The history of language and mathematical notation is filled with chance and serendipity.
I want readers to see mathematics as a living organism, a living subject, with a history. Mathematics is all about logical truth, and the discipline seems forged in steel. But in fact, mathematics evolved through different phases. It had its childhood, where problems were solved rhetorically or by using appropriate geometrical constructions. Then, during its adolescence, brilliant mathematicians like Fran莽ois Vi猫te pioneered notational shortcuts, and it reached full maturity when the logical foundations of mathematics were developed rigorously.
Your book, The Language of Mathematics: The Stories behind the Symbols, has just been published. What is it about?
RR: Well, if you open a mathematics book today, it is immediately clear that it is about mathematics because of the symbolism. Over the centuries, mathematics has developed its own peculiar language. This is now taught in school, from basic arithmetic to algebra and calculus. But we are rarely told by our teachers how this special notation came about: where and when symbols such as the equality sign were introduced, and by whom. I think that every symbol has its own story, and my book collects them as if the symbols themselves were telling them. Together, these stories form a tapestry that highlights the quirks and idiosyncrasies of mathematical notation.
In earlier times, mathematics did not yet have this distinctive language. If you open a medieval algebra book, symbolism was scarce and it takes time to realize that this work is actually about mathematics. The solutions to numerical problems were described entirely in words: a style known as 鈥渞hetorical mathematics鈥�. My book traces the evolution of mathematics from these humble beginnings to the rich array of symbols we use today, giving mathematics its unique flavor among the sciences.
What inspired you to write this book? Was there a particular moment or idea that sparked your interest in the history of mathematical notation?
RR: I have always been fascinated by the history of mathematical notation, and nearly 30 years ago, I had the opportunity to explore this interest by organizing a seminar on the subject. Each of my students researched the history of a specific symbol, tracing its origins, and we even collaborated with a graphic designer who created stunning vignettes of the symbols for stained-glass windows.
The students, many of whom had never considered the 鈥渆tymology鈥� of mathematical symbols before, became deeply engaged. That was the first surprise. We also explored the remarkable lives of the mathematicians who introduced the symbols, and the fascinating journey from an initial notation idea to its widespread acceptance. It’s intriguing to discover the many symbols that didn’t quite make the cut, a testament to the rigorous process of mathematical development. My manuscript grew through different iterations of these seminars. It took many years.
What was the most surprising or unexpected thing you learned about the adoption of mathematical symbols during your research for the book?
RR: The 鈥渇or all symbol鈥� is one that anyone studying mathematics and engineering should know. It looks like an inverted capital A, without the serifs, but I hadn’t noticed that before I started working on this book. It is obvious after some tells you. But our professors just told us 鈥渢his is the for all symbol鈥�. Then, I learned that it had been proposed by the mathematician Gerhard Gentzen for his PhD thesis. It turns out that Gentzen was actually following in the footsteps of the Italian Giuseppe Peano, who came up with the 鈥渆xists鈥� symbol by mirroring a capital E. I also learned about the life of Gentzen, which is quite a story. To advance his academic career, he became enmeshed in the Nazi machinery during WWII, which is a tragic fact, but something important to be said, and a warning for all scientists.
It seems like the adoption of mathematical symbols must involve a lot of arbitrary decisions or historical accidents. Did you come across any examples where the reasoning behind their widespread use seemed unclear or dubious?
RR: The history of language and mathematical notation is filled with chance and serendipity. For instance, the symbols + and 鈭� were first used by merchants to label sacks containing more or less than the standard weight. In contrast, Italian mathematicians used the letters 鈥減鈥� and 鈥渕鈥� instead (for the words piu and meno). However, as the cross and hyphen gained broader acceptance among mathematicians, they became the universal symbols we know today.
There was a time when most Europeans were unable to read or perform arithmetic鈥攖hey were illiterate and innumerate. But with the rise of mercantilism, particularly in trading hubs like Venice, the demand for literacy and numeracy surged. Schools emerged to teach basic arithmetic using paper and ink, and those skilled in these techniques became highly valued, especially as masters of accounting. This transformation from widespread illiteracy to literacy was further accelerated by the invention of the printing press, making this transition especially remarkable.
Mathematical symbols, however, are certainly not logically inevitable. As I explore in my book, special 鈥渕athematical regions鈥� (such as England, France, Central Europe, and Italy) initially developed their own unique notations. It took centuries of gradual unification before these diverse symbols converged into the standard mathematical language we use today. Sometimes we adopted two symbols coming from different quarters for the same thing, for example / and 梅 for division, or the cross and point for multiplication.
Do you see any modern parallels in how digital tools or technology influence the development of new mathematical or scientific symbols?
RR: With the advent of digital typography, the range of possible symbols has expanded dramatically. The Bourbaki school of mathematics, for instance, introduced the 鈥渄angerous bend鈥� symbol, resembling an S shaped like the spur of a car skidding on a highway (this symbol warns readers that a difficult passage lies ahead) which is now used widely. I’ve also encountered many new symbols created by individuals designing their own digital fonts. While this newfound freedom fosters creativity, mathematical journals play a crucial role in maintaining standardization, ensuring that innovation doesn鈥檛 spiral into chaos.
It’s fascinating how this creative freedom coexists with the need for consistency. Mathematical journals serve as gatekeepers, striking a delicate balance between tradition and innovation. New symbols are welcomed, but only if they adhere to the conventions necessary for clear and universal communication, preserving the integrity of mathematical language.
That sounds like a gradual process. Why the violence?
RR: The revolutionaries took a complex demographic of religions and dialects and localities and produced a simple binary: the Greeks and the Turks. It鈥檚 fascinating to see it happen, and it truly was revolutionary. They hurled at each other in a shockingly violent decade. It probably killed off about a quarter of the population of what became independent Greece. This is what made the Greek Revolution a demographic revolution, first and foremost. And then this model spread. The model traveled up the Balkans to create largely Christian states, across the Ottoman space to produce the Middle Eastern states and republican Turkey as mainly Muslim peoples. Of course, they all claimed to be unique and exceptional. And then it spread across the world.
Who do you see as the primary audience for your book?
RR: As I mention in the preface, this book is well-suited for high school students or those beginning college. In fact, it鈥檚 intended for anyone with a basic familiarity with mathematics who has ever wondered about the origins of its unique notation. My hope is that understanding the development of this symbolic language will help students approach mathematics with less apprehension. I truly believe that everyone who studies mathematics should, at least once in their lifetime, encounter these fascinating stories.
If readers take one key insight or lesson from your book, what would you want it to be?
RR: I want readers to see mathematics as a living organism, a living subject, with a history. Mathematics is all about logical truth, and the discipline seems forged in steel. But in fact, mathematics evolved through different phases. It had its childhood, where problems were solved rhetorically or by using appropriate geometrical constructions. Then, during its adolescence, brilliant mathematicians like Fran莽ois Vi猫te pioneered notational shortcuts, and it reached full maturity when the logical foundations of mathematics were developed rigorously. Amazingly, we’ve only had standardized mathematical notation for less than 150 years in the 22 centuries, or more, that we have had mathematics! We progressed from the 鈥済eometrization of mathematics鈥�, a feature of Greek math, to the algebraization of mathematics and even of geometry, which reached its pinnacle with Descartes. Then, we went through to the exploration of the infinite in the work of Newton and Leibniz. Peano, Russell and Whitehead took this to new heights, and we now find ourselves with a universal mathematical notation.
I really want to get this message across to readers: mathematics is an exciting subject with a rich history and a bright future! The next stop could be some kind of notation that can be processed efficiently by computers to verify theorems. There has been some progress in that direction in recent years, and the future holds the answer. Maybe in 300 years, we will be surprised at how simple our current notation still is. The evolution of the living never stops!
About the Author
Ra煤l Rojas is professor of mathematics and statistics at the University of Nevada, Reno, and professor emeritus of computer science and mathematics at the Free University of Berlin. A world-renowned expert in artificial intelligence, he is the author of the seminal book Neural Networks and the editor (with Ulf Hashagen) of The First Computers.