave boy who has never been taught any mathematics figures out how to double a square (Meno 82a–85e). Second, concerning educational appeal, what evidence do we really have that making mathematics seem like a human creation will make it easier to teach or (perhaps more importantly) make it something that people will want to learn? Undoubtedly, humanizing mathematics will appeal to some, but it will also put off others. A significant number of people find mathematics appealing precisely because it is non-human. Bertrand Russell speaks for many when he says:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. . . . Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home. (1919, 60–61)

Many concur with Russell's view that the mathematical realm's “other-worldliness” is an incentive to study mathematics, including prominent mathematicians such as G.H. Hardy.

When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, “one at least of our nobler impulses can most easily take refuge from the dreary exile of the actual world.” (1992, 143)

Canadians play hockey and Cubans don't. The reason has nothing to do with class or culture, but rather with climate. Many children learn to play the violin. Learning the violin does have something to do with social class, since it is the rich who have the money for concerts and lessons. There is perhaps some justice in calling it an elitist activity. But if violin lessons were freely available to all (something for which any genuinely civilized society would surely strive) and if hockey rinks were to be built in abundance in Cuba, the learning processes would be similar. Hockey and the violin both take an awful lot of practice to master. I doubt it would make any difference to novices whether they think hockey is a human creation or that there is a Platonic form of hockey playing (which I'm sometimes tempted to believe). The learning will be the same either way. And the same, I suspect, will be true of foreign languages and mathematics.

Hersh takes his ideological critique of Platonism somewhat further than Lakoff and Núñez. Not only does Platonism oppress by bungling mathematical education, it is also, according to him, directly tied to odious political tendencies. Hersh classifies various philosophies as “left wing” or “right wing,” and characterizes these terms as, respectively, promoting or restricting “popular political rights,” though he does not say what these rights are (Hersh 1997, 239). Hersh works through a list of prominent philosophers who have said something about mathematics: Plato, Descartes, Leibniz, Frege, Brouwer, Quine, and Lakatos, among others are classed as right wing, while Aristotle, Spinoza, Locke, Hume, Kant, Mill, and Russell, among others, are left wing. Then he classifies them again as advocating a humanistic (Aristotle, Hume, Locke, Mill, and Lakatos) or anti-humanistic (Plato, Descartes, Leibniz, Frege, and Russell) philosophy of mathematics. Hersh notes a strong correlation: Anti-humanists are right wing and the humanists are on the left.

The classification of these various individuals is debatable (e.g., on the right he classifies Plato, who is often called a communist, and on the left he has Aristotle, who thought that women are inherently inferior and that some people are naturally slaves, and Hume, who was a staunch Tory). And the sample is certainly not random; lots of great mathematicians (not to mention ordinary mathematicians) have been omitted. But we will leave these obvious objections aside, even though they are enough to sink his potty view. What is Hersh's explanation for the correlation? He does not shy away from speculation: “Political conservatism opposes change. Mathematical Platonism says the world of mathematics never changes. Political conservatism favours the elite over the lower orders. In mathematics teaching, Platonism suggests that the elite student can ‘see’ mathematical reality or that she/he can't” (1997, 245–246).

This is not the place for an extended discussion on the politics of the realism versus anti-realism debate. It should be noted, however, that the Hersh view is mirrored in the so-called “science wars.” Social constructivists often claim that “objective science” is politically oppressive, but many on the political left (most famously Alan Sokal of the notorious hoax), claim the very opposite. They argue that left-wing social causes are much better served by an objective attitude towards science than by relativism. The Left does not have guns or money; the only weapons it has are objectivity and rationality. It would be folly to give up them up. Speaking for myself, at any rate, progressive politics has nothing to fear from Platonism.¹²

In sum, the naturalistic account of mathematics presented by George Lakoff and Rafael Núñez has little to commend it. I haven't discussed every aspect of their work, but examination of a reasonable sample suggests it fails in every important way. There is certainly a role for metaphor, simile, and analogy in mathematics, but it is decidedly not the role assigned to it by them. On all points Platonism and perhaps other non-naturalistic accounts of mathematics do a superior job.