Mathematical platonists, awake! Here, in a book written to put the final nail in the coffin of platonism, we have the beginnings of a response to the one serious philosophical challenge to platonism: how can human beings, finite physical beings, ever develop an understanding of mathematics, abstract and infinite, disjoint from physical experience?
George Lakoff, a linguist, and Rafael Núñez, a cognitive psychologist, have long-standing interests in mathematics. They have written a unique and fascinating, if flawed, book attempting to "apply the science of mind to human mathematical ideas" (p. xi) to discover where our mathematical ideas come from. This book introduces the discipline of "mathematical idea analysis:" how our understanding of human cognitive processes can account for the development of mathematical ideas.
Their thesis is that virtually all mathematical ideas arise as metaphors. Virtually all, because a bit of mathematics, called subitizing — the ability to recognize very small numbers — involves innate capacities of our brains. But subitizing cannot account even for arithmetic: while we can subitize numbers as large as 4, we can't subitize 4 + 3. For this we need metaphor. "Metaphor is not a mere embellishment; it is the basic means by which abstract thought is made possible. One of the principal results in cognitive science is that abstract concepts are typically understood, via metaphor, in terms of more concrete concepts." (p. 39) Their metaphors, essentially isomorphisms, consist of a source domain, generally concrete; a target domain (the domain of the new objects being developed) and a mapping between the two. A very early example is Arithmetic As Object Collection (p. 54-55): the source domain is object collection, the target domain is arithmetic, collections of objects of the same size correspond to numbers, the size of the collection corresponds to the size of a number, the smallest possible collection to the number 1, etc.
Arithmetic As Object Collection is the simplest type of mathematical metaphor, a "grounding metaphor," using everyday experiences to ground abstract concepts such as addition. There are also "linking metaphors" which have both source and target domains within mathematics. An example is the metaphor Numbers Are Points on a Line (p. 279), with source domain points on a line, and target domain numbers; a point P corresponds to a number P', the origin to the number 0; a designated unit distance pointI to the number 1; etc. More sophisticated mathematical ideas may involve several metaphors at once, in a "conceptual blend." For example, Dedekind's Number-Line blend "uses two metaphors: Spaces Are Sets and Numbers are Points on a Line." (p. 295)
Their most important metaphor is the Basic Metaphor of Infinity (BMI). They start with finite but continuous iterative processes, involving a beginning state, intermediate states, and a final resultant state, and map them to iterative processes that go on and on, but the final resultant state now becomes actual infinity. (p. 159) This is no longer quite an isomorphism. Which is, in some ways, a good thing: their earlier metaphors which are simply isomorphisms seem rather sterile. But now that we're getting something genuinely new, the ambiguity of how to go from the intermediate states to the final state leaves a gap that needs more explanation than they give. In particular, it can lead to paradoxes such as those exposed by Zeno, or one they discuss that involves the limit of arc-lengths of semicircles with centers at ((2m-1)/2n,0), m=1,2,...,2n-1: for each n, the semicircles' lengths add up to pi/2, yet the pointwise limit of the semicircles is the interval [0,1](p. 325 - 333).
Beginning with the BMI, they also start making rather frequent mathematical errors, particularly in their discussion of infinitesimals. For example, they develop the set of infinitesimals by constructing sets S(n) which are "the set of all numbers greater than zero and less than 1/n, satisfying the first nine axioms for the real numbers" (p. 228). Among these axioms (p. 200) are "the existence of identity elements for both addition and multiplication" (axiom 4) and the existence of additive (axiom 5) and multiplicative (axiom 6) inverses. Unfortunately, there is no set of real numbers greater than 0 and less than 1 containing identities and inverses: their sets are inconsistent! This can be fixed, of course, by being a bit more detailed. But it gets worse when they attempt to develop the "granular numbers", which are basically the smallest subfield of the hyperreal numbers generated by the real numbers and one infinitesimal. They construct this field by attempting to pick out "the first infinitesimal number produced by the BMI" (sic, p. 235) — but there can be no first such number! (They believe the granular numbers are a new mathematical object that they've discovered, and go on, p. 254, to discuss why mathematicians could have been so blind as not to have found them sooner!!)
The book deteriorates from the BMI onward. In the chapter on "Real Numbers and Limits", the authors observe, correctly, that our usual epsilon-delta definition of limit doesn't really capture how we conceptualize limits, the process of a function approaching a limit, partly because we allow all possible real numbers as values for epsilon, resulting in acceptable epsilons which are irrelevant to the limit. They decide, therefore, to use a sequential definition of limit. However, rather than following a standard treatment of this topic (involving all subsequences of the sequence under consideration), they introduce the concept of a sequence of "critical elements," which are "those terms of the sequence that must converge in order for the sequence as a whole to converge" (p. 195); their notion is incoherent. A bit later they confuse how the standard definition of limit works, believing they can choose epsilons as they please (p. 199); similar errors continue through much of the remaining mathematical content.
Not satisfied with introducing a new field of intellectual inquiry, the authors devote the penultimate part of the book (the last part is an extended explanation, in terms of their metaphors, of the mathematical basis of ei + 1 = 0) to the introduction of a new philosophy of mathematics which they believe to be implied by their conclusions. They assert that they have dealt a fatal blow to what they call the "Romance of Mathematics" (p. 339), roughly what is often referred to as platonism: "Mathematics is an objective feature of the universe ... What human beings believe about mathematics therefore has no effect on what mathematics really is. ... Since logic itself can be formalized as mathematical logic, mathematics characterizes the very nature of rationality. ..." As with many social constructivists (e.g., Reuben Hersh), they dislike this romance because "It intimidates people. ... It helps to maintain an elite and then justify it." (p. 341) Their arguments in favor of "human mathematics" are briefer and no more eloquent than those in Hersh's What is Mathematics, Really? and have little direct connection with the rest of the book. While I have no more sympathy than the authors for this elitism of mathematicians (and have devoted most of my life to undoing its effects), the elitism of mathematicians is no more a consequence of a belief that mathematical facts are an objective feature of the universe than the elitism of physicists is a consequence of their belief that physical facts are an objective feature of the universe. While their description of how humans develop concepts of mathematics is consistent with the restricted social constructivism of Hersh, it is also consistent with any reasonable version of platonism that distinguishes between mathematical facts and human knowledge of those mathematical facts. Indeed, as we begin to describe how human understanding of mathematical ideas is consonant with human understanding of other abstract systems, platonists will be able to respond to the challenge from philosophers of how finite corporeal beings can have contact with, and knowledge of, an infinite abstract branch of knowledge.
One small annoyance: the references are broken into 6 categories: to find "Narayanan ," you may have to look through them all before finding the full reference.
Despite its flaws, this book is a significant contribution to our understanding of mathematics' relation to people. Although the analysis has some defects, most first attempts to introduce a new discipline involve some important insights but also some stumbling around in the dark. The insights these authors introduce make at least the first half of the book well worth reading for anyone (advanced undergraduate and up) interested in the philosophy of mathematics, or in the genesis of mathematical ideas.
The authors' reply to this review appears as one of the "reader reviews" below.
Bonnie Gold (firstname.lastname@example.org) is chair of the Mathematics Department at Monmouth University. Her interests include alternative pedagogies in undergraduate mathematics education and the philosophy of mathematics. She is editor of MAA Online's Innovative Teaching Exchange, and co-editor of Assessment Practices in Undergraduate Mathematics (MAA Notes #49). On the philosophical side, she is the author of "What is the Philosophy of Mathematics and What Should It Be" (Mathematical Intelligencer, 1994) and was the co-organizer of a session on the philosophy of mathematics at the January 2001 joint meetings in New Orleans.