David Foster Wallace's new book explains why studying abstract math hurts so good.
Even those who actually enjoyed college mathematics probably wouldn't want to repeat the experience, let alone reread the textbook. Few of us deal with complex equations and abstractions enough to induce that frisson of intellectual terror that is, after all, the reason someone might consider consulting a math book. No wonder so many folks who write on the topic fill their pages with tales of wacky mathematicians instead of actual math.
Fortunately, David Foster Wallace, author of the brainy best seller Infinite Jest, sees beauty in numbers and has found an engaging way to discuss them. His latest book, Everything and More: A Compact History of ∞, chronicles the mathematical concept of infinity, from the ancient Greeks to Georg Cantor, the 19th-century German mathematician who blew the subject wide open by showing not only that infinity exists but that there are an infinite number of infinites. Despite Wallace's novelistic credentials, he spends relatively little time on the life stories of his subjects. His interest lies in mathematical abstractions - and the pleasure and pain they can bring. Once you begin to ponder concepts like infinity or four-dimensional space, Wallace writes, you can feel "the first popped threads of a mind starting to give at the seams."
Madness is, Wallace points out, an occupational hazard for mathematicians. Cantor was a frequent guest at sanitariums. Kurt Gédel starved himself to death. Isaac Newton traveled in a pretty eccentric orbit. And John Nash spent years scaring MIT and Princeton undergrads. To help the reader understand why, Wallace cites early 20th-century thinker G. K. Chesterton, who wrote that mathematicians are more likely than poets to lose their minds, because the real danger "lies in logic, not in imagination."
The first chapters of the book are like an extended set of preflight instructions - here are the oxygen masks, here the emergency exits. And yes, there are footnotes. Wallace's fans have come to know them as one of his most enjoyable stylistic tics. But here, with the reader balancing difficult concepts like a tray of overfilled drinks, they are often an unwelcome distraction.
Still, those who take the time to work through this thin volume will be rewarded for their effort. The Greek mathematician Euclid said that "there is no royal road to geometry," and compared to unraveling the infinite, geometry is a walk in the park. When the concrete idea of numbers, conceived by the Babylonians as a way to keep track of their sheep, became the abstract toy of the Greeks, "incalculable displacements" were inevitable. Infinities emerged unbidden, and with them a set of paradoxes, like those of Zeno, that showed what troublesome creatures those infinities could be, making the simple task of crossing the street a philosophical poser. In fact, Zeno's paradoxes caused mathematicians and philosophers to snub infinity for thousands of years, until Cantor demonstrated the proper way to think about it.
That Cantor's ideas spun off their own brain-busting anomalies and rocked the foundation of all mathematics is only to be expected. As Wallace demonstrates in his book (though some of the details are fuzzy), the "abstract math that's banished superstition and ignorance and unreason and birthed the modern world is also the abstract math that is shot through with unreason and paradox and conundrum and has been as it were trying to tie its shoes on the run ever since the beginning of its status as a real language." We should be grateful that people like Wallace have raced alongside and taken notes.
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