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.
Ultimately, Lakoff and Nunez return to the nature of the existence of mathematical objects. Whether they are mental constructions, facets of an idealized reality, or just rule-governed manipulations (like the game of chess) is an issue that has resonated all through the history of philosophy and is certainly not settled in this book.
Whatever one's views on the nature of mathematical entities and truths, however, the book is provocative and beneficial in its emphasis on the metaphorical aspects of mathematical concepts. A deeper appreciation of the sometimes unconscious, usually mundane sources of mathematical ideas can only help us learn and teach mathematics.
To demonstrate this, the book ends with an extended case study of the authors' approach to mathematical idea analysis. In it they clearly explain all the layers and interconnections among the metaphors necessary to develop an intuitive grasp of Euler's famous equation, eπ*i+ 1 = 0, relating five of the most significant numbers in mathematics.
Bertrand Russell wrote of the "cold, austere beauty" of mathematics. In very different ways A Beautiful Mind and Where Mathematics Comes From remind us of the warm bodies from which this beauty arises.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears every month.