Hidden figures: giving history’s most overlooked mathematicians their due

Pythagoras. Isaac Newton. Alan Turing. John Nash. Mathematicians rarely become famous but those who do receive celebrity treatment are invariably white men. Turing was played by Benedict Cumberbatch on the big screen; Nash by Russell Crowe.

That lens has been missing an infinitely richer, more nuanced, more multicultural story. A new book, The Secret Lives of Numbers, by Kate Kitagawa and Timothy Revell, shines a light on overlooked contributions to maths by women and men in China, India, the Arabian peninsula and other parts of the world.

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“When we think of the history of mathematics, it is not just about ancient Greeks and bearded white men,” says Revell, 34, a British journalist, speaking via Zoom from London. “This isn’t about tearing anyone down. This is about explaining that the history of mathematics is way more complex, chaotic and amazing than you may have known. My hope is that our book goes some way to illuminating that.”

Kitagawa, 44, a maths historian from Japan, adds via Zoom from New York: “People already know about big figures and we do not want to challenge that idea: truth is truth. But we want to make it richer and so it’s about integration of knowledge as well.

“Personally I enjoyed bringing in my background – raised in east Asia, reading in Chinese; I went to school in Canada and was in the US. It was precious for me recalling my time in the US and presenting those challenges that Black people and women faced in academia.”

Kitagawa and Revell came up with the idea of a history over a cup of tea at a bookshop in Charing Cross, London. They thought it would be straightforward but it was anything but. They found the origin of ideas to be as beautiful, varied and elusive as the most elegant of mathematical problems.

The authors write: “As we worked our way through thousands of years of mathematics, almost everything we thought we knew was challenged in one way or another. Some well-known stories ended up being misrepresentations and others complete fabrications. Many mathematicians and mathematics have wrongly been excluded from history.”

For example, the invention of calculus – the theory for describing and determining how things change over time – is typically credited to Newton and Gottfried Leibniz, who each developed their own version in the 17th century. But Kitagawa and Revell contend that neither of them got their first, tracing the roots of calculus instead to 14th-century India and a a school in Kerala, where the mathematician Madhava of Sangamagrama used elements of calculus in his teaching.

Revell, who is executive editor of New Scientist, says: “The origins of calculus are normally told as this battle between two titans of mathematics – Newton and Leibniz – and of course both of those people did an awful lot of work on calculus.

“There’s a fun part of that story where Newton says, well, the person who will settle who got there first will be the Royal Society. The Royal Society decides it’s Newton. But of course Newton was head of the Royal Society – not the world’s most independent report on that front.”

He continues: “But hundreds of years before, in the 14th century, there was a mathematician called Madhava, and he was part of a school in Kerala, India, where they had loads of fantastic mathematicians. They worked on something that if you looked at it today, you would say, that’s calculus.

“Now, it doesn’t have all the polish of the modern calculus but it has the crucial parts of it. It has infinite series, which are absolutely crucial to calculus, and it also has some of the rules that they must have known, which you can infer from some of their writings also suggests they had a better understanding of the theory. For us that is part of the origin story of calculus.”

The book chronicles the life and work of extraordinary female mathematicians. Hypatia, who lived in the fourth to fifth century in Alexandria, was an astronomer, philosopher and mathematician whose talks about the geometry of the universe drew audiences from far and wide.

Revell says: “She had this great school that she took over from her father and then she also reworked some of the classic texts of the time, which was how mathematicians then showed their acumen as mathematicians.

“They improved on what had come before. We’re not entirely sure because a lot of these books were then lost but we think some of the work that Hypatia did then went on and was basically rediscovered in the Renaissance period by European mathematicians later on.”

But Hypatia was accused of political meddling and met a grisly end. A Christian mob dragged her from her carriage and into a church, where she was stripped and beaten to death with broken pieces of pottery. Her body was then paraded through the streets and burned.

Kitagawa explains: “She was accused of having a mythical sort of power. It’s not like mathematics as we can see now. So she had this special ability even to attract people and it was like a witch-hunt and very sadly she had to meet this terrible death. Her story has been retold many times but not in a fair way until recently. There has been ages of misunderstanding and also miswriting about her character.”

There is a chapter on Sophie Kowalevski, born in Moscow in 1850, the daughter of a patriarch who thought that women needed education only to participate in fine society. But her uncle Pyotr often talked with her about maths. She later wrote in her memoir: “The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science, which opens up to its initiates a new world of wonders.”

When Kowalevski was 18, she entered into a “white marriage” (a marriage of convenience) so she could escape her father’s control to move abroad and study maths. At first he would not agree to the union but, “inspired by Dostoevsky’s novels, she made a scene”, locking herself in her future husband’s apartment until her father agreed.

At the University of Berlin, Kowalevski was in effect barred from gaining a PhD because women were not allowed to take part in the standard oral defence of their work in front of a panel of experts. Eventually, she managed to get a PhD from the University of Göttingen.

She devoted much of her time as a maths professor at the University College of Stockholm to a problem she called “the mathematical mermaid”. As the authors put it, ballet dancers intuitively time their rotations to perfection by tweaking the variables of shape, acceleration or speed. But mathematicians could not figure out how to express this in an equation. Even a spinning top that was not completely round defied them.

Revell says: “The thing they couldn’t crack was when it was slightly oddly shaped, when it wasn’t symmetrical. What Sophie Kowalevski did was make breakthroughs on it and that ultimately won her the Prix Bordin [a prestigious annual prize awarded by the French Academy of Sciences]. There’s this amazing moment where she turns out to be the winner of it, even though almost all the mathematicians who would have been winning those kind of prizes were men.”

The book also tells the stories of China’s Ban Zhao, one of the earliest known female mathematicians who taught maths and astronomy to Empress Deng Sui, and Euphemia Lofton Haynes, who became the first Black woman to obtain a PhD in mathematics and fought systemic racism in education.

Then there were the scholars of the “House of Wisdom”, a library and temple of knowledge founded in Baghdad in the eighth century, associated with Muhammad ibn Musa al-Khwarizmi, who introduced decimal numbers and the first hints of algorithms and algebra. The House of Wisdom’s destruction in the 13th-century siege of Baghdad was a loss arguably on par with that of the Library of Alexandria.

Kitagawa reflects: “They have translators and scholars lining up and they gather so much information and books, trying to examine what kind of things they can believe. They are trying to examine without prejudice. They did not just choose one culture but many cultures from all directions. Maybe that’s why this place was destroyed because it had so much power, so much wisdom. It was a very important place that synthesised all the work so far.”

The Secret Lives of Numbers makes readers reconsider the origin stories of concepts such as pi or zero. It is a reminder that maths, like any branch of science, is a human activity undertaken in a social context. It is a collaboration between the living and the dead, often spanning continents and millennia. The book’s authors would not claim that theirs is the last word on the subject.

“My hope is that it’s one of many,” Revell comments. “This is a starting point. Like we say in the book, there’s no such thing as a complete history. There never can be.

“But we’re now in a new era where we can view these things slightly differently and that means when you look at the history of mathematics through this lens you can see it for what it really is: beautifully chaotic, intricate. Sometimes ideas spring up and then they disappear, sometimes someone else takes on the mantle and continues them but from different parts of the world from different people.

“We hadn’t anticipated to the extent that would be when we first started talking about this. But looking back at it now, that’s the journey we went on and the one we try and tell.”

• The Secret Lives of Numbers is out now