Milo Beckman was studying advanced algebra when he was 8. By 15, he was a Harvard prodigy. Now aged just 25, he's written a brilliant book that takes everything we know (and fear) about maths out of the equation – starting with numbers.
Sometimes, in the school corridors, Milo Beckman's fellow pupils would accost him, wondering what someone so young was doing there. "It was annoying," he says. "They would pinch my cheeks and ask me where I was going."
Later, as a 15-year-old at Harvard, he confesses it also took him time to adjust. No one pinched his cheeks, but, "My first year was not my favourite year of my life, you know. There are social aspects to college that, no matter how much schooling you've done, you aren't necessarily prepared for."
While he might not have necessarily been the frat boy type, he clearly did adjust. In 2015 in his final year, at the same age of many of those in their first, he wrote a column in the student newspaper, The Harvard Crimson.
Current headlines in the Cambridge, England, student newspaper are: "Fez club to close permanently" and "Investigation: Cambridge best city in zombie apocalypse". Its counterpart in Cambridge, Massachusetts, is not like that.
In his column he advised, from his lofty experience, that the new crop of freshmen be wary of specialising too soon. "Academia is in desperate need of cross-pollination," the 19-year-old Beckman wrote. "Institutional barriers beget intellectual barriers."
Milo Beckman has spent his life being years ahead of his peers. Now though, at last, his timing appears bang on. At the ripe old age of 25 he has published a book, Math Without Numbers. It is an attempt to break down some of the intellectual barriers most of the world has erected between themselves and the advanced mathematics he loves.
Its release coincides with a year when maths is having a bit of a moment. No longer is captaining your city's maths team (as he did at the age of 13) or going to math camp (which he also did – "It's math, but it's centred around socialisation") an ever so slightly shameful geeky secret.
In a world where we talk about R rates and logarithmic scales, where our future is mapped out by Bayesian infection models and statistical analyses of lagged data, mathematics is mainstream.
But his book is a reminder that beyond that world of spreadsheets and celebrity modellers (of the mathematical rather than catwalk kind), there lies another mathematical land that is altogether more mysterious – so mysterious that numbers themselves are left far behind.
What is a circle and why isn't it a square? This may, to the non-mathematician, sound like a silly question. A circle is a circle. A square is a square. But it is also one of those silly questions that gets clever mathematicians overthinking. Is their circliness really that circly? A topologist's answer is, no. A circle is merely a square that's been squished a bit, or an oval that hasn't been stretched yet. The difference, they would add, is superficial – and they are all the same.
Or here's another question, a little more obviously mathsy. What is infinity? One answer is that it is the largest thing there is, a number line stretching into the distance that never ends. Another is that it's not the largest thing at all. There is "the continuum" and it is larger. It is larger than infinity in the same way as infinity is larger than one.
What, finally, is a child prodigy? This one is harder. "In math, definitions are always extremely precise; when you have a word, it means a very specific thing," says Beckman. "You know: this is a torus; this is a manifold." Torus and manifold being, one should add, words more commonly found in conversations with those who were once child prodigies. "When people use the word 'prodigy', I think there's this danger that it does try to create this sort of separate class."
Prodigies can, just like a circle, be squeezed into shape. They can be tutored and pushed and prodded. They exist not as a class apart, but on a continuum – an infinity of different kinds. There is the Mozart kind, the Tiger Woods kind, the Tom Lehrer kind, and all the kinds in between – their edges and corners lost in the creation of a single category called "prodigy". So, argues Beckman, it is with him, and he doesn't like the popular perception of what "prodigy" means. "The more you create this trope of the little eight-year-old math genius, the more you create this image in people's heads that that is what math is supposed to be," he says.
All this may be true. This stereotype of the maths prodigy might well be off-putting for the rest of us; it may reinforce an idea that there are areas of intellectual endeavour for ever cut off to those of us who didn't start advanced algebra at primary school. The problem is, of course, Beckman was also once an eight-year-old maths genius. And he did indeed start advanced algebra at primary school.
There is a familiarity to the start of Beckman's story. It fits in that genre of child maths-prodigy tales that, normally, end satisfyingly badly – with the overschooled child running away to the circus, becoming a pole dancer or joining the Moonies.
Beckman had, he says, been interested in maths "as long as I can remember". His parents were not particularly academic – they founded a children's film festival – but valued education. Thanks to a family friend who spotted something in Beckman, when he was eight he started doing some classes at Stuyvesant High School, a highly competitive New York grammar school.
Initially, it was just the odd class here and there. "I would go for one class in the morning and then I would walk two blocks and do the rest of my fourth-grade day at the elementary school down the block." Then, when he was 11, he skipped middle school entirely and applied to Stuyvesant as a full student. "In what would have been seventh grade, I enrolled in ninth grade," says Beckman, who is speaking to me from his home in Brooklyn, New York. Four years later his fellow ninth-graders graduated, and so he did too. Having captained the maths team alongside 18-year-olds, he saw no reason not to go to university alongside them too.
This is when, traditionally, the prodigy realises that they are a misfit, that in gaining academically they have also lost out socially. "A couple of schools I applied to basically said, 'Come back in two years.' They're worried they're gonna have to change your diapers or something. There were some admissions officers who gave very frank and good advice on that and basically said, 'You know there's a lot more to school than just the schooling.' "
Circuses and pole dancing are not, though, where Beckman's journey takes us. Yes, he struggled to make friends among his fellow Harvard mathematicians. But that wasn't because he was younger than them. It was because they were – not to put it too delicately – mathematicians. "I would try to talk to the person next to me and say, 'Hey, this work is due Thursday. Do you want to get together on Wednesday?' and they would always be like, 'Mumble mumble.' It's exhausting. How do you make friends like that?"
So, he had a look at other courses too.
There are mathematicians who study maps. Not the kind of maps with contours and roads and "PH" for pub. Instead, these are maps that chart the spatial or temporal change and relationships in objects – visualising them as a linked "graph".
The object that is Milo Beckman has a very confusing map. Like contours that cross, or a PH not on a road, the relations between his properties do not make sense.
At Harvard, where students are encouraged to have a broad education and experiment with subjects, rather than being the insular maths prodigy, he did just that. "I ended up taking a real crazy scattering of things in my freshman year. Introduction to linguistics, jazz harmony, history of sexuality." He kept up maths but, eventually, he chose to major in a subject called "government" because it had the least onerous lecture schedule – leaving more room to study other courses too. Then he graduated, and became no easier to classify.
In his author's biography, it explains: "He worked for three tech companies, two banks, and a US senator before retiring at age 19 to teach math in New York, China and Brazil, and to work on this book." What that idiosyncratic CV left out was a diversion to graduate school to write a thesis on the philosophy of physics, and a sideline writing crosswords for The New York Times.
His teaching in Brazil was mainly statistics. He was giving conservationists a grounding in numbers and data. This is the maths that, today, is enjoying a certain celebrity. That is the school of maths to which R belongs. It is about using numbers to tell stories and explain the world – and the viruses within it.
There is a kind of mathematician, though, who views such work as a little too practical. They prefer things that barely exist in, and certainly don't explain, the world. They like to ask how many shapes there are, or whether there is more than one kind of infinity.
"A mathematician," said the great interwar number theorist GH Hardy, "is a maker of patterns, like a painter or poet." He argued that the non-applicability of their work was its very point. "It is undeniable that a good deal of elementary mathematics – and I use the word 'elementary' in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus – has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts that have the least aesthetic value. The 'real' mathematics of the 'real' mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless'."
It is this poetic maths that is the subject of Beckman's book. It is clear why Beckman's publisher liked the title, Math Without Numbers. It sounds comforting and unthreatening. This is mathematics minus the bit most people disliked at school. Actually, though, the deeper you go into mathematics and the harder it gets, the less likely it is that numbers are going to pop up at all. "I had this experience when I was taking a complex analysis class," says Beckman. "And I remember seeing the professor writing on the board and the whole board was filled with chalk, and then at the end she wrote '= 2', and the whole class sat back and chuckled because, wow, we hadn't seen a number in a while."
Calling it "math without numbers" is not some trick, then, to make the subject accessible. It is simply a description of what happens when maths gets really hard. The odd thing, though, is the book is not.
According to documents held by the University and Colleges Admissions Service, in the late Nineties I read a lot of maths books. Go into the UCAS archives and you will see there is a Tom Whipple who is applying for a maths degree, who is "passionate" about maths and who found several popular mathematics books from the time variously "fascinating", "enthralling" and "eye-opening".
I was not exactly lying. I did indeed anticipate that by the time of my university interview I would have both read those books and also been enthralled, fascinated and – in retrospect, the bare minimum condition – kept my eyes open. Alas, when the interviews came I had not even managed the latter. I finished none of the books.
In this context, then, I consider it worth mentioning that Beckman's book is not only fascinating and enthralling but also one I actually kept my eyes open long enough to finish (in part because – and this is a massively underrated virtue in popular science books – it is short).
He wrote it because of what he sees as a flaw in US mathematics teaching. And, since US mathematics teaching is based on UK mathematics teaching, a flaw in ours too. "If you talk to mathematicians, there's kind of a universal agreement that the math education system is almost perfectly calibrated to make students terrified and not interested in it.
"For the most part we're teaching this curriculum that was solidified in, like, 1923, and it's based on this idea of what it takes to be a scholar." So you learn trigonometry, Euclidean geometry and the precursors of calculus – but not so much on subjects that are, today, arguably more useful, such as statistics.
"We try to do this compromise between practical math and more abstract math and in the end we're kind of not teaching either that well."
He is right that this view is common among some mathematicians. When politicians periodically lament the numeracy of the country, they typically call for more rigour in the mathematics syllabus. There are many mathematicians who want less. The last time Michael Gove suggested shaking up the curriculum, Timothy Gowers, a Cambridge professor who won the Fields Medal – the mathematical equivalent of the Nobel – lamented, "At the moment, there are large numbers of people who give up mathematics after GCSE with great relief and go through the rest of their lives saying how bad they were at it." Yet as the pandemic has shown, when there is a clear application and purpose, people can become interested and proficient in quite advanced concepts.
"The stuff that you really need to be teaching everyone is the way that math interacts with our daily life: taxes and credit and personal finance, and just the ways we interact with numbers," says Beckman. "People should learn coding; everyone should be required to take some kind of intro to computer science."
Then there's another kind of maths, the kind he likes, that most people will never find useful – but which should be available to them if they want it. This is the kind in his book, a kind he hopes will not be useful, but instead be inspiring.
"Most people when they hear the word math think of school math: multiplication tables, long division, trigonometry. That's great, but I want people to know that there is this whole other type of math, and that when people who love math are talking about math, they're really not talking about that stuff. We don't just sit around doing long multiplication problems by hand. We're not just trying to be a sort of less efficient calculator."
Instead, they think about things with no purpose at all. "What's crazy about it is that at the beginning there is no point. It's just questions like, 'How many shapes are there?' or, 'What's bigger than infinity?' "
But, inevitably, those tediously worldly applications intrude in a way you couldn't have predicted. GH Hardy's abstract number theory of the Twenties became the internet cryptography of the Noughties. "And then, usually 30 years or 200 years later, someone comes up with an application and then they invent an iPhone," Beckman says.
He smiles and, ever so briefly, a little bit of me wants to pinch his cheek.
The world's finest maths brains
By Monique Rivalland
Grigori Perelman, 54
A multi-award-rejecting Russian recluse famous for his contributions to geometric topology. Perelman turned down prestigious jobs at Princeton and Stanford and in 2006 won (and rejected) the Fields Medal, seen as the Nobel prize for maths, because he did not "want to be on display like an animal in a zoo". He has rebuffed every award since, including one worth $1 million.
Alex Gerko, 41
A Russian maths prodigy, Gerko founded an AI-run fintech firm in London in 2015. XTX Markets has no human traders and, according to the Bloomberg Billionaires Index, his algorithms have taken Gerko's net worth to $700 million. Last year XTX was one of the biggest British donors of the pandemic, giving £21.5 million to charities.
Demis Hassabis, 44
Founder of DeepMind, the AI company bought by Google for $400 million in 2014. The son of a Greek Cypriot taxi driver from north London, Hassabis was a master chess player by the time he was 13, sat his A-levels at 15, and in his gap year programmed a world-famous computer game called Theme Park that bought him a Porsche, which he used to drive to freshers week at Cambridge University.
Hou Yifan, 26
The best female chess player in the world. Hou started playing at three years old; by nine she was on the Chinese national team, becoming the youngest female grandmaster in history at 14 and women's world chess champion at 16.
Jim Simons, 83
Simons has been called "the most successful hedge fund manager of all time". His quantitative investment methods – using mathematical models and algorithms – have made him an estimated $23.5 billion. Before trading he was known for his contributions to pattern recognition and string theory.
Sir Roger Penrose, 89
Penrose is Britain's maths poster boy and last year jointly won the Nobel prize for showing that the theory of general relativity does form black holes. He has also shared an award with Stephen Hawking, and created the "impossible" Penrose triangle.
Wang Pok Lo, 16
Pok, as he is known, is on course to become Britain's youngest PhD holder, from Edinburgh University. A Scotsman from Hong Kong, while at his state school in South Queensferry he achieved a first-class degree in maths by the time he was 13 and a master's in statistics by 15.
Alexandra Botez, 25
A Beth Harmon for the 21st century. The American-Canadian chess whizz is the daughter of Romanian refugees and competed in her first national chess championship aged eight. She has more than one million followers across Instagram, YouTube and her Twitch channel BotezLize, where she streams her chess matches.
Sir Martin Hairer, 45
The British-Austrian winner of this year's £2.3 million Breakthrough Prize in Mathematics (Mark Zuckerberg's science award) for his contributions to stochastic partial differential equations – which is to say, he used mind-boggling maths to explain the random effects of things like stirring a cup of tea.
Terence Tao, 45
An Australian-American sensation who has won both the Fields Medal and the Breakthrough Prize. Tao was the youngest person, at 13 years old, to win gold in the International Mathematical Olympiad.
Milo Beckman's Math Without Numbers is out now.
Written by: Tom Whipple
© The Times of London
