Review: March 2009

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In The Assayer, an eloquent discourse on the scientific method published in 1623, Galileo Galilei observed that God’s design for the universe “is written in the language of mathematics, and the characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to comprehend a single word of it.” Those sentiments may have been controversial in an era when most learned men venerated Aristotle, who had described the world quite cogently using neither formulas nor equations. But from the perspective of the twenty-first century, the unique explanatory power of mathematics seems self-evident. We know that mathematics is essential to the design of every device we use to augment our senses or control our environment, from iPods to self-defrosting refrigerators.

Why should this be? The effectiveness of mathematics in formulating the laws of nature is as puzzling the laws are themselves. If, as Galileo’s comments imply, scientific laws exist independent of human consciousness, then scientists are, like explorers skirting the shores of an unknown continent, gradually mapping the preexisting contours of nature through observation, experiment, and reflection. It is a perspective that inspires scientific research to this day: “Our universe is not just described by mathematics,” says cosmologist Max Tegmark of MIT, “it is mathematics.”

There is much to commend this attitude, according to astrophysicist Mario Livio of the Space Telescope Science Institute in Baltimore, Maryland, whose latest book, despite the theological resonance of its title, is actually a well-crafted popular history of mathematical philosophy. Many theories that were originally thought to be pure abstractions, he notes, later turned out to describe aspects of the natural world. For instance, group theory, devised in the 1800s as a description of the relations between various mathematical operations, provided the key to understanding subatomic structure a century later.

Yet there are those who argue that the mathematical structure of the universe is mere appearance, a human invention that results from our brain’s natural tendency to impose order on the raw experience of our senses. This point of view gained credence in the 1800s, when Euclid’s geometry, long thought to be the only way to describe the world, was found to be only one among many axiomatic systems that did the job. Mathematics, to cognitive scientists George Lakoff and Rafael Nuñez, tells us more about the organizing principles of our minds than the underlying structure of nature.

So which is it? If you are impatient for answers, you might want to skip the various historical discourses on statistics, knot theory, symbolic logic, etc., and head straight for the final chapter, where Livio sets forth his own views on the matter. Avoid such temptation, however: the merit of this book is, as with much of mathematics, not so much with the “Q.E.D.” at the bottom line, but with the steps taken to get there.