John Allen Paulos Explains How Math Is Rooted in Metaphors

Feb. 1 -- I hear it all the time: Mathematics is impossibly esoteric.

You're born with mathematical talent or you're not. One solves math problems instantaneously. The source of mathematical insight is unfathomable. And so on and on.

The movie A Beautiful Mind tells the fascinating story of mathematician John Nash, but unfortunately it also suggests to many that the above beliefs are true.

It may not be the intent of the recently released Where Mathematics Comes From to combat these widespread misconceptions, but happily that is one of its effects.

The book's authors, linguist George Lakoff and psychologist Rafael Nunez, analyze the cognitive basis of mathematical ideas and in the process suggest new avenues of educational research.

So where does mathematics come from? Not surprisingly, none of us start out with a knowledge of differential equations. Instead the authors contend that from a rather puny set of inborn skills — an ability to distinguish objects, to recognize very small numbers at a glance and, in effect, to add and subtract numbers up to three — people extend their mathematical powers via an ever-growing collection of metaphors.

Our common experiences of standing up straight, pushing and pulling objects, and moving about in the world lead us to form more complicated ideas and to internalize the associations among them.

In fact, the authors argue that we understand most abstract concepts by projecting our physical responses onto them. The notion of a conceptual metaphor is well known from Lakoff's earlier work, particularly The Metaphors We Live By, a book that underscored how metaphors pervade our everyday thinking about the world. Physical warmth, for example, helps elucidate our understanding of affection: "She was cool to him." "He shot her an icy stare." "They had the hots for each other."

What Are Metaphors?

Lakoff and Nunez take a metaphor to be an association between a familiar realm, something like temperature, construction, or movement, and a less familiar one, something like arithmetic, geometry, or calculus.