An Introduction to Einstein | Weirdest Maths – Excerpt

In Weirdest Maths, David Darling and Agnijo Banerjee reveal the mathematics at the farthest reaches of our world – from its role in the plots of novels to how animals employ numerical skills to survive. Along the way they explore what makes a genius, and one particular genius…

Ask someone to name a genius and the chances are they’ll say ‘Einstein’ and have in mind the image of an old man with smiling eyes and long, wild hair. Einstein’s adult years are well documented, his childhood less so. It’s sometimes said that he was a late developer, that he couldn’t speak well until he was about nine, and that his teachers thought he might have learning difficulties. What seems more likely, as some psychiatrists have suggested, is that Einstein had Asperger’s syndrome, the traits of which include a preoccupation with narrow, often arcane interests, disregard for social mores, lack of interest in general conversation, and sometimes unkempt appearance. The diagnosis is bound to be uncertain and controversial because Einstein is no longer with us and the condition wasn’t fully recognised until after his death. But certainly, as a child, Einstein was intensely focused on maths. Aged twelve he was given a geometry textbook by a friend of the family, Max Talmey, a medical student at the time, who mentored the young Albert on weekly visits to his home. Talmey later recalled that, in the space of one summer: ‘[He] had worked through the whole book. He thereupon devoted himself to higher mathematics… Soon the flight of his mathematical genius was so high I could not follow.’ In the same year, Einstein started teaching himself calculus, a subject he mastered within a year or two. Over the same period, he absorbed Immanuel Kant’s Critique of Pure Reason, a dense, difficult treatise that would baffle most adults. In other school subjects, Einstein was no more than average, which is why he didn’t gain entrance to the Federal Institute of Technology in Zurich (ETH) at his first attempt. As well as being narrowly focused, he was also, as fits an Asperger’s profile, socially remote. In his own words:

I am truly a lone traveller and have never belonged to my country, my home, my friends, or even my immediate family, with my whole heart; in the face of all these ties, I have never lost a sense of distance and a need for solitude…

When he was finally accepted at ETH, after earning a diploma from a minor college in Switzerland, Einstein didn’t always impress those who taught him. One of his lecturers, Hermann Minkowski, called him a ‘lazy dog’ who ‘never bothered about mathematics at all’. Of course, that wasn’t really true. Einstein cared deeply about maths and physics – just not always those aspects or problems presented to him as part of his formal education.

After he graduated, Einstein, like many independent thinkers, found it hard to find a job but after a couple of years was hired as a clerk in a Swiss patent office thanks to the father of one of his academic friends. The work wasn’t too demanding so it left Einstein with plenty of time to develop his own ideas. In 1905, his annus mirabilis, he published a series of papers, on the photoelectric effect, Brownian motion, the special theory of relativity, and the equivalence of mass and energy, any one of which might have won him a Nobel Prize (although, in fact, only the first did). Einstein was now twenty-six and at the peak of his powers. He’d remain at that summit for perhaps another decade during which he hatched a radically new theory of gravity – the general theory of relativity. But after 1915 his creativity fell away and he pioneered no more ground-breaking science for the rest of his life.

In his 1940 memoir, A Mathematician’s Apology, G. H. Hardy wrote: ‘No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game.’ Hardy’s sentiment is commonly extended to include physics, especially theoretical physics, which is highly mathematical. Certainly there are many examples to bolster the argument – Einstein being a case in point. His genius burned intensely for a dozen or so years after the turn of the century, but then flickered out.

By the time Einstein arrived at his final academic home, the Institute for Advanced Study (IAS), in Princeton, New Jersey, in 1933, he’d begun a long and futile quest for a unified theory of gravity and electromagnetism. At the IAS he found himself in the company of two other intellectual giants, both unusual in their own ways: the Austrian Kurt Gödel, a logician, who became Einstein’s closest friend, and John von Neumann.

As premier geniuses of the twentieth century, Einstein and von Neumann make an interesting contrast. Einstein today is overwhelmingly the better known, but von Neumann’s achievements span a broader range and began earlier in his career. By the age of nineteen, von Neumann had published two major mathematical papers, the second of which gave the modern definition of so-called ordinal numbers – numbers that can be used to generalise the concept of natural numbers. He later pioneered game theory and early electronic computers and played a prominent role in the Manhattan Project, the top-secret US programme to develop an atomic bomb.

One of von Neumann’s colleagues on the Manhattan Project was fellow high-IQ Hungarian Eugene Wigner. The two had been a year apart in the same elite school in Budapest. When asked why the Hungary of his generation had spawned so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that von Neumann was the only genius. Perhaps being a close friend from childhood, Wigner was biased but he said of von Neumann ‘only he was fully awake.’ Comparing him with Einstein, however, he commented:

Einstein’s understanding was deeper even than von Neumann’s. His mind was both more penetrating and more original than von Neumann’s.

Various factors, it seems, contribute to what we call genius and the forms it may take: speed of thought (at which von Neumann, by all accounts, was exceptional), depth of understanding (at which, according to Wigner, Einstein excelled), originality, creativity, and so forth. Sometimes, too, genius may be narrow in its focus – as in the case of Einstein or Ramanujan – while at other times, as illustrated by von Neumann, and to an even greater extent by some Renaissance figures such as Leonardo da Vinci, it can range over many subjects. For all his recognition during life, von Neumann doesn’t have the celebrity status today of Einstein whose office at the IAS was just down the hall. Yet while Einstein essentially stagnated after his arrival at the Institute, von Neumann continued to flourish, taking on one massively difficult challenge after another right up until the end of his relatively short life. From the maths of quantum mechanics he’d pivot to practical problems in weather prediction or hydrology, the foundations of computing, or cellular automata. He had an outstanding mastery of many branches of mathematics, which he could bring to bear in his work in physics and computation. Also, he had an astonishing, apparently photographic memory. The mathematician and computer scientist Herman Goldstine, a fellow collaborator on the ENIAC computer project, wrote:

As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation… On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.