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.
The size of a collection (of stones, grapes, toys), for example, is associated with the size of a number. Putting collections together is associated with adding numbers, and so on.
Another metaphor associates the familiar realm of measuring sticks (small branches, say, or pieces of string) with the more abstract one of arithmetic. The length of a stick is associated with the size of a number once some specified segment is associated with the number one. Scores and scores of such metaphors underlying other more advanced mathematical disciplines are then developed.
Demystifying Mathematical Ideas
Throughout the book the authors attempt to demystify mathematical thought. They stress that mathematical ideas do not gush out of some pipeline to the Truth (such as John Nash's schizophrenia), but have a source similar to that of other, more prosaic notions. The root of some of our mystification, they argue, is the "numbers equals things" metaphor, which leads to the Platonic idea that numbers are "up there" somewhere. Lakoff and Nunez are intent on debunking this belief and others linked to it.
The second half of Where Mathematics Comes From is a bit more technical and deals largely with infinity and the metaphors that animate our understanding of the ideas in calculus such as limits and infinite series.