The Structure of Everything

Did we invent math? Or did we discover it?

(Author Mario Livio)
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Reviewed by Marc Kaufman
Sunday, February 8, 2009; Page BW02


By Mario Livio

Simon & Schuster. 308 pp. $26

Did you know that 365 -- the number of days in a year -- is equal to 10 times 10, plus 11 times 11, plus 12 times 12?

Or that the sum of any successive odd numbers always equals a square number -- as in 1 + 3 = 4 (2 squared), while 1 + 3 + 5 = 9 (3 squared), and 1 + 3 + 5 + 7 = 16 (4 squared)?

Those are just the start of a remarkable number of magical patterns, coincidences and constants in mathematics. No wonder philosophers and mathematicians have been arguing for centuries over whether math is a system that humans invented or a cosmic -- possibly divine -- order that we simply discovered. That's the fundamental question Mario Livio probes in his engrossing book Is God a Mathematician?

Livio, an astrophysicist at the Hubble Space Telescope Science Institute in Baltimore, explains the invention-vs.-discovery debate largely through the work and personalities of great figures in math history, from Pythagoras and Plato to Isaac Newton, Bertrand Russell and Albert Einstein. At times, Livio's theorems, proofs and conundrums may be challenging for readers who struggled through algebra, but he makes most of this material not only comprehensible but downright intriguing. Often, he gives a relatively complex explanation of a mathematical problem or insight, then follows it with a "simply put" distillation.

An extended section on knot theory is, well, pretty knotty. But it ultimately sheds light on the workings of the DNA double helix, and Livio illustrates the theory with a concrete example: Two teams taking different approaches to the notoriously difficult problem of how many knots could be formed with a specific number of crossings -- in this case, 16 or fewer -- came up with the same answer: 1,701,936.

The author's enthusiasm is infectious. But it also leads him to refer again and again to his subjects as "famous" and "great" and to their work as "monumental" and "miraculous." He has a weakness as well for extended quotes from these men (and they are all men) that slow the narrative without adding much. There are exceptions, including the tale of how Albert Einstein and mathematician Oskar Morgenstern tried to guide Kurt Gödel, a fellow mathematician and exile from Nazi Germany, through the U.S. immigration process.

A deep-thinking and intense man, Gödel threw himself into preparing for his citizenship test, including an extremely close reading of the U.S. Constitution. In his rigorously logical analysis, he found constitutional weaknesses that he thought could allow for the rise of a fascist dictatorship in America. His colleagues told him to keep that reading to himself, but he blurted it out during his naturalization exam. He was allowed to stay anyway.

The interplay of mini-biography and the march of mathematical knowledge serves the author well. It does not, however, ultimately help him to answer the big question, Is God a mathematician?

On one side of the debate are all those remarkable constants that crop up, the makings of the ideal yet hidden world posited by Plato. In addition, there's what the physicist Eugene Wigner, in a seminal 1960 essay, called the "unreasonable effectiveness" of mathematical theorems: the astounding ability of math to predict unimagined results. Wigner was picking up on ideas explored earlier by Einstein, and Einstein's general theory of relativity remains one of the best examples: His predictions about how gravity can cause ripples in space-time was recently corroborated by measuring radio waves from a distant set of compact, high-energy stars called double pulsars, using technology unknown in Einstein's day. Doesn't all this indicate that the mathematical structure of the world is out there waiting to be discovered?

On the other hand, math cannot explain many situations, and chaos theory suggests that it may never be possible to predict the weather or the stock market with accuracy. Recent research has pointed to basic mathematical constructs in the human brain, suggesting that we impose numbers and forms on the world, not vice versa. In addition, mathematics is less stable than it appears to us in grade school. At the higher reaches of the field, there is constant ferment and debate. If the "truths" discovered through mathematics are always changing, doesn't that indicate they are a product of human study and manipulation, rather than something fixed and eternal?

As explained by Livio, the history of mathematics is partly a struggle between these points of view: that math is how God (or nature) organizes the world, or it is simply a human tool to understand that world.

Livio comes down in the middle, contending that math may well be both invented and discovered. He points, for instance, to the eternal truth contained in the geometry formulated by Euclid 2,400 years ago. By the 19th century, however, iconoclasts had posited and established a whole new world of non-Euclidian geometry. Livio writes about the symmetries of the universe: the immutable, if incompletely understood, laws of math and physics that make a hydrogen atom, for instance, behave in the same way on Earth as it acts 10 billion light years away. Another sign of universal structure, as teased apart with the help of math? No, Livio writes, it is more likely a sign that "to some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment."

The author acknowledges that some readers will find his inconclusive conclusion to be unsatisfying. I didn't. Sometimes the adventure, the intellectual ride, is more important than the final destination. ·

Marc Kaufman, a science reporter for The Washington Post, is writing a book about astrobiology.

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