Book Review : Should We Count on Mathematics?

Descartes’ Dream: The World According to Mathematics by Philip J. Davis and Reuben Hersh (Harcourt Brace Jovanovich: $22.95)

One of the basic assumptions of Western civilization is that knowledge is good and humanity is better off knowing things than not knowing them.

For nearly five centuries, this proposition went essentially unchallenged. Its truth was seen as self-evident in the higher standard of living that science brought many people and promised to bring to many more.

But World War II gave the world another truth to think about: Knowledge can lead to Auschwitz and Hiroshima, as well as to penicillin.

In the four decades since, we have become much more cautious about the blessings of science. We have learned that knowledge can harm as well as help, and myriad public policy debates turn on the difference. Supposedly beneficial substances can have toxic side-effects. Science does not have the solution to all of the world’s problems within its grasp. Whether the net effect of science is good or bad seems to have become an open question.

Descartes Gets the ‘Blame’

What does all this have to do with a book subtitled “The World According to Mathematics”? Plenty, according to Philip J. Davis and Reuben Hersh, mathematicians at Brown University and the University of New Mexico respectively. Their book is a description and discussion of the increasing mathematization of the world that has occurred without interruption since the 17th Century. And they are not persuaded that people are better off for it.

Rene Descartes set the world on this course, which has proved powerful beyond his or anyone’s wildest dreams. In the Age of Science, we take it as a given that objective reality can be measured, counted, quantified and analyzed by the tools of mathematics. If anything, the computer--another invention of World War II--has accelerated this trend.

But Davis and Hersh find the mathematization of reality both wrong and harmful. Wrong, they say, because the world isn’t really mathematical at its core. People impose mathematics on reality, they say. Worse, it’s harmful because mathematics and science have no values; they must be tempered by humanism, which mathematicians frequently lack.

Abstraction Vs. Compassion

Mathematics is the science of abstraction, the authors say. “The spirit of abstraction and the spirit of compassion are often antithetical,” they write. As a result, “Whereas World War I was a chemists’ war and World War II a physicists’ war, World War III will be a mathematicians’ war.”

Five years ago, Davis and Hersh collaborated on a marvelous book called “The Mathematical Experience,” which explained for non-mathematicians what contemporary mathematics is all about. They easily and authoritatively discussed the current problems in the field and tackled the deep philosophical issues of knowing that are inherent in mathematical knowledge. Their new book, which looks out at the world from inside mathematics, is just as wonderful.

In their earlier work and in this new volume, Davis and Hersh reject the idea that mathematics consists of timeless truths that remain true whether or not anyone has discovered them.

The other side argues that “2 plus 2 equals 4” is an eternal truth that would be true even if the Big Bang had never occurred and the universe did not exist. Davis and Hersh reject that view. To them, mathematics is a creation of the human mind, a point brought home by the discovery of non-Euclidean geometries in the 19th Century. If even the geometry of Euclid is contextual and not built into the basic structure of the universe, what eternal truths can mathematics possibly contain?

Truth as Time-Dependent

“Are the truths of mathematics independent of time?” the authors ask. “Well, there must be a sense in which they are. I will not assert that while 2 plus 3 was 5 in the days of old Pythagoras, now, due to certain adjustments that have to be made, it is slightly more. . . . But construing ‘truth’ in a larger sense, the truths of mathematics are time-dependent.”

This idea remains the source of great controversy in mathematical and epistemological circles, but Davis and Hersh make a compelling case for it. Part of the pleasure of their book is the ease with which they convey complicated ideas, their masterful grasp of their material and the balanced tone of their argument.

For what they are ultimately doing is decrying blind faith in reason, logic and mathematics. “Part of the conflict between scientists and humanists derives from a feeling on the part of the humanists that there ought to be a portion of the world that is immune from mathematization, and an opposing feeling of the scientists that all must be grist for the mathematical mill,” they write.

“We should never forget,” they say, “that a stroll in the woods or a deep conversation with a new or old friend are beyond mathematics. And then, when we go back to our jobs, as administrators, teachers, or whatever, will we still remember that numbers are only the shadow, that life is the reality.”

From Axiom Via Logic to Truth

When I was a sophomore in high school, I studied geometry, as all high-school sophomores do. A few weeks into the course, I was struck by the beauty of the method. I remember the moment very clearly. Aha, I thought, this is the way to find the truth. Start with axioms and draw conclusions from them based on the rules of logic.

A few years later I discovered that Descartes had already tried this approach. This is the “dream” of Davis and Hersh’s title, and, they say, it is a false god. Reason, math and logic can do only so much. To follow them further is to invite catastrophe.

“The world . . . is full of paradoxes and contradictions, unresolved and unresolvable,” the authors write. “Can the mathematical attitude resolve the paradox of Job? If it attempted to do so, it might do considerable damage. . . .

“The solution . . . lies in the cultivation of strong values that lie outside science. We must have more education for scientists in the humanities, in history. We cannot afford to be ignorant technicians. . . . We must remember that though Mathematics may be the Queen of the Sciences, Science is not the only principle of life. ‘The tree of life is larger than the tree of thought,’ said Immanuel Kant. I believe it.”

So do I.