Thinking in Numbers - critical summary review

About eight years before he published his third book, “Thinking In Numbers,” in 2014, Daniel Tammet was diagnosed with high-functioning autistic savant syndrome. In other words, the connections in his brain, ever since birth, had formed unusual circuits: he wasn’t painfully shy or hypersensitive, but a strange and rare combination of disabilities and abilities.

Now, most people with similar ailments – if one can call them like that – can’t explain to the world how they do what they do. Tammet is unique because he can. In “Thinking In Numbers,” he does this through 25 unconnected essays, most of which revolve around something he calls the “math of life.” For our microbook, we’ve selected five.

So, get ready to learn what Shakespeare did after discovering the number zero and how Archimedes calculated the number of grains of sand in the universe – and prepare to hear a lot about extremely big numbers!

Classroom intuitions

According to Tammet at least, Mr. Baxter – his secondary-school math teacher – wasn’t a good teacher. In fact, the only two things he acquired from his class were a lifelong indifference to algebra and a pretty definite idea how not to teach others. This latter came in handy when, during his gap year, he was offered an opportunity to teach English in Lithuania. He realized not only that he liked it, but also that he was good at it. So, two years after leaving school, he applied for a tutoring job and started teaching primary-school level math.

Almost all of his students were children. His favorite one was a brown-haired, freckled 8-year-old boy with a fondness for collecting football stickers and remembering the names of the players depicted on them.

One time, because of a mistake, “the boy hit upon a small but clever insight.” While copying down his homework, he wrote down 19+2 rather than 12+9. He was amazed to learn that the result didn’t change: 12 plus 9, and 19 plus 2, both equal 21. This not only pleased him but made him think. A bit later, Tammet asked the boy for the answer to a much larger sum, say something like adding 83 and 9. The boy closed his eyes and started counting: “Eighty-nine, 90, 91.” Tammet writes – “I knew then that he had understood.”

His only adult student was “a housewife with copper-colored skin, and a long name with vowels and consonants that he had never before seen in such a permutation.” She wanted to become a professional accountant, but she had problems understanding some basic mathematical concepts.

However, when she would finally grasp them, she seemed capable of illustrating each in such a vivid real-world-based manner that Tammet remembers some of these comparisons to this very day. For example, when Tammet tried to explain to her negative numbers, she had problems understanding how can one subtract something from nothing. And then it hit her: “You mean, like a mortgage?”

Another time, after a lesson on improper fractions, the two started discussing halving. Both student and teacher were amazed to think that, theoretically speaking, you can continue halving something indefinitely. And it was then that, in an attempt to explain this phenomenon, the housewife came to “a beautiful conclusion about fractions” that Tammet has never forgotten to this very day: “There is no thing that half of it is nothing,” she said.

Shakespeare’s zero

“Few things,” writes Tammet in possibly the most interesting essay of the book, “so fascinated William Shakespeare as the presence of absence: the lacuna where there ought to be abundance, of will, or judgment, or understanding.” For example, when King Lear asks Cordelia what she can say to him to express her love, she says no more than a single word: “Nothing.” To which King Lear replies – “Nothing will come of nothing.” He is gravely mistaken: everything that occurs afterward in the play happens because of this nothing.

Of course, Shakespeare’s contemporaries were pretty familiar with the idea of nothingness. However – and this is something only people like Tammet would probably notice – not many of them would have been familiar with mathematical nothingness, that is, the number zero. Believe it or not, the future bard became one of the first generations of English schoolboys to learn about this digit (also “figure”) – thanks to Robert Recorde’s 1543 textbook, “The Ground of Artes.”

Before Recorde’s book, only one type of number was in used in England: Roman, which the Elizabethans called “German.” Either way, as you already know, these numbers are essentially letters: 1 is the letter I; 5, the letter V; 10, the letter X; 100, the letter C. Perhaps because of this, Recorde compares the digit 0 to the letter O. Years later, Shakespeare would use this comparison (and an appropriate pun) in “King Lear”: “Thou art an O without a figure,” says the Fool to the King. “Thou art nothing.” However, it’s more interesting the way he uses the knowledge appropriated from his classes in “Henry V.”

Namely, at one place in Recorde’s book, the author demonstrates how, in the Arabic system of numbers, any number becomes ten times bigger than itself once a 0 is added at the end. And this is because all of these zeros acquire meaning concerning their place in the number: a 0 in the middle of a three-digit number denotes zero tens, but a 0 at the second place of a five-digit number signifies zero thousands.

Zero, in plain terms, denotes size. In “Cymbeline” you can find a poetic paraphrase of this analysis concerning the number 3,000, but it is four lines from Shakespeare’s prologue to “Henry V” that really resonate: “O, pardon! Since a crooked figure [digit] may/ Attest in little place a million,/ And let us, ciphers to this great accompt,/ On your imaginary forces work.” In translation, even a “nothing,” when placed properly, can amount to something immense. After all, the difference between 1 and 1,000,000 is just six nothings.

On big numbers

Now, let us go back some 2,000 years and south some two thousand miles – and spend a few minutes with Archimedes in the middle of the third century B.C. 

About 200 years before him, one of ancient Greece’s most remarkable lyric poets, Pindar, proclaimed, in a memorable verse (imitated numerous times since) that “the sand escapes numbering.” Intuitively, this seemed so truthful to his fellow Greeks that they started using the phrase “sand hundred” to denote “inconceivably great quantity.” Probably not that happy that a poetic metaphor had evolved into a literal claim, Archimedes decided to demonstrate to the king of Syracuse that not only the number of sand grains in the world isn’t innumerable, but that it wouldn’t have been even if the entire universe was filled with sand. The result? The very first academic paper in recorded history, titled “The Sand Reckoner.”

“Some people believe, King Gelon,” it started, “that the grains of sand are infinite in number… I will attempt to prove to you through geometrical demonstrations, which you will follow, that some of the numbers named by us… exceed… the number of grains of sand having a magnitude equal to the earth filled up.” To achieve this, Archimedes first supposed (and, to make his assignment more difficult, was quite generous in his estimations) that a single poppy seed could hold ten thousand grains of sand (or “myriad” in the language of the ancient Greeks). Then he patiently lined the poppy seeds along a smooth ruler and uncovered that 40of them equaled the length of one inch (2.54 centimeters). So, he was now able to claim that the maximum number of grains of sand that could fill one square inch was 16 million (10,000 x 40 x 40).

Supposing that the universe had a diameter no greater than 100 trillion stadia (about two light-years), Archimedes calculated that it would take no more than 10 to the power of 63(1063) grains of sand to fill the entire universe. Now, Archimedes promised to prove that this number already existed. And that’s what he did next. Demonstrating to King Gelon that the largest number in use back then (“myriad myriads” or 100 million in modern terminology) is already a coinage (myriad square) and can be multiplied by itself to form a coinage twice as big (“myriad myriad myriad myriads” or 10 quadrillions), Archimedes concluded that there would never be more than 7 times “myriad myriads” grains of sand in the universe. Or, in other words, that there is far less and, consequently, this number can be both totaled and formulated into language.

Are we alone?

Archimedes’ calculations – and his tinkering with big numbers – might be dizzying, but it is nothing compared to similar modern-world experiments. After all, Archimedes’ universe was nothing more but a few planets; these, to us, are just a grain of sand in an almost inconceivably bigger universe, the observable part of which is about 46 billion light-years in diameter (or, in other words, about 23 billion times bigger than the one imagined by Archimedes). The question asks itself: how is it possible that, in such a gigantic universe, we, the humans, are the only intelligent species, and our planet, Earth – a tiny dot – the only habitable planet?

Already in the 17th century, people started challenging the dogmatic beliefs of the church that there was nobody but us in the universe. In 1895, American astronomer Percival Lowell wrote that “modesty, backed by a probability little short of demonstration, forbids the thought that we are the sole thinkers in this great universe.” It was the use of the word “probability” – “a sesame word” that opens ears and minds every time it’s mentioned – that incited thousands of people around the world to believe Lowell’s claim that Mars is hospitable and populated. 

The biologist Alfred Russel Wallace – who independently of Darwin discovered the principle of natural selection – was not among them. In his view, the “exceptional and exceptionally complex combination and sequence of events – physical, chemical, cosmological – permitting the origin of life on Earth made the prospect of finding other beings elsewhere in the universe immeasurably remote.” In other words, according to Wallace, the formation of intelligent life might have been a once-in-a-universe event.

About half a century after Wallace’s death, American astronomer and astrophysicist Frank Drake set his sights on developing a formula to estimate the probable number of communicative extraterrestrial civilizations in the Milky Way galaxy alone. The result? Apparently, Lowell might have been right: Drake’s equation (which takes into consideration numerous factors) demonstrates that other intelligent civilizations exist and that we should be able to contact them with almost absolute certainty. Ironically, the very same year that Drake predicted that we should be able to contact our first interstellar neighbors – 2000 – NASA’s supercomputers “performed 50 million tests per second on the data from the largest ever and most sophisticated radio scan of the heavens. They found nothing.”

So, are we alone? “The only thing that we can know for certain,” writes Tammet, “is that the probability of intelligent life in our universe is above zero… The rest is speculation.” In other words, there might have been millions of civilizations since the big bang. But just as well, we might be the only one.

The admirable number pi

In her beautiful poem, “The admirable number pi,” Polish poet and winner of the 1996 Nobel Prize in literature, Wisława Szymborska, admires the fact that even though snakes and comet tails might end, “the pageant of digits comprising the number pi” never does, “always nudging a sluggish eternity to continue.” This is what Szymborska deems admirable about pi. For Tammet, it’s much more literal than this.

You see, he is one of the very few people in the world with a remarkable condition called synesthesia, in which stimulation of one sense mixes with involuntary experiences in a second sense. Because of this, Tammet can see the color of numbers and feel their texture. To him, the number 289 is, quite literally, hideous, and the number 333 very appealing. And pi… Well, pi is magical, a thing of pure beauty. 

“Long after my school days ended,” writes Tammet, “pi’s beauty stayed with me. The digits insinuated themselves into my mind. Those digits seemed to speak of endless possibility, illimitable adventure.” This adventure, for him, started when he first murmured “3.141,” and culminated on March 14, 2004, when, for 5 hours and 9 minutes – to a packed room of spectators in an Oxford museum – he recited pi from memory to 22,514 digits. It remained the European record for the next decade.

Final Notes

If we need to describe “Thinking In Numbers” with a single word, we will opt for “unique.” 

Described by one reviewer as “a strange, hit-and-miss little book,” Daniel Tammet’s exploration of the “math of life” doesn’t always come up with the goods, but when it does, it does so in an exciting and profoundly meaningful manner.

Even if you are not mathematically inclined, we challenge you to read this book without a sense of delight and astonishment.

12min Tip

Study math: rather than being elements of incomprehensible mathematical equations, numbers, and numerical patterns are a matter of the world and belong in life. Learn to enjoy them.