How Mathematics Created Civilization
Michael Brooks
Scribe Rrp $32.99
There ain’t half been some clever bastards
Probably got help from their mum
(who had help from her mum)
There ain’t half been some clever bastards
Now that we’ve had some
Let’s hope that there’s lots more to come
Ian Dury
Michael Brooks holds a PhD in Quantum Physics (University of Sussex). He is a journalist, broadcaster, public speaker and author. The Art of More is a broad history of mathematics, starting from the start in the Fertile Crescent through to our modern digital age.
“Humans are born only with what is now known as an ‘approximate number sense’…Our natural counting system is ‘1, 2, 3, more’…Once our brains are schooled in the art of ‘more’, they become able to cope with complicated abstractions.” From geometry and algebra to calculus, the abstractions becoming increasingly complicated and challenging. Brooks notes we don’t have the same facility with maths as our forebears, now with the electronic convenience of pushing a button for an instant answer. “I can’t help feeling that, in some ways, we might be the poorer for it. Some researchers argue, for instance, that our disconnection from geometry stymies our creativity.” “Without technological assistance…we find ourselves able to reliably add and subtract only relatively small numbers, and maybe to multiply and divide a little…We might even become ‘maths-phobic’, actively avoiding an encounter with numbers.” Brooks hopes his book will reverse this perception, as “…almost every aspect of our existence is built on mathematical foundations.”
There follows a very long list of clever bastards and their accomplishments. Under the sub-heading “The Tortured Minds of Calculus”, the author writes “You might, at this point, be asking yourself what kind of person can come up with all this. The answer is: not people you’d want to knock around with, for the most part.” Definitely not people-persons, some bristly, secretive and cantankerous loners, and often nerds like (“in the best possible sense”) “the unsung hero” John Parr Snyder who devised (1976) the system for every sort of digital mapping in use today “applying 82 equations to each of the data points”…creating a Mercator projection …from a moving vantage point…with only minimal distortion of the area directly below the satellite…
We can’t even begin to contemplate how it works here, but it is intriguing to note that Snyder’s paper … nvolves a complex array of sines, cosines and tangents. Thousands of years after we first discovered its properties, we are still harnessing the power of the triangle.”
“Our earliest evidence of commercial accounting …around 4,000 years ago, when Mesopotamian traders began making records of agreements to sell sheep. Each agreement was represented by a clay ball” and a quantity of balls could be baked and sealed inside a hollow container, marking the number of them in it as a record. This was then simplified by marking clay tablets and baking them. King Shulgi of Ur in southwest Iran (2074 B.C.) founded “what scholars have termed the ‘first mathematical state’”. King Shulgi, not surprisingly, declared himself a divinity and his subjects worshipped him, praising his mathematical ability. Unfortunately, Brooks doesn’t provide us the King’s formula for his metamorphosis, but “Within a generation, mathematics became the highest art in the land, an essential component of a scribe’s training. By the turn of the second millennium B.C., a fully qualified scribe would be able to read and write in Sumerian and Babylonian, and know about music and mathematics…the mathematics in question was not the utilitarian number-wrangling of accountants, but the manipulation of numbers to do extremely difficult—and ostensibly useless—calculation.” It was maths for the sake of maths.
Brooks argues that mathematics “first found its place in the… humanities”, on the basis that “an educated scribe” required “mathematical prowess” to satisfy the criteria for “nam-lu-ulu”, Sumerian for ‘the condition of being human’.” The many thousands of clay tablets unearthed reveal a knowledge of “fractions and algebra, and geometric tools such as an approximate value for pi and the square root of 2.” Brooks states “when what we call civilization began, numbers lay at the heart of society.”
Brooks might have mentioned Gobekli Tepe, in south eastern Anatolia, dating from 9,500 to 8,000 B.C. (the world’s oldest known megaliths), the early dates confounding our previous understanding of human history and “civilization”. Prior to the development of agriculture, mathematics had already been architecturally employed by these hunter-gatherers: three round structures (the largest being 20 metres in diameter) were positioned on the geometry of an equilateral triangle.
Brooks notes that oral cultures are “under-represented” in his discussion, citing the West African Akan. The Akan “operated a sophisticated mathematical system for weighing gold used in trade…one for Arab and Portuguese and another for Dutch and English measures. “No wonder, then, that the captains of slave-trading ships who made bargains with African slave-dealers, described them as ‘sharp arithmeticians’”. The procurers and sellers of slaves were Africans turning a profit on their fellows. Some researchers suggested the Akan’s system was “so breathtakingly complex that it should be given UNESCO World Heritage status”: the dealers transact the sale by reducing “them (the slaves) immediately by the head into bars, coppers, ounces, according to the medium of exchange that prevails in the part of the country in which he resides, and immediately strikes the balance.”
The Incas, as another example, hadn’t a written language either, but recorded data and accounts on “knotted strings called quipus. Every town had a ‘keeper of the knots’ who was appointed by the king and acted as a government statistician…” Brooks gives us a broad history, but with a necessarily restricted focus, but maybe the maths behind the knotting is more important than standard histories value. As an example, measured from Kim MacQuarrie’s wider, perspective, mathematics presents as ancillary, rather than central, to the Inca theocracy of Sun-kings, so abruptly brought to end by smallpox, influenza, typhus and that small band of ruthless conquistadores:
The Inca empire straddled the Andes, a mountain chain formed by the continuing collision of a giant tectonic plate called the Nazca plate that slowly smashed into the South American plate, whose western edge also forms the western edge of South America. The Incas thus built their empire within the Pacific’s “ring of fire” where volcanoes periodically erupt. Because of the colliding plates, violent earthquakes are common, destroying cities and towns. In addition, the empire was beset by the climatic havoc wreaked by El Niños every seven years, resulting in savage floods that disrupted food supplies.
The Incas did their best to fathom what, at the time, was unfathomable – the violent, unpredictable catastrophes of nature which, in some cases, had ended cultures that preceded them. To their credit, the Incas did their best to ensure the survival of their people and empire by paying close attention to nature and doing their best to use every means at their disposal, including human sacrifice, to gain control over it…Like other agriculturally based empires, Inca rule was built on reciprocity between the Inca elite and peasants, who were expected to pay taxes in the form of goods and labour; in return, the state was expected to provide the empire’s citizens with security, laws and administration and also with emergency relief in times of famine or natural catastrophe.
The Incas constructed huge storehouses filled with foods and goods. If one area of the empire suffered drought or some other form of calamity, the Incas withdrew food and supplies from the storehouses and replaced them when local production increased again. If another area was attacked by marauding tribes, Inca armies soon arrived to repel the attackers and restore order. Through their labour tax, a succession of Inca rulers built new cities, constructed networks of roads, marshalled vast armies, erected and filled storehouses, and enlarged their empire.
Brook’s prefaces his book with a note: “All are welcome here, whether you love mathematics, have always hated it, or just wish you understood it better…there are bits of actual maths in here: some graphs, equations, and calculations that I’ll walk you through gently. But if any of it bothers you, and you don’t feel like being bothered, just skip that bit. Life’s too short already.”
In truth, there is a lot of math in the book. Whether or not the reader fully grasps the math examples provided by the author, he/she will come away with a better understanding that mathematics is not the tyrannical and mechanical subject some of us were taught (where the answer is either right or wrong), but a fluidly creative, expansive mental exercise.
The author’s references to nature are generated from the standpoint of quantum physics, instead of a standard ecological framework; “imaginary numbers had to be discovered because they are an essential part of the description of nature”. “Nature, it seems, is written in sets of numbers, but not infinitely many of them. Mathematicians have proved that, with the octonions (from the famous epiphany of Irish mathematician William Rowan Hamilton while strolling with his wife along Dublin’s Royal Canal 16 October 1843 and then further collaboration with fellow John Graves) we now have the full set of possible systems with which humans can perform the unreasonably effective work of describing the universe in numbers.”
Brooks only briefly discusses “…the environmental crisis—catastrophic climate change, unprecedented extinction rates, accelerating deforestation, and severely reduced soil fertility (among other concerns)—also has its roots in double-entry bookkeeping. Our obsession with the power of numbers has led us to value only what we can write down as figures on a spreadsheet. We thus reduce the economy of nations to a single, arbitrarily defined number—the gross domestic product—which, according to orthodox economists…must be maximised at all costs. Meanwhile, we operate the global economy through the institutions ruled by numbers: the banks, which operate with near-impunity as they control the fortunes of entire nation states. But at the same time we fail to account for the value of our soils, our forests, our wildlife—especially the insects—and transnational assets such as the polar ice-caps. It is, some have suggested, an algorithm for environmental collapse.”
Influence of Logarithms
Brooks notes that “Logarithms could plausibly be cited as the single most influential invention in modern history”, owing their “existence” to Edinburgh-born John Napier, raised in sectarian violence and “a committed Protestant…consumed by hatred for Catholics”. “When transferred onto a set of wooden sticks known as a slide rule, logarithms powered centuries of science and engineering. The slide rule facilitated the Enlightenment, the Industrial Revolution, the nuclear age and the space race.”
The author broaches the human inability to fathom the term “exponential” despite it being so commonly used in our vocabulary. He cites a lecture given “thousands of times” by “the American physicist Allan Albert Bartlett” who began his talk with “The greatest shortcoming of the human race is our inability to understand the exponential function”. This mental inadequacy is known as exponential-growth-bias (EGB). “When we’re told things are growing exponentially…we vastly underestimate the growth. That’s because our brains shy away from extremes, normalising the growth in our imagination to make it more or less linear.” “This, Bartlett said, is the tragic story of our understanding of exponentials, whether it’s applied to outbreaks of disease or the unsustainable growth of human populations.”
The Wonderful Prof Shannon
Brooks considers Claude Elwood Shannon’s Information Theory “the culmination of tens of thousands of years of human insight, invention, and ingenuity; the pinnacle of the art of more.” Fellow MIT Professor emeritus Rolbert G. Gallagher stated “No-one had come close to this idea before: the information content of a message consists simply of the number of 1s and 0s it takes to transmit it.” This revolutionary concept was taken up by communication engineers who went on to develop the seemingly magical technology of our information age. Professor Shannon’s concept of “channel capacity” led to communication lines measured in bits per second, also enabling the use of bits for storing pictures, voice streams and other data on computers.
Unlike Newton and Descartes (“less than enticing characters”), mathematical geniuses aren’t “always disagreeable”. Although preoccupied with a “head full of thoughts” and therefore “sometimes…difficult to engage”, Shannon, a distant relative of Thomas Edison, was also a likeable larrikin. As a child he fancied himself a circus performer, learning first how to juggle, then do so while riding a unicycle, and then doing this on a steel cable. At Bell Laboratories, Shannon was renowned for riding down the halls while juggling three balls on his unicycle. He constructed styrofoam shoes so he could walk across water, and “developed a juggling machine, rocket-powered Frisbees, motorised Pogo sticks, a mind-reading machine, a mechanical mouse that could navigate a maze and a device that could solve the Rubik’s Cube puzzle. He invented a “Useless Machine”, which turned itself off every time it was turned on. “The science fiction writer Arthur C. Clarke found it disturbing. ‘There is something unspeakably sinister about a machine that does nothing—absolutely nothing—except switch itself off’, he said.” It is a fitting symbol of a self-exterminating species.
Shannon invented the first wearable computer, about the size of a cigarette packet, to “analyse the speed and trajectory of a ball on a roulette wheel”. In the summer of 1961, Shannon and his graduate student, Edward Thorpe (the co-developer), visited Las Vegas to give it a go, with their wives “on the lookout”. The device worked well and they planned to return after growing their hair longer to better conceal the earpiece, but never did so, we are informed. Shannon continued to invest in start-up companies from people he knew and respected and some of these went on to become major corporations, like HP, Teledyne and Motorola. As his wealth increased, he slowly disappeared from “public view” but his popularity never diminished. He appeared, age 69, at The International Symposium on Information Theory in Brighton, 1985, where someone recognised him and passed the word around. Strong-armed to the podium during the evening dinner to make a speech, and worried about saying something boring, Shannon “produced some juggling balls and turned his speech into a cabaret performance. The night ended with physicists, who generally have no interest in celebrities, joining a long cue to get Shannon’s autograph.” Tragically, this wonderful, eccentric genius died in a nursing home after a long battle with Alzheimer’s, 24th February 2001, age 84.
Brooks informs us “There seems to be no area of life that can’t benefit from Shannon’s insights. It has taught us the secrets of the solar system and how to enjoy worry-free online shopping. It brings movies on demand and (hopefully) the final theory of physics. It ties together the internet and the I-Ching. Whether we consider war-winning computers, data-loaded mobile phone signals or songs streamed and beamed through the air and into our ears, our world would be unrecognisable without Shannon’s contribution to mathematics.” I’d never heard of MIT Professor Claude Shannon before I read Brook’s book, a most compelling reason to read it.
There were, in fact, a host of other clever bastards I’d never heard about before. Yes, let’s hope there’s lots more to come…
As the author concludes, brushing aside concerns of dehumanisation, detachment from reality, and devolution:
“We saw how the properties of triangles and circles would bring hitherto impossible calculations to heel, and used the resulting tools to engineer our way into the twentieth century. We understood that abstract ideas like information and imaginary numbers were the key to unlocking atomic, electrical, and electronic power; you are living through the marvel-filled consequences of that. Mathematics has shaped the very experience of what it means to be human, and left its mark on all of us—we just didn’t see it until now. So while we may never agree whether we discovered mathematics or created it, perhaps we can now all agree on something: that mathematics created us.”
Dr John Stockard OAM
Wingham NSW.