Where Mathematics Comes From

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a 2000 book by cognitive linguist George Lakoff and cognitive scientist Rafael E. Núñez. The book seeks to establish a cognitive science of mathematics, or a theory of embodied mathematics.

The book calls for (and attempts to begin) a cognitive idea analysis of mathematics which analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations and other cognitive mechanisms which gave rise to them. Ultimately, it is held, mathematics is a result of the human cognitive apparatus and must therefore be understood in cognitive terms. This idea analysis is distinct from mathematics itself and cannot be performed by mathematicians not sufficiently trained in the cognitive sciences.

Lakoff and Núñez attempt to reject the position of platonism in the philosophy of mathematics. They emphasize that all we know and can ever know is human mathematics, the mathematics arising from our brains, and the question whether a transcendent mathematics objectively exists is thus unanswerable and close to meaningless.

Human cognition and mathematics
Lakoff and Núñez start by recognizing that humans have what appears to be an innate ability (what is called subitizing) to identify small numbers (up to about 5) and to add or subtract such small numbers. They describe recent experiments with infants, and how infants quickly become excited or curious when presented with "impossible" concepts, such as having three toys appear when only two were initially present.

Lakoff and Núñez argue that mathematics extends further through the use of metaphor. As an example, they argue that the Pythagorean position that all is number, and the associated crisis of confidence that came about with the discovery of the irrationality of the square root of two, arises soly from the human mind's strictly metaphorical relation between length (of the diagonal of a square) and number (of objects.)

The book likewise critiques mathemetician's emphasis on the concept of closure. Lakoff and Núñez argue that the demand for closure is an artefact of the human mind's ability to metaphorically relate fundamentally different concepts.

Critical response
Reviews by mathematicians of Lakoff and Núñez's book have been mixed. The book's focus on conceptual strategies and metaphors for understanding mathematics is generally welcomed, but mathematical reviewers have tended to critique the book as containing mathematical errors. Moreover, many mathematicians critiqued some of the book's philosophical arguments on the grounds that mathematical statements have objective meanings such that, for example, Fermat's last theorem can mean exactly the same thing now that it did 300 years ago when initially proposed. Mathematical reviewers have also pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person. The metaphor and the conceptual strategy is not the same as the mathematical definition that mathematicians work with.

The authors seem to dismiss some negative commentary by mathematicians on the grounds that mathematicians are not taking the view of cognitive science, which in a book whose subject matter involves mathematics and philosophy as well as cognitive science is a strategy which cuts both ways. The questions of what mathematics is about and where (in terms of cognitive science) it comes from are to many people distinct ones.

Mathematics educators have found the book interesting for what it suggests about the process of learning mathematics.