Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (review)

This is an extremely ambitious book. Its goal is to launch the discipline of ‘mathematical idea analysis’, a branch of cognitive science devoted to the understanding of the concepts of mathematics, considered as human creations, not as disembodied notions awaiting discovery by intelligent creatures. Lakoff and Núñez refer to the latter view as the ‘romance of mathematics’ (xv). More specifically, L&N maintain that ‘[t]here is no way to know whether theorems proved by human mathematicians have any objective truth, external to human beings or any other beings’ (2); they choose instead to focus on a question that they think they can begin to answer, namely ‘[W]hat mechanisms of the human brain and mind allow human beings to formulate mathematical ideas and reason mathematically?’ (1) Here’s how they try to do it.¹

First, in Ch. 1, L&N summarize research that shows that humans have certain innate mathematical abilities including simple arithmetic and ‘subitizing’, the ability to determine the size of small collections of objects without counting them. Second, in Ch. 2, they claim that humans are also endowed with conceptual primitives they call ‘image schemas’, for building systems of spatial relations that are expressed in human language. These can be combined in various ways to form what they call conceptual schemas, some of which, such as the ‘In schema’, are said to be fundamental to mathematical thought. Third, also in Ch. 2, they invoke ‘metaphor . . . the basic means by which abstract thought is made possible’ (39). The rest of the book is primarily taken up with the metaphors that L&N identify as providing the meaning of central mathematical ideas, ranging from the concepts of arithmetic (Chs. 3–4); algebra, logic, and set theory (Chs. 5–7); infinity (Chs. 8–11); and calculus (Chs. 12–14). Chs. 15–16 deal with the theory and philosophy of what they call ‘embodied mathematics’, and two other sections discuss the resolution of a particular paradox of infinity (325–33) and the meaning of Euler’s equation e^πi + 1 = 0 (383–451).

There is little discussion of linguistics in this book, but what there is is crucial to L&N’s arguments. First, as I have already observed, L&N ground certain basic mathematical ideas on semantic primitives of natural languages and some notion of compositionality. Second, L&N argue that a specific linguistic notion is at the source of our understanding of infinity: ‘To begin to see the embodied source of the idea of infinity, we must look to . . . what linguists call the aspectual system’ (155). Specifically, L&N maintain that the device of iteration with imperfective verbs (as in flew and flew and flew on and on) denotes continuous processes by means of a metaphor that equates continuous processes with iterative ones (157). To arrive at the notion of ‘actual infinity—infinity conceptualized as a thing, not merely as an unending process’ (xii), one must go one step further, by creating ‘a metaphorical result of a process without end’ (158). This is what L&N call ‘the basic metaphor of infinity’ (159).

However, the result of an unending process can be understood without the use of metaphor. Understanding of ordinary universal quantification is sufficient. One who understands a simple English sentence such as every number is interesting, and also understands that there is no end to the number sequence 1,2,...,thereby understands the concept of actual (denumerable) infinity without the use of metaphor.² So the use of metaphor is not necessary for the understanding of such mathematical concepts as absolute infinity.

It is also not sufficient for the understanding of mathematical concepts, despite L&N’s airy claim that ‘[o]ne of the principal results in cognitive science is that abstract concepts are typically understood, via metaphor, in terms of more concrete concepts’ (39). L&amp...