Meet the author

# Duly Noted

Familiar math symbols are surprisingly recent inventions.
June 17, 2014

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.

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.”

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.”

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.”

Recent Books from the MIT Community

Making a Global Immigrant Neighborhood: Brooklyn’s Sunset Park
By Tarry Hum, MCP ’87
Temple University Press, 2014, \$32.95

#! (pronounced “Shebang”)
By Nick Montfort, associate professor of digital media
Counterpath, 2014, \$20

The Girl in the Road
By Monica Byrne, SM ’05
Crown, 2014, \$26

The Forms of the Affects
By Eugenie Brinkema, assistant
professor of contemporary literature and media
Duke University Press, 2014, \$25.95

Networks of Rebellion: Explaining Insurgent Cohesion and Collapse
By Paul Staniland, PhD ’10
Cornell University Press, 2014, \$27.95

Building Information Modeling
By Karen M. Kensek ’84
Routledge, 2014, \$29.95

The Politics of Adoption: Gender and the Making of French Citizenship
By Bruno Perreau, associate professor of French studies
MIT Press, 2014, \$29

By Stephen Yablo, professor of philosophy
Princeton University Press, 2014, \$45

Please submit titles of books and papers published in 2013 and 2014 to be considered for this column.

Contact MIT News
E-mail mitnews@technologyreview.com
Write MIT News, One Main Street, 13th Floor, Cambridge, MA 02142
Fax 617-475-8043

### 10 Breakthrough Technologies 2024

Every year, we look for promising technologies poised to have a real impact on the world. Here are the advances that we think matter most right now.

### Scientists are finding signals of long covid in blood. They could lead to new treatments.

Faults in a certain part of the immune system might be at the root of some long covid cases, new research suggests.

### AI for everything: 10 Breakthrough Technologies 2024

Generative AI tools like ChatGPT reached mass adoption in record time, and reset the course of an entire industry.

### What’s next for AI in 2024

Our writers look at the four hot trends to watch out for this year

### Stay connected

Illustration by Rose Wong

## Get the latest updates fromMIT Technology Review

Discover special offers, top stories, upcoming events, and more.

Thank you for submitting your email!