It all started in a wine bar in Italy. During a monthlong residency at the Rockefeller Foundation’s Bellagio Center in 2008, Joseph Mazur, PhD ’72, spent an evening chatting with a musician and a psychologist over glasses of Barolo. The conversation turned to ancient symbols and their significance in recording human thought. Mazur, now a professor emeritus of mathematics at Marlboro College in Vermont, mentioned that mathematical symbols as we know them were relatively recently developed. His assertion was met with disbelief.
Enlightening Symbols
By Joseph Mazur, PhD ’72
Princeton University Press, 2014, $29.95
Mazur had been considering tackling the history of mathematical notation for some time, he says, but realizing that even his well-informed companions were not so knowledgeable about the subject prompted him to move it to the top of his to-write list. In Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, Mazur explains the evolution of now-familiar systems that were largely nonexistent before the 16th century. (Algebra, for example, was written primarily in words until then.) The “hidden powers” of the subtitle refers to the way properly designed symbols can help us process what Mazur calls “complicated notions.”
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In researching the book, Mazur realized that not all notation is created equal. “There is good notation and bad notation,” he says. “When you have notation that’s kind of willy-nilly put in, a squiggle to say that ‘This is something,’ you get in trouble with that later. Bad notation can make obstacles for you or later mathematicians when they try to use it.” Take the expression x2-3x+2. In a publication from 1553, it was written like this: 1Zm.3Rp.2. “So the x2 would be simply Z, the m stood for minus, and the R stood for x,” Mazur explains. “All is fine until you get to cubic polynomials, and you lose the sense that x3, x2, and x are all of the same genus.”
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The master of the art of notation, Mazur says, was the 17th-century German mathematician Gottfried Leibniz, who invented more than 200 new symbols, including those used in differential and integral calculus. Mazur regards Leibniz’s notation and his integral sign as particularly brilliant. His symbols, he writes, “made life in the calculus world far easier than it would have been had Newton’s or Fermat’s symbols survived.”
The last comprehensive book covering mathematical notation came out in 1927, long before many of the primary sources Mazur used came to light, and was structured as an encyclopedic history of symbols. Like his earlier works, which include What’s Luck Got to Do with It? and Euclid in the Rainforest, this book is designed to be much more approachable; in writing it, Mazur viewed himself as a “journalist embedded with mathematicians.”
“I’m trying to popularize math and science,” he says. “[I] tend to be memoirish; I bring myself into it. I think that pulls people into it … I don’t water anything down, I just try to simplify it to make sure that people are understanding.”
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