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After Tamerlane: The Rise and Fall of Global Empires 1400 – 2000 by John Darwin (2007)

Empires exist to accumulate power on an extensive scale…
(After Tamerlane: The Rise and Fall of Global Empires 1400 – 2000 page 483)

Questions

Why did the nations of Western Europe rise through the 18th and 19th centuries to create empires which stretched around the world, how did they manage to subjugate ancient nations like China and Japan, to turn vast India into a colonial possession, to carve up Africa between them?

How did white European cultures come to dominate not only the territories and peoples who they colonised, but to create the modern mindset – a vast mental framework which encompasses capitalist economics, science and technology and engineering, which dominates the world right down to the present day?

Why did the maritime states of Europe (Britain, France, the Dutch, Spanish and Portuguese) end up either settling from scratch the relatively empty places of the world (America, Australia), or bringing all the other cultures of the world (the Ottoman Empire, Hindu India, Confucian China and Shinto Japan) under their domination?

Answers

For at least two hundred years politicians, historians, economists and all kinds of academics and theoreticians have been writing books trying to explain ‘the rise of the West’.

Some attribute it to the superiority of the Protestant religion (some explicitly said it was God’s plan). Some that it was something to do with the highly fragmented nature of Europe, full of squabbling nations vying to outdo each other, and that this rivalry spilled out into unceasing competition for trade, at first across the Atlantic, then along new routes to India and the Far East, eventually encompassing the entire globe.

Some credit the Scientific Revolution, with its proliferation of new technologies from compasses to cannons, an unprecedented explosion of discoveries and inventions. Some credit the slave trade and the enormous profits made from working to death millions and millions of African slaves which fuelled the industrial revolution and paid for the armies which subjugated India.

Lenin thought it was the unique way European capitalism had first perfected techniques to exploit the proletariat in the home countries and then applied the same techniques to subjugate less advanced nations, and that the process would inevitably lead to a global capitalist war once the whole world was colonised.

John Darwin

So John Darwin’s book, which sets out to answer all these questions and many more, is hardly a pioneering work; it is following an extremely well-trodden path. BUT it does so in a way which feels wonderfully new, refreshing and exciting. This is a brilliant book. If you were only going to read one book about imperialism, this is probably The One.

For at least three reasons:

1. Darwin appears to have mastered the enormous revisionist literature generated over the past thirty years or more, which rubbishes any idea of innate European superiority, which looks for far more subtle and persuasive reasons – so that reading this book means you can feel yourself reaping the benefits of hundreds of other more detailed & specific studies. He is not himself oppressively politically correct, but he is on the right side of all the modern trends in historical thought (i.e. is aware of feminist, BAME and post-colonial studies).

2. Darwin pays a lot more attention than is usual to all the other cultures which co-existed alongside Europe for so long (Islam, the Ottoman Empire, the Mughal Empire, the Safavid Empire, the Chinese Empire, Japan, all are treated in fascinating detail and given almost as much space as Europe, more, in the earlier chapters) so that reading this book you learn an immense amount about the history of these other cultures over the same period.

3. Above all, Darwin paints a far more believable and plausible picture than the traditional legend of one smooth, consistent and inevitable ‘Rise of the West’. On the contrary, in Darwin’s version:

the passage from Tamerlane’s times to our own has been far more contested, confused and chance-ridden than the legend suggests – an obvious enough point. But [this book places] Europe (and the West) in a much larger context: amid the empire-, state- and culture-building projects of other parts of Eurasia. Only thus, it is argued, can the course, nature, scale and limits of Europe’s expansion be properly grasped, and the jumbled origins of our contemporary world become a little clearer.

‘Jumbled origins’, my God yes. And what a jumble!

Why start with Tamerlane?

Tamerlane the Eurasian conqueror died in 1405. Darwin takes his death as marking the end of an epoch, an era inaugurated by the vast wave of conquest led across central Asia by Genghis Khan starting around 1200, an era in which one ruler could, potentially, aspire to rule the entire Eurasian landmass.

When Tamerlane was born the ‘known world’ still stretched from China in the East, across central Asia, through the Middle East, along the north African shore and including Europe. Domination of all of China, central Asia, northern India, the Middle East and Europe was, at least in theory, possible, had been achieved by Genghis Khan and his successors, and was the dream which had inspired Tamerlane.

Map of the Mongol Empire created by Genghis Khan

But by the death of Tamerlane the political situation across Eurasia had changed. The growth in organisation, power and sophistication of the Ottoman Empire, the Mamluk state in Egypt and Syria, the Muslim sultanate in north India and above all the resilience of the new Ming dynasty in China, meant this kind of ‘global’ domination was no longer possible. For centuries nomadic tribes had ravaged through Eurasia (before the Mongols it had been the Turks who emerged out of Asia to seize the Middle East and found the Ottoman Dynasty). Now that era was ending.

It was no longer possible to rule the sown from the steppe (p.5)

Moreover, within a few decades of Tamerlane’s demise, Portuguese mariners had begun to explore westwards, first on a small scale colonising the Azores and Canary Islands, but with the long-term result that the Eurasian landmass would never again constitute the ‘entire world’.

What was different about European empires?

Empires are the oldest and most widespread form of government. They are by far the commonest way that human societies have organised themselves: the Assyrians, Babylonians, Egyptians, Persians, the Greek and Roman Empires, the Aztec Empire, the Inca Empire, the Mali Empire, Great Zimbabwe, the Chinese empire, the Nguyễn empire in Vietnam, the Japanese Empire, the Ottoman empire, the Mughal empire, the Russian empire, the Austro-Hungarian empire, to name just a few.

Given this elementary fact about history, why do the west European empires come in for such fierce criticism these days?

Because, Darwin explains, they were qualitatively different.

  1. Because they affected far more parts of the world across far more widespread areas than ever before, and so ‘the constituency of the aggrieved’ is simply larger – much larger – than ever before.
  2. Because they were much more systematic in their rapaciousness. The worst example was surely the Belgian Empire in the Congo, European imperialism stripped of all pretence and exposed as naked greed backed up by appalling brutality. But arguably all the European empires mulcted their colonies of raw materials, treasures and of people more efficiently (brutally) than any others in history.

The result is that it is going to take some time, maybe a lot of time, for the trauma of the impact of the European empires to die down and become what Darwin calls ‘the past’ i.e. the realm of shadowy past events which we don’t think of as affecting us any more.

The imperial legacy is going to affect lots of people, in lots of post-colonial nations, for a long time to come, and they are not going to let us in the old European colonial countries forget it.

Structure

After Tamerlane is divided into nine chapters:

  1. Orientations
  2. Eurasia and the Age of Discovery
  3. The Early Modern Equilibrium (1750s – 1800)
  4. The Eurasian Revolution (1800 – 1830)
  5. The Race Against Time (1830 – 1880)
  6. The Limits of Empire (1880 – 1914)
  7. Towards The Crisis of The World (1914 – 42)
  8. Empire Denied (1945 – 2000)
  9. Tamerlane’s Shadow

A flood of insights

It sounds like reviewer hyperbole but there really is a burst of insights on every page of this book.

It’s awe-inspiring, dazzling, how Darwin can take the elements of tremendously well-known stories (Columbus and the discovery of America, or the Portuguese finding a sea route to India, the first trading stations on the coasts of India or the unequal treaties imposed on China, or the real consequences of the American Revolution) and present them from an entirely new perspective. Again and again on every page he unveils insight after insight. For example:

American

Take the fact – which I knew but had never seen stated so baldly – that the American War of Independence wasn’t about ‘liberty’, it was about land. In the aftermath of the Seven Years War (1756 – 63) the British government had banned the colonists from migrating across the Appalachians into the Mississippi valley (so as to protect the Native Americans and because policing this huge area would be ruinously expensive). The colonists simply wanted to overthrow these restrictions and, as soon as the War of Independence was over (i.e. after the British gave up struggling to retain the rebel colonies in 1783), the rebels set about opening the floodgates to colonising westward.

India

Victorian apologists claimed the British were able to colonise huge India relatively easily because of the superiority of British organisation and energy compared with Oriental sloth and backwardness. In actual fact, Darwin explains it was in part the opposite: it was because the Indians had a relatively advanced agrarian economy, with good routes of communication, business hubs and merchants – an open and well-organised economy, which the British just barged their way into (p.264).

(This reminds me of the case made in The Penguin History of Latin America by Edwin Williamson that Cortés was able to conquer the Aztec and Pissarro the Incas, not because the Indians were backward but precisely because they were the most advanced, centralised and well organised states in Central and South America. The Spanish just installed themselves at the top of a well-ordered and effective administrative system. Against genuinely backward people, like the tribes who lived in the arid Arizona desert or the swamps of Florida or hid in the impenetrable Amazon jungle, the Spanish were helpless, because there was no one emperor to take hostage, or huge administrative bureaucracy to take over – which explains why those areas remained uncolonised for centuries.)

Cultural conservatism

Until about 1830 there was still a theoretical possibility that a resurgent Ottoman or Persian empire, China or Japan, might have reorganised and repelled European colonisers. But a decisive factor which in the end prevented them was the intrinsic conservatism of these cultures. For example, both Chinese and Muslim culture venerated wisdom set down by a wise man (Mohammed, Confucius) at least a millennium earlier, and teachers, professors, civil servants were promoted insofar as they endorsed and parroted these conservative values. At key moments, when they could have adopted more forward-looking ideologies of change, all the other Eurasian cultures plumped for conservatism and sticking to the Old.

Thus, even as it dawned on both China and Japan that they needed to react to the encroachments of the Europeans in the mid-nineteenth century, both countries did so by undertaking not innovations but what they called restorations – the T’ung-chih (‘Union for Order’) restoration in China and the Meiji (‘Enlightened rule’) restoration in Japan (p.270). (Darwin’s description of the background and enactment of both these restorations is riveting.)

The Western concept of Time

Darwin has a fascinating passage about how the Europeans developed a completely new theory of Time (p.208). It was the exploration of America which did this (p.209) because here Europeans encountered, traded and warred with Stone Age people who used bows and arrows and (to start with) had no horses or wheeled vehicles and had never developed anything like a technology. This led European intellectuals to reflect that maybe these people came from an earlier phase of historical development, to develop the new notion that maybe societies evolve and develop and change.

European thinkers quickly invented numerous ‘systems’ suggesting the various ‘stages of development’ which societies progressed through, from the X Age to the Y Age and then on to the Z Age – but they all agreed that the native Americans (and even more so, the Australian aborigines when they were discovered in the 1760s) represented the very earliest stages of society, and that, by contrast, Western society had evolved through all the intervening stages to reach its present state of highly evolved ‘perfection’.

Once you have created mental models like this, it is easy to categorise all the other cultures you encounter (Ottomans, Hindus, China, Japan, Siam, Annamite etc) as somewhere lower or backward on these paths or stages of development.

And being at the top of the tree, why, naturally that gave white Europeans the right to intervene, invade, conquer and administer all the other people of the world in order to ‘raise’ them to the same wonderful level of civilisation as themselves.

18th and 19th

I’ve always been a bit puzzled by the way that, if you read accounts of the European empires, there is this huge difference between the rather amateurish 18th century and the fiercely efficient 19th century. Darwin explains why: in the eighteenth century there were still multiple European players in the imperial game: France was the strongest power on the continent, but she was balanced out by Prussia, Austria and also Spain and Portugal and the Dutch. France’s position as top dog in Europe was admittedly damaged by the Seven Years War but it wasn’t this, it was the Napoleonic Wars which in the end abolished the 18th century balance of power in Europe. Britain emerged from the Napoleonic Wars as the new top dog, with a navy which could beat all-comers, which had hammered the French at the Battle of the Nile and Trafalgar, and which now ruled the waves.

The nineteenth century feels different because Britain’s world-encompassing dominance was different in kind from any empire which ever preceded it.

The absence of Africa

If I have one quibble it’s that I’d like to have learned more about Africa. I take the point that his book is focused on Eurasia and the Eurasian empires (and I did learn a huge amount about Persia, the Moghul empire, China and Japan) and that all sub-Saharan Africa was cut off from Eurasia by the Sahara, but still… it feels like an omission.

And a woke reader might well object to the relative rareness of Darwin’s references to the African slave trade. He refers to it a few times, but his interest is not there; it’s in identifying exactly where Europe was like or unlike the rival empires of Eurasia, in culture and science and social organisation and economics. That’s his focus.

The expansion of the Russian empire

If Africa is disappointingly absent, an unexpected emphasis is placed in each chapter on the imperial growth of Russia. I knew next to nothing about this. A quick surf on Amazon suggests that almost all the books you can get about the Russian ’empire’ are about the fall of the Romanovs and the Bolshevik Revolution and then Lenin or Stalin’s creation of a Bolshevik empire which expanded into Eastern Europe after the war. That’s to say it’s almost all about twentieth century Russia (with the exception of a crop of ad hoc biographies of Peter the Great or Catherine the Great).

So it was thrilling to read Darwin give what amounts to a sustained account and explanation of the growth of the Kingdom of Muscovy from the 1400s onwards, describing how it expanded west (against Poland, the Baltic states, Sweden), south towards the Black Sea, south-west into the Balkans – but most of all how Russian power was steadily expanded East across the vast inhospitable tundra of Siberia until Russian power reached the Pacific.

It is odd, isn’t it, bizarre, uncanny, that a nation that likes to think of itself as ‘European’ has a huge coastline on the Pacific Ocean and to this day squabbles about the ownership of small islands with Japan!

The process of Russian expansion involved just as much conquering of the ‘primitive’ tribal peoples who hunted and trapped in the huge landmass of Siberia as the conquest of, say, Canada or America, but you never read about it, do you? Can you name any of the many native tribes the Russians fought and conquered? No. Are there any books about the Settling of the East as there are thousands and thousands about the conquest of the American West? Nope. It is a historical black hole.

But Darwin’s account of the growth of the Russian Empire is not only interesting as filling in what – for me at any rate – is a big hole in my knowledge. It is also fascinating because of the role Russian expansion played again and again in the game of Eurasian Risk which his book describes. At key moments Russian pressure from the North distracted the attention of the Ottoman Empire from making more offensive thrusts into Europe (the Ottomans famously encroached right up to the walls of Vienna in 1526 and then again in 1683).

When the Russians finally achieved one of their territorial goals and seized the Crimea in 1783, as a result of the Russo-Turkish War, it had the effect, Darwin explains, of cracking the Ottoman Empire open ‘like an oyster’. For centuries the Black Sea had been an Ottoman lake and a cheaply defensible frontier. Now, at a stroke, it became a massive vulnerability which needed costly defence (p.175).

And suddenly, seeing it all from the Russian perspective, this sheds new light on the timeworn story of the decline of the Ottoman Empire which I only know about from the later 19th century and from the British perspective. For Darwin the role of Russian expansionism was vital not only in itself, but for the hemming in and attritional impact it had on the other Eurasian empires – undermining the Ottomans, making the Chinese paranoid because Russian expansion around its northern borders added to China’s sense of being encircled and endangered, a sense that contributed even more to its risk-averse policy of doubling down on its traditional cultural and political and economic traditions, and refusing to see anything of merit in the Westerners’ technology or crude diplomacy. A policy which eventually led to the Chinese empire’s complete collapse in 1911.

And of course the Russians actually went to war with imperial Japan in 1905.

Numbered lists

Darwin likes making numbered lists. There’s one on almost every page. They rarely go higher than three. Here are some examples to give a flavour of his careful, forensic and yet thrillingly insightful way of explaining things.

The 18th century geopolitical equilibrium

The geopolitical revolution which ended the long equilibrium of the 18th century had three major effects:

  1. The North American interior and the new lands in the Pacific would soon become huge extensions of European territory, the ‘new Europes’.
  2. As a result of the Napoleonic war, the mercantile ‘zoning’ system which had reflected the delicate balance of power among European powers was swept away and replaced with almost complete control of the world’s oceans by the British Navy.
  3. Darwin gives a detailed description of why Mughal control of North India was disrupted by invasions by conquerors from the north, first Iran then Afghanistan, who weakened central Indian power at just the moment the British started expanding from their base in Bengal. Complex geopolitical interactions.

The so-called stagnation of the other Eurasian powers can be characterised by:

  1. In both China and the Islamic world classical, literary cultures dominated the intellectual and administrative elites – the test of intellectual acumen was fitting all new observations into the existing mindset, prizes went to those who could do so with the least disruption possible.
  2. Cultural and intellectual authority was vested in scribal elites backed up by political power, both valuing stasis.
  3. Both China and the Islamic world were profoundly indifferent and incurious about the outside world.

The knowledge revolution

Compare and contrast the East’s incuriosity with the ‘West’, which underwent a cognitive and scientific revolution in which merit went to the most disruptive inventors of new theories and technologies, and where Darwin describes an almost obsessive fascination with maps. This was supercharged by Captain Cook’s three huge expeditions around the Pacific, resulting in books and maps which were widely bought and discussed, and which formed the basis of the trade routes which followed in his wake, and then the transportation of large numbers of convicts to populate Australia’s big empty spaces (about 164,000 convicts were transported to the Australian colonies between 1788 and 1868).

Traumatic impact of the Napoleonic Wars

I hadn’t quite realised that the Napoleonic Wars had such a traumatising effect on the governments of the main European powers who emerged in its aftermath: Britain, France, Prussia, Austria and Russia. Very broadly speaking there was peace between the European powers between the 1830s and 1880s. Of course there was the Crimean War (Britain, France and Turkey containing Russia’s imperial expansion), war between Austria and Prussia (1866) and the Franco-Prussian War. But all these were contained by the system, were mostly of short duration and never threatened to unravel into the kind of general conflict which ravaged Europe under Napoleon.

Thus, from the imperial point of view, the long peace had four results:

  1. The Royal Navy’s policing of all trade routes across the Atlantic and between Europe and Asia kept trade routes open throughout the era and kept costs down for everyone.
  2. The balance of power which the European powers maintained among themselves discouraged intervention in either North or South America and allowed America to develop economically as if it had no enemies – a rare occurrence for any nation in history.
  3. The post-Napoleonic balance of power in Europe encouraged everyone to tread carefully in their imperial rivalries.
  4. Geo-political stability in Europe allowed the growth across the continent of something like a European ideology. This was ‘liberalism’ – a nexus of beliefs involving the need for old-style autocratic power to be tempered by the advice of representatives of the new middle class, and the importance of that middle class in the new technologies and economics unleashed by the industrial revolution and in founding and administering the growing colonies abroad.

Emigration

Emigration from Europe to the New World was a trickle in the 1830s but had become a flood by the 1850s. Between 1850 and 1880 over eight million people left Europe, mostly for America.

  1. This mass emigration relieved the Old World of its rural overcrowding and transferred people to an environment where they could be much more productive.
  2. Many of the emigrants were in fact skilled artisans. Moving to an exceptionally benign environment, a vast empty continent rich in resources, turbo-charged the American economy with the result that by the 1880s it was the largest in the world.

Fast

His chapter The Race Against Time brings out a whole area, an entire concept, I’ve never come across before, which is that part of the reason European colonisation was successful was it was so fast. Not just that Western advances in military technology – the lightning advances in ships and artillery and guns – ran far ahead of anything the other empires could come up with – but that the entire package of international finance, trade routes, complex webs sending raw materials back home and re-exporting manufactured goods, the sudden flinging of railways all across the world’s landmasses, the erection of telegraphs to flash knowledge of markets, prices of goods, or political turmoil back from colonies to the European centre – all of this happened too quickly for the rival empires (Ottoman, Japan, China etc) to stand any chance of catching up.

Gold rushes

This sense of leaping, hurtling speed was turbo-charged by literal gold rushes, whether in the American West in the 1840s or in South Africa where it was first gold then diamonds. Suddenly tens of thousands of white men turned up, quickly followed by townships full of traders and artisans, then the railway, the telegraph, the sheriffs with their guns – all far faster than any native American or South African cultures could hope to match or even understand.

Shallow

And this leads onto another massive idea which reverberates through the rest of the book and which really changed my understanding. This is that, as the spread of empire became faster and faster, reaching a kind of hysterical speed in the so-called Scramble For Africa in the 1880s (the phrase was, apparently, coined by the London Times in 1884) it meant that there was something increasingly shallow about its rule, especially in Africa.

The Scramble for Africa

Darwin says that most radical woke historians take the quick division of Africa in the 1880s and 1890s as a kind of epitome of European imperialism, but that it was in fact the opposite, and extremely unrepresentative of the development of the European imperialisms.

The Scramble happened very quickly, markedly unlike the piecemeal conquest of Central, Southern of North America, or India, which took centuries.

The Scramble took place with almost no conflict between the European powers – in fact they agreed to partitions and drew up lines in a very equable way at the Congress of Berlin in 1885. Other colonies (from the Incas to India) were colonised because there were organised civilisations which could be co-opted, whereas a distinctive feature about Africa (‘historians broadly agree about one vital fact’ p.314) was that people were in short supply. Africa was undermanned or underpeopled. There were few organised states or kingdoms because there simply wasn’t the density of population which lends itself to trading routes, settled farmers and merchants – all the groups who can be taxed to create a king and aristocracy.

Africans hadn’t progressed to centralised states as humans had in Eurasia or central America because there weren’t enough of them. Hence the poverty and the lack of resistance which most of the conquerors encountered in most of Africa.

In fact the result of all this was that most of the European governments weren’t that keen on colonising Africa. It was going to cost a lot of money and there weren’t the obvious revenue streams that they had found in a well-established economy like India.

What drove the Scramble for Africa more than anything else was adventurers on the ground – dreamers and fantasists and ambitious army officers and business men and empire builders who kept on taking unilateral action which then pitched the home government into a quandary – deny their adventurers and pass up the opportunity to win territory to a rival, or reluctantly support them and get enmeshed in all kinds of messy responsibilities.

For example, in the mid-1880s a huge swathe of West Africa between the desert and the forest was seized by a buccaneering group of French marine officers under Commandant Louis Archinard, and their black rank and file. In a few years these adventurers brought some two million square miles into France’s empire. The government back in Paris felt compelled to back them up which meant sending out more troops, police and so on, which would cost money.

Meanwhile, modern communications had been invented, the era of mass media had arrived, and the adventuring soldiers and privateers had friends and boosters in the popular press who could be counted on to write leading articles about ‘the white man’s burden’ and the torch of civilisation and ask: ‘Isn’t the government going to defend our brave boys?’, until reluctant democratic governments were forced to cough up support. Modern-day liberals often forget that imperialism was wildly popular. It often wasn’t imperialist or rapacious governments or the ruling class which prompted conquest, but popular sentiment, jingoism, which couldn’t be ignored in modern democracies.

Darwin on every page, describes and explains the deep economic, trade and financial structures which the West put in place during the nineteenth century and which eventually underpinned an unstoppable steamroller of annexation, protectorates, short colonial wars and long-term occupation.

The Congress of Berlin

The Congress of Berlin helped to formalise the carving up of Africa, and so it has come to be thought of as evil and iniquitous, particularly by BAME and woke historians. But once again Darwin makes you stop and think when he compares the success of the congress at reaching peaceful agreements between the squabbling European powers – and what happened in 1914 over a flare-up in the Balkans.

If only Bismarck had been around in 1914 to suggest that, instead of rapidly mobilising to confront each other, the powers of Europe had once again been invited for tea and cake at the Reichstag to discuss their differences like gentlemen and come to an equable agreement.

Seen from this perspective, the Berlin Congress is not so much an evil colonialist conspiracy, but an extremely successful event which avoided any wars between the European powers for nearly thirty years. Africa was going to be colonised anyway because human events have a logic of their own: the success was in doing so without sparking a European conflagration.

The Scramble for China

The Scramble for China is not as well known as its African counterpart,  the competition to gain ‘treaty ports’ on the Chinese coast, impose unfair trading terms on the Chinese and so on.

As usual, though, Darwin comes at it from a much wider angle and makes one massive point I hadn’t registered before, which is that Russia very much wanted to seize the northern part of China to add to its far eastern domains; Russia really wanted to carve China up, but Britain didn’t. And if Britain, the greatest trading, economic and naval power in the world, wasn’t onside, then it wouldn’t happen. There wasn’t a genuine Scramble for China because Britain didn’t want one.

Why not? Darwin quotes a Foreign Office official simply saying, ‘We don’t want another India.’ One enormous third world country to try and administer with its hundreds of ethnic groups and parties growing more restive by the year, was quite enough.

Also, by the turn of the century, the Brits had become paranoid about Russia’s intentions to conquer Afghanistan and march into North India. If they partitioned China with Russia, that would mean policing an even longer frontier even further way against an aggressive imperialist power ready to pounce the moment our guard was down.

Summary

This is an absolutely brilliant book. I don’t think I’ve ever come across so many dazzling insights and revelations and entirely new ways of thinking about a time-worn subject in one volume.

This is the book to give anyone who’s interested not just in ‘the rise of the West’ but how the whole concept of ‘the West’ emerged, for a fascinating description not just of the European empires but of all the empires across Eurasia – Ottoman, Persian, Moghul, Chinese and Japanese – and how history – at this level – consists of the endless juggling for power of these enduring power blocs, the endless and endlessly

complex history of empire-, state- and culture-building. (p.490)

And of course it all leads up to where we are today: a resurgent Russia flexing its muscles in Ukraine and Crimea; China wielding its vast economic power and brutally oppressing its colonial subjects in Tibet and Xinkiang, while buying land, resources and influence across Africa. And both Russia and China using social media and the internet in ways we don’t yet fully understand in order to undermine the West.

And Turkey, keen as its rulers of all colours have been since the Ottoman days, to keep the Kurds down. And Iran, as its rulers have done for a thousand years, continually seeking new ways to extend its influence around the Gulf, across Syria and to the Mediterranean, in eternal rivalry with the Arab world which, in our time, means Saudi Arabia, against whom Iran is fighting a proxy war in the Yemen.

Darwin’s books really drives home the way the faces and the ideologies may change, but the fundamental geopolitical realities endure, and with them the crudeness and brutality of the tools each empire employs.

If you let ‘morality’, especially modern woke morality, interfere with your analysis of this level of geopolitics, you will understand nothing. At this level it always has and always will be about power and influence, dominating trade and ensuring raw resources, and behind it all the never-ending quest for ‘security’.

At this level, it isn’t about following narrow, English notions of morality. Getting hung up on that only gets in the way of grasping the utterly amoral forces at play everywhere in the world today, just as they’ve always been.

Darwin stands up for intelligence and insight, for careful analysis and, above all, for a realistic grasp of human nature and human society – deeply, profoundly flawed and sometimes pitiful and wretched though both routinely are. He takes an adult view. It is absolutely thrilling and a privilege to be at his side as he explains and analysis this enormous history with such confidence and with so many brilliant ideas and insights.

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Posted by Simon on November 15, 2019

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Alex’s Adventures In Numberland by Alex Bellos (2010)

Alexander Bellos (born in 1969) is a British writer and broadcaster. He is the author of books about Brazil and mathematics, as well as having a column in The Guardian newspaper. After adventures in Brazil (see his Wikipedia page) he returned to England in 2007 and wrote this, his first book. It spent four months in the Sunday Times bestseller list and led on to five more popular maths books.

It’s a hugely enjoyable read for three reasons:

  1. Bellos immediately establishes a candid, open, good bloke persona, sharing stories from his early job as a reporter on the Brighton Argus, telling some colourful anecdotes about his time in Brazil and then being surprisingly open about the way that, when he moved back to Britain, he had no idea what to do. The tone of the book is immediately modern, accessible and friendly.
  2. However this doesn’t mean he is verbose. The opposite. The book is packed with fascinating information. Every single paragraph, almost every sentence contains a fact or insight which makes you sit up and marvel. It is stufffed with good things.
  3. Lastly, although its central theme is mathematics, it approaches this through a wealth of information from the humanities. There is as much history and psychology and anthropology and cultural studies and philosophy as there is actual maths, and these are all subjects which the average humanities graduate can immediately relate to and assimilate.

Chapter Zero – A Head for Numbers

Alex meets Pierre Pica, a linguist who’s studied the Munduruku people of the Amazon and discovered they have little or no sense of numbers. They only have names for numbers up to five. Also, they cluster numbers together logarithmically i.e. the higher the number, the closer together they clustered them. Same thing is done by kindergarten children who only slowly learn that numbers are evenly spaced, in a linear way.

This may be because small children and the Munduruku don’t count so much as estimate using the ratios between numbers.

It may also be because above a certain number (five) Stone Age man needed to make quick estimates along the lines of, Are there more wild animals / members of the other gang, than us?

Another possibility is that distance appears to us to be logarithmic due to perspective: the first fifty yards we see in close detail, the next fifty yards not so detailed, beyond 100 yards looking smaller, and so on.

It appears that we have to be actively taught when young to overcome our logarithmic instincts, and to apply the rule that each successive whole number is an equal distance from its predecessor and successor i.e. the rational numbers lies along a straight line at regular intervals.

More proof that the logarithmic approach is the deep, hard-wired one is the way most of us revert to its perspective when considering big numbers. As John Allen Paulos laments, people make no end of fuss about discrepancies between 2 or 3 or 4 – but are often merrily oblivious to the difference between a million or a billion, let alone a trillion. For most of us these numbers are just ‘big’.

He goes on to describe experiments done on chimpanzees, monkeys and lions which appear to show that animals have the ability to estimate numbers. And then onto experiments with small babies which appear to show that as soon as they can focus on the outside world, babies can detect changes in number of objects.

And it appears that we also have a further number skill, that guesstimating things – the journey takes 30 or 40 minutes, there were twenty or thirty people at the party, you get a hundred, maybe hundred and fifty peas in a sack. When it comes to these figures almost all of us give rough estimates.

To summarise:

  • we are sensitive to small numbers, acutely so of 1, 2, 3, 4, less so of 5, 6, 7, 8, 9
  • left to our own devices we think logarithmically about larger numbers i.e lose the sense of distinction between them, clump them together
  • we have a good ability to guesstimate medium size numbers – 30, 40, 100

But it was only with the invention of notation, a way of writing numbers down, that we were able to create the linear system of counting (where every number is 1 larger than its predecessor, laid out in a straight line, at regular intervals).

And that this cultural invention enabled human beings to transcend our vague guesstimating abilities, and laid the basis for the systematic manipulation of the world which followed

Chapter One – The Counter Culture

The probable origins of counting lie in stock taking in the early agricultural revolution some 8,000 years ago.

We nowadays count using a number base 10 i.e. the decimal system. But other bases have their virtues, especially base 12. It has more factors i.e. is easier to divide: 12 can be divided neatly by 2, 3, 4 and 6. A quarter of 10 is 2.5 but of 12 is 3. A third of 10 is 3.333 but of 12 is 4. Striking that a version of the duodecimal system (pounds, shillings and pence) hung on in Britain till we finally went metric in the 1970s. There is even a Duodecimal Society of America which still actively campaigns for the superiority of a base 12 counting scheme.

Bellos describes a bewildering variety of other counting systems and bases. In 1716 King Charles XII of Sweden asked Emmanuel Swedenborg to devise a new counting system with a base of 64. The Arara in the Amazon count in pairs, the Renaissance author Luca Paccioli was just one of hundreds who have devised finger-based systems of counting – indeed, the widespread use of base 10 probably stems from the fact that we have ten fingers and toes.

He describes a complicated Chinese system where every part of the hand and fingers has a value which allows you to count up to nearly a billion – on one hand!

The Yupno system which attributes a different value for parts of the body up to its highest number, 33, represented by the penis.

Diagram showing numbers attributed to parts of the body by the Yupno tribe

Diagram showing numbers attributed to parts of the body by the Yupno tribe

There’s another point to make about his whole approach which comes out if we compare him with the popular maths books by John Allen Paulos which I’ve just read.

Paulos clearly sees the need to leaven his explanations of comparative probability and Arrow’s Theorem and so on with lighter material and so his strategy is to chuck into his text things which interest him: corny jokes, anecdotes about baseball, casual random digressions which occur to him in mid-flow. But al his examples clearly 1. emanate from Paulos’s own interests and hobby horses (especially baseball) and 2. they are tacked onto the subjects being discussed.

Bellos, also, has grasped that the general reader needs to be spoonfed maths via generous helpings of other, more easily digestible material. But Bellos’s choice of material arises naturally from the topic under discussion. The humour emerges naturally and easily from the subject matter instead of being tacked on in the form of bad jokes.

You feel yourself in the hands of a master storyteller who has all sorts of wonderful things to explain to you.

In fourth millennium BC, an early counting system was created by pressing a reed into soft clay. By 2700 BC the Sumerians were using cuneiform. And they had number symbols for 1, 10, 60 and 3,600 – a mix of decimal and sexagesimal systems.

Why the Sumerians grouped their numbers in 60s has been described as one of the greatest unresolved mysteries in the history of arithmetic. (p.58)

Measuring in 60s was inherited by the Babylonians, the Egyptians and the Greeks and is why we still measure hours in 60 minutes and the divisions of a circle by 360 degrees.

I didn’t know that after the French Revolution, when the National Convention introduced the decimal system of weights and measures, it also tried to decimalise time, introducing a new system whereby every day would be divided into ten hours, each of a hundred minutes, each divided into 100 seconds. Thus there were a very neat 10 x 100 x 100 = 100,000 seconds in a day. But it failed. An hour of 60 minutes turns out to be a deeply useful division of time, intuitively measurable, and a reasonable amount of time to spend on tasks. The reform was quietly dropped after six months, although revolutionary decimal clocks still exist.

Studies consistently show that Chinese children find it easier to count than European children. This may be because of our system of notation, or the structure of number names. Instead of eleven or twelve, Chinese, Japanese and Koreans say the equivalent of ten one, ten two. 21 and 22 become two ten one and two ten two. It has been shown that this makes it a lot simpler and more intuitive to do basic addition and subtraction.

Bellos goes on to describe the various systems of abacuses which have developed in different cultures, before explaining the phenomenal popularity of abacus counting, abacus clubs, and abacus championships in Japan which helps kids develop the ability to perform anzan, using the mental image of an abacus to help its practitioners to sums at phenomenal speed.

Chapter Two – Behold!

The mystical sense of the deep meaning of numbers, from Pythagoras with his vegetarian religious cult of numbers in 4th century BC Athens to Jerome Carter who advises leading rap stars about the numerological significance of their names.

Euclid and the elegant and pure way he deduced mathematical theorems from a handful of basic axioms.

A description of the basic Platonic shapes leads into the nature of tessalating tiles, and the Arab pioneering of abstract design. The complex designs of the Sierpinski carpet and the Menger sponge. And then the complex and sophisticated world of origami, which has its traditionalists, its pioneers and surprising applications to various fields of advanced science, introducing us to the American guru of modern origami, Robert Lang, and the Japanese rebel, Kazuo Haga, father of Haga’s Theorem.

Chapter Three – Something About Nothing

A bombardment of information about the counting systems of ancient Hindus, Buddhists, about number symbols in Sanskrit, Hebrew, Greek and Latin. How the concept of zero was slowly evolved in India and moved to the Muslim world with the result that the symbols we use nowadays are known as the Arabic numerals.

A digression into ‘a set of arithmetical tricks known as Vedic Mathematics ‘ devised by a young Indian swami at the start of the twentieth century, Bharati Krishna Tirthaji, based on a series of 16 aphorisms which he found in the ancient holy texts known as the Vedas.

Shankaracharya is a commonly used title of heads of monasteries called mathas in the Advaita Vedanta tradition. Tirthaji was the Shankaracharya of the monastery at Puri. Bellos goes to visit the current Shankaracharya who explains the closeness, in fact the identity, of mathematics and Hindu spirituality.

Chapter Four – Life of Pi

An entire chapter about pi which turns out not only to be a fundamental aspect of calculating radiuses and diameters and volumes of circles and cubes, but also to have a long history of mathematicians vying with each other to work out its value to as many decimal places as possible (we currently know the value of pi to 2.7 trillion decimal places) and the surprising history of people who have set records reciting the value if pi.

Thus, in 2006, retired Japanese engineer Akira Haraguchi set a world record for reciting the value of pi to the first 100,000 decimal places from memory! It took 16 hours with five minute beaks every two hours to eat rice balls and drink some water.

There are several types or classes of numbers:

  • natural numbers – 1, 2, 3, 4, 5, 6, 7…
  • integers – all the natural numbers, but including the negative ones as well – …-3, -2, -1, 0, 1, 2, 3…
  • fractions
  • which are also called rational numbers
  • numbers which cannot be written as fractions are called irrational numbers
  • transcendent numbers – ‘a transcendental number is an irrational number that cannot be described by an equation with a finite number of terms’

The qualities of the heptagonal 50p coin and the related qualities of the Reuleux triangle.

Chapter Five – The x-factor

The origin of algebra (in Arab mathematicians).

Bellos makes the big historical point that for the Greeks (Pythagoras, Plato, Euclid) maths was geometric. They thought of maths as being about shapes – circles, triangles, squares and so on. These shapes had hidden properties which maths revealed, thus giving – the Pythagoreans thought – insight into the secret deeper values of the world.

It is only with the introduction of algebra in the 17th century (Bellos attributes its widespread adoption to Descartes’s Method in the 1640s) that it is possible to fly free of shapes into whole new worlds of abstract numbers and formulae.

Logarithms turn the difficult operation of multiplication into the simpler operation of addition. If X x Y = Z, then log X + log Y = log Z. They were invented by a Scottish laird John Napier, and publicised in a huge book of logarithmic tables published in 1614. Englishman Henry Briggs established logarithms to base 10 in 1628. In 1620 Englishman Edmund Gunter marked logarithms on a ruler. Later in the 1620s Englishman William Oughtred placed two logarithmic rulers next to each other to create the slide rule.

Three hundred years of dominance by the slide rule was brought to a screeching halt by the launch of the first pocket calculator in 1972.

Quadratic equations are equations with an x and an x², e.g. 3x² + 2x – 4 = 0. ‘Quadratics have become so crucial to the understanding of the world, that it is no exaggeration to say that they underpin modern science’ (p.200).

Chapter Six – Playtime

Number games. The origin of Sudoku, which is Japanese for ‘the number must appear only once’. There are some 5 billion ways for numbers to be arranged in a table of nine cells so that the sum of any row or column is the same.

There have, apparently, only been four international puzzle crazes with a mathematical slant – the tangram, the Fifteen puzzle, Rubik’s cube and Sudoku – and Bellos describes the origin and nature and solutions to all four. More than 300 million cubes have seen sold since Ernö Rubik came up with the idea in 1974. Bellos gives us the latest records set in the hyper-competitive sport of speedcubing: the current record of restoring a copletely scrambled cube to order (i.e. all the faces of one colour) is 7.08 seconds, a record held by Erik Akkersdijk, a 19-year-old Dutch student.

A visit to the annual Gathering for Gardner, honouring Martin Gardner, one of the greatest popularisers of mathematical games and puzzles who Bellos visits. The origin of the ambigram, and the computer game Tetris.

Chapter Seven – Secrets of Succession

The joy of sequences. Prime numbers.

The fundamental theorem of arithmetic – In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers.

The Goldbach conjecture – one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that, Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.

Neil Sloane’s idea of persistence – The number of steps it takes to get to a single digit by multiplying all the digits of the preceding number to obtain a second number, then multiplying all the digits of that number to get a third number, and so on until you get down to a single digit. 88 has a persistence of three.

88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8

John Horton Conway’s idea of the powertrain – For any number abcd its powertrain goes to abcd, in the case of numbers with an odd number of digits the final one has no power, abcde’s powertrain is abcde.

The Recamán sequence Subtract if you can, unless a) it would result in a negative number or b) the number is already in the sequence. The result is:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11….

Gijswijt’s sequence a self-describing sequence where each term counts the maximum number of repeated blocks of numbers in the sequence immediately preceding that term.

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, …

Perfect number A perfect number is any number that is equal to the sum of its factors. Thus 6 – its factors (the numbers which divided into it) are 1, 2 and 3. Which also add up to (are the sum of) 6. The next perfect number is 28 because its factors – 1, 2, 4, 7, 14 – add up to 28. And so on.

Amicable numbers A number is amicable if the sum of the factors of the first number equals the second number, and if the sum of the factors of the second number equals the first. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Added together these make 284. The factors of 284 are 1, 2, 4, 71 and 142. Added together they make 220!

Sociable numbers In 1918 Paul Poulet invented the term sociable numbers. ‘The members of aliquot cycles of length greater than 2 are often called sociable numbers. The smallest two such cycles have length 5 and 28’

Mersenne’s prime A prime number which can be written in the form 2n – 1 a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, … and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, …

These and every other sequence ever created by humankind are documented on The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane’s. This is an online database of integer sequences, created and maintained by Neil Sloane while a researcher at AT&T Labs.

Chapter Eight – Gold Finger

The golden section a number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Phi The number is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form:

a/b = (a+b)/a = 1.6180339887498948420 …

As with pi (the ratio of the circumference of a circle to its diameter), the digits go on and on, theoretically into infinity. Phi is usually rounded off to 1.618.

The Fibonnaci sequence Each number in the sequence is the sum of the two numbers that precede it. So the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn.

as the basis of seeds in flowerheads, arrangement of leaves round a stem, design of nautilus shell and much more.

Chapter Nine – Chance Is A Fine Thing

A chapter about probability and gambling.

Impossibility has a value 0, certainty a value 1, everything else is in between. Probabilities can be expressed as fractions e.g. 1/6 chance of rolling a 6 on a die, or as percentages, 16.6%, or as decimals, 0.16…

The probability is something not happening is 1 minus the probability of that thing happening.

Probability was defined and given mathematical form in 17th century. One contribution was the questions the Chevalier de Méré asked the mathematical prodigy Blaise Pascal. Pascal corresponded with his friend, Pierre de Fermat, and they worked out the bases of probability theory.

Expected value is what you can expect to get out of a bet. Bellos takes us on a tour of the usual suspects – rolling dice, tossing coins, and roulette (invented in France).

Payback percentage if you bet £10 at craps, you can expect – over time – to receive an average of about £9.86 back. In other words craps has a payback percentage of 98.6 percent. European roulette has a payback percentage of 97.3 percent. American roulette, 94.7 percent. On other words, gambling is a fancy way of giving your money away. A miserly slot machine has a payback percentage of 85%. The National Lottery has a payback percentage of 50%.

The law of large numbers The more you play a game of chance, the more likely the results will approach the statistical probability. Toss a coin three times, you might get three heads. Toss a coin a thousand times, the chances are you will get very close the statistical probability of 50% heads.

The law of very large numbers With a large enough sample, outrageous coincidences become likely.

The gambler’s fallacy The mistaken belief that, if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In other words, that a random process becomes less random, and more predictable, the more it is repeated.

The birthday paradox The probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. (These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.) In other words you only need a group of 23 people to have an evens chance that two of them share a birthday.

The drunkard’s walk

The difficulty of attaining true randomness and the human addiction to finding meaning in anything.

The distinction between playing strategy (best strategy to win a game) and betting strategy (best strategy to maximise your winnings), not always the same.

Chapter Ten – Situation Normal

Carl Friedrich Gauss, the bell curve, normal distribution aka Gaussian distribution. Normal or Gaurrian distribution results in a bell curve. Bellos describes the invention and refinement of the bell curve (he explains that ‘the long tail’ results from a mathematician who envisioned a thin bell curve as looking like two kangaroos facing each other with their long tails heading off in opposite directions). And why

Regression to the mean – if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. And recent devastating analyses which show how startlingly random sports achievements are, from leading baseball hitters to Simon Kuper and Stefan Szymanski’s analysis of the form of the England soccer team.

Chapter Eleven – The End of the Line

Two breakthroughs which paved the way for modern i.e. 20th century, maths: the invention of non-Euclidean geometry, specifically the concept of hyperbolic geometry. To picture this draw a triangle on a Pringle. it is recognisably a triangle but all its angles do not add up to 180°, therefore it defies, escapes, eludes all the rule of Euclidean geometry, which were designed for flat 2D surfaces.

Bellos introduces us to Daina Taimina, a maths prof at Cornell University, who invented a way of crocheting hyperbolic surfaces. The result looks curly, like curly kale or the surface of coral.

Anyway, the breakaway from flat 2-D Euclidean space led to theories about curved geometry, either convex like a sphere, or hyperbolic like the pringle. It was this notion of curved space, which paved the way for Einstein’s breakthrough ideas in the early 20th century.

The second big breakthrough was Georg Cantor’s discovery that you can have many different types of infinity. Until Cantor the mathematical tradition from the ancient Greeks to Galileo and Newton had fought shy of infinity which threatened to disrupt so many formulae.

Cantor’s breakthrough was to stop thinking about numbers, and instead think of sets. This is demonstrated through the paradoxes of Hilbert’s Hotel. You need to buckle your safety belt to understand it.

Thoughts

This is easily the best book about maths I’ve ever read. It gives you a panoramic history of the subject which starts with innumerate cavemen and takes us to the edge of Einstein’s great discoveries. But Bellos adds to it all kinds of levels and abilities.

He is engaging and candid and funny. He is fantastically authoritative, taking us gently into forests of daunting mathematical theory without placing a foot wrong. He’s a great explainer. He knows a good story when he sees one, and how to tell it engagingly. And in every chapter there is a ‘human angle’ as he describes his own personal meetings and interviews with many of the (living) key players in the world of contemporary maths, games and puzzles.

Like the Ian Stewart book but on a vastly bigger scale, Bellos makes you feel what it is like to be a mathematician, not just interested in nature’s patterns (the basis of Stewart’s book, Nature’s Numbers) but in the beauty of mathematical theories and discoveries for their own sakes. (This comes over very strongly in chapter seven with its description of some of the weirdest and wackiest number sequences dreamed up by the human mind.) I’ve often read scientists describing the beauty of mathematical theories, but Bellos’s book really helps you develop a feel for this kind of beauty.

For me, I think three broad conclusions emerged:

1. Most mathematicians are in it for the fun. Setting yourself, and solving, mathematical puzzles is obviously extremely rewarding. Maths includes the vast territory of puzzles and games, such as the Sudoku and so on he describes in chapter six. Obviously it has all sorts of real-world application in physics, engineering and so on, but Bellos’s book really brings over that a true understanding of maths begins in puzzles, games and patterns, and often remains there for a lifetime. Like everything else maths is no highly professionalised the property of tenured professors in universities; and yet even to this day – as throughout its history – contributions can be made by enthusiastic amateurs.

2. As he points out repeatedly, many insights which started out as the hobby horses of obsessives, or arcane breakthroughs on the borders of our understanding, and which have been airily dismissed by the professionals, often end up being useful, having applications no-one dreamed of. Either they help unravel aspects of the physical universe undreamed of when they were discovered, or have been useful to human artificers. Thus the development of random number sequences seemed utterly pointless in the 19th century, but now underlies much internet security.

On a profounder note, Bellos expresses the eerie, mystical sense many mathematicians have that it seems so strange, so pregnant with meaning, that so many of these arcane numbers end up explaining aspects of the world their inventors knew nothing of. Ian Stewart has an admirably pragmatic explanation for this: he speculates that nature uses everything it can find in order to build efficient life forms. Or, to be less teleological, over the past 3 and a half billion years, every combination of useful patterns has been tried out. Given this length of time, and the incalculable variety of life forms which have evolved on this planet, it would be strange if every number system conceivable by one of those life forms – humankind – had not been tried out at one time or another.

3. My third conclusion is that, despite John Allen Paulos’s and Bellos’s insistence, I do not live in a world ever-more bombarded by maths. I don’t gamble on anything, and I don’t follow sports – the two biggest popular areas where maths is important – and the third is the twin areas of surveys and opinion polls (55% of Americans believe in alien abductions etc etc) and the daily blizzard of reports (for example, I see in today’s paper that the ‘Number of primary school children at referral units soars’).

I register their existence but they don’t impact on me for the simple reason that I don’t believe any of them. In 1992 every opinion poll said John Major would lose the general election, but he won with a thumping majority. Since then I haven’t believed any poll about anything. For example almost all the opinion polls predicted a win for Remain in the Brexit vote. Why does any sane person believe opinion polls?

And ‘new and shocking’ reports come out at the rate of a dozen a day and, on closer examination, lots of them turn out to be recycled information, or much much more mundane releases of data sets from which journalists are paid to draw the most shocking and extreme conclusions. Some may be of fleeting interest but once you really grasp that the people reporting them to you are paid to exaggerate and horrify, you soon learn to ignore them.

If you reject or ignore these areas – sport, gambling and the news (made up of rehashed opinion polls, surveys and reports) – then unless you’re in a profession which actively requires the sophisticated manipulation of figures, I’d speculate that most of the rest of us barely come into contact with numbers from one day to the next.

I think that’s the answer to Paulos and Bellos when they are in their ‘why aren’t more people mathematically numerate?’ mode. It’s because maths is difficult, and counter-intuitive, and hard to understand and follow, it is a lot of work, it does make your head ache. Even trying to solve a simple binomial equation hurt my brain.

But I think the biggest reason that ‘we’ are so innumerate is simply that – beautiful, elegant, satisfying and thought-provoking though maths may be to the professionals – maths is more or less irrelevant to most of our day to day lives, most of the time.

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