War in Europe is a staple topic in the study of history, but there’s one major conflict most history books won’t teach you—the battle of the equals sign, “=.” These two parallel lines were, in fact, the source of major conflict between European mathematicians in the mid 1500s. This is just one of many untold histories in The Language of Mathematics: The Stories behind the Symbols by author and mathematician Raúl Rojas. In it, Rojas explores the complex, and sometimes uncertain, history of mathematical symbolism.
Another debate has been raging for centuries, for instance, over who invented the symbol for zero, “0.” Over the years, competing camps have debated adopting one or another notation for many different aspects of mathematics. Rojas’s tale guides us along the historical arc of mathematics, intertwining its evolution with the cultural, philosophical and practical needs of the societies that shaped and relied on it.
Scientific American spoke to Rojas about this history, the deeply engaging humanity of mathematics and the egos at play in defining the mathematical language we take for granted today.
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[An edited transcript of the interview follows.]
What inspired you to write this book about the stories behind these symbols?
I started teaching in 1977, and across my nearly 50 years, I noticed that students were always interested in the history of mathematics. When you teach linear algebra or calculus, it’s important to tell students about the people who developed the concepts and how those concepts came to be. I started doing seminars on the history of mathematical notation and had every student study one symbol and explain its origin. I found that those students who are falling asleep in class suddenly wake up when you add a human story behind the abstract symbols.
Throughout the book, you discuss symbols that ultimately failed to become the standard en route to the notation we know today. How were these things decided?
One of the interesting things about the history of mathematical notation is its regional variation over the centuries. There was one kind of notation in Italy, another in Germany, the U.K. and France. All these different regions were producing symbols, and with the advent of the printing press, there was an explosion of proposals. So how did it happen that a single symbol could become standardized?
One good example is the symbol of equality, “=.” This relation was mostly expressed with words in the beginning. Later [René Descartes in France started using the rotated Taurus symbol, “∝,” while [Gottfried Wilhelm] Leibniz in Germany used a wedgelike shape. And [before Descartes and Leibniz] Robert Recorde in [the U.K.] invented the equality sign we use today, though in an elongated form. Mathematicians found themselves in a kind of battle over arithmetical symbols based on popularity. A notable contest was between “+” and “–” versus “p” and “m,” which the Italians preferred for denoting operations. Eventually, the plus and minus signs became universal, as did the English symbol for equality, but only after decades of famous mathematicians competing in these popularity contests to set the trends.
Is there a particular symbol in the history of mathematics that significantly influenced how we think about abstract concepts?
There is one symbol, which has an incredibly long history that has not yet been fully written: “0.” How did it arise? We know it was used by the Babylonians, but they didn’t write a “0” as we know it. They worked with a positional base-60 system and simply left a blank where we would write 0 today. This was their natural way of showing zero: if it’s nothing, then you don’t have to write anything.
Later, through the conquests of Alexander the Great, the Greeks took the positional number system to India, where we believe the Hindu culture developed the first representation of “0.” There’s a friendly competition between anthropologists working to find the oldest instances of “0” in writing. Every five or six years, someone finds an older engraving. It’s fascinating because this simple symbol we use every single day without thinking about it has a history that encompasses … thousands of years.
You describe Gerhard Gentzen’s “for all” symbol (∀) as a “cubist tear flowing from an eye that Picasso could have painted.” What’s the story behind that notation?
Gentzen’s life is deeply tragic to me. He was an exceptional mathematician who, like many others in Nazi Germany, compromised with the regime. Although he was never a political person, he became a member of the Nazi Party and later joined the SS—the most criminal arm of the regime. Absorbed in his work, he made these compromises to advance his career. Even after the war, he expressed no guilt, claiming that he was neither a soldier nor doing anything wrong. He had taken a position at the University of Prague, however, displacing others under Nazi occupation. Ultimately, he chose not to flee after the war, was captured, and died of starvation in prison.
There is no excusing his actions, but his life remains tragic from beginning to end, especially when considering what he might have accomplished had he taken a different path. It’s fascinating that such a simple symbol—this upside-down “A”—carries such a complex and poignant history.
What do you hope readers—especially those outside the math community—might take away from your book?
It’s important to understand that mathematics is a historical process, just like any social science or politics. Mathematics didn’t arise complete and finished through the work of just one mathematician; it has a cultural history that spans many years. For centuries, we’ve been looking at the sky or computing. In school, they teach addition and multiplication but rarely explain the origins or history of the symbols. This vast history is untold, but the excitement of doing mathematics comes from this knowledge that you are building on a framework developed by fascinating people over thousands of years.
