[2] The origins of the idea are much older, though. In Crito, Socrates argued powerfully that he must stay in prison and accept the death penalty. The Laws of Athens, he says, are what made his entire way of life possible, from the marriage of his parents to his education and upbringing. So he is obliged to live, and die, by the laws he has benefited from. Plato argued that people want government to ensure justice and avoid being treated unjustly by others.
[3] Forty years ago, American philosopher John Rawls proposed a variation of the social contract in line with game theory, in which rational, entirely self-serving people in an ‘original position’ (which was his take on the state of nature) agreed on government in their own best interests. Irish philosopher Philip Pettit argues that because consent can always be ‘manufactured’ by governments, agreement to the contract should be signified instead by lack of rebellion against it.
#25 Calculus
In his book Taming the Infinite, Professor Ian Stewart declares that calculus was ‘the most significant advance in the history of mathematics’. It is a bold claim, but there is a strong case to be made in its favour. Calculus was the mathematics that drove the scientific revolution which began in the seventeenth century and continues to this day. It turned out to be the mathematics of the natural world, and allowed everything from the trajectory of planets to the growth of ant populations to be analysed mathematically.
Previously, mathematics had only been really good at coping with things that don’t move, or that stay moving at exactly the same rate, which is why, before calculus, mathematics had pretty much only been used for abstract logic, financial and architectural calculations and basic mechanical calculations. Calculus opened the way for maths to explore things that are changing and moving at a varying pace. Since that applies to just about everything in nature, this change was profound and far-reaching. It gave scientists a uniquely powerful tool for exploring and explaining the natural world.
The door to calculus was pushed open in the early seventeenth century when Fermat and Descartes realised that algebra and geometry could be tied together simply by using co-ordinates – two or more numbers that pinpoint a location in relation to reference lines, such as the axes on a graph. Using co-ordinates, all geometry can be summed up in algebraic equations and, in theory, every equation can be conveyed geometrically as curves and lines on a graph.
Bringing algebra and geometry together in an entirely new kind of mathematics called analytic geometry allowed, for the first time, mathematics to be used to analyse how things move and change. But there were two fundamental missing elements. Analytic geometry could not study the rate at which things were changing nor analyse the amount of change that happened. It could show, for instance, how far an object travels in a particular time, but not its velocity and acceleration at a particular point. That’s like studying a bird’s flight by observing only where it takes off and lands. That’s where calculus comes in. It allows you to analyse what’s going on in between.
Imagine knocking a flowerpot from your windowsill. Newton’s law of gravity tells you that it accelerates earthwards before smashing on the pavement below with some force. But how fast will the pot be travelling after falling for a second? You can work out its average speed by measuring how far it travels in a second. But that doesn’t tell you how fast it is going – the pot’s instantaneous speed. Calculus provides the answer by using limits. If you measure the average speed between two limits – or points either side of the point it reaches after a second – you get very close to the instantaneous speed. The closer your limits are, the closer you get to the instantaneous speed.
You might think that if you were to reduce the limits to nothing you would then get the instantaneous speed exactly. But if the limits were nothing, then the pot wouldn’t have moved at all in between them. It would have moved no distance and taken no time to do it. That would give you a speed of zero.[1] Irish philosopher Bishop Berkeley famously criticised calculus because of this problem, describing it as ‘the ghosts of departed quantities’. Solving the conundrum took mathematicians two centuries, but by then calculus had more than proved its worth.
The credit for the invention of calculus was fiercely debated almost immediately. Was it the German polymath and mathematical genius Gottfried Wilhelm Leibniz?[2] Or was it the towering figure of English science, the creator of the laws of motion and the theory of gravity, Sir Isaac Newton? Probably in terms of sheer chronological priority, the winner was Leibniz. It was Leibniz, too, who gave calculus the notation it uses today, such as the terms ‘delta’ x and ‘delta’ y. But his writings were so fragmented and obscure that even one of his supporters, the Italian Bernoulli brothers Jacob and Johann, wrote of his first description of calculus that it was ‘an enigma rather than an explanation’.
Newton created a much clearer and fuller description. His ideas were first summarised in 1671 in a book entitled Method of Fluxions and Infinite Series. He called calculus ‘fluxions’ because the idea of the limits was of a quantity ‘flowing’ towards zero, the instantaneous point, but not actually getting there. Newton was notoriously cautious and cagey, so used only simple maths in his great Principia outlining the laws of motion and gravity, but there is every chance that he created calculus in order to develop these great theories – since mathematically they need calculus in order to be fully expressed.
Calculus, as developed by Leibniz and Newton, involves two principal operations: differentiation and integration. Differentiation comes from the idea of taking apart and integration of bringing together. They are opposite sides of the same coin, even though each has led to its own branch of calculus: differential calculus and integral calculus. Differentiation is about calculating rates of change, just as in the case of our precipitous flowerpot. In terms of the geometry of graphs, it’s also a way of finding tangents to curves. Integration is about the opposite; finding the quantity of something, if you know only its rate of change. In terms of the geometry of graphs, it’s about finding the area under the curve between two points.[3]
The fight over who first thought of calculus, Leibniz or Newton, created an intellectual rivalry between Britain and Europe that was great for British empirical philosophy but something of a disaster for British mathematics. Although Newton’s system of calculus was much more coherent and fully developed than Leibniz’s, its focus on geometry proved, ironically, to be a handicap. In the meantime, continental mathematicians made more rapid progress with Leibniz’s more algebraic calculus. It was only once the differences were resolved, though, and the problem of Berkeley’s ghostly zero was solved by the development of analysis in the early 1800s, that the full power of calculus was finally unleashed. From then on, calculus was at the forefront of science.
The roll call of scientific breakthroughs made possible by calculus is extraordinary. Without it, there would probably have been no Laws of Motion and no law of gravity. There would have been no theory of electromagnetism, relativity or quantum physics. No Big Bang theory or knowledge of black holes. All of these and many more key scientific theories depend on analysing rates of change.
Calculus is equally important in technology, too. The process of calculating the trajectory of space probes depends on it. So does calculating how an aircraft will fly. Or how a bomb will explode. Even our understanding of how animal populations change, or how financial instruments will fare, can be explored using calculus. In medicine, calculus is used to plan the most effective way to intervene and stop epidemics spreading, or to calculate the timing and dosage of drugs, or the angle to insert an artificial blood vessel to ensure the best flow.
Of course, calculus is a great idea not just because of the sheer range of uses it has, especially in science, or even the key role it played in bringing the key scientific theories of the last two centuries to fruition – even though this is quite astounding. What makes it so great is the beauty of such an ingenious, brilliant mathematical tool that can explain to us the laws of nature – and reveal
patterns we just never knew were there. Newton and Leibniz are rightly considered geniuses for their creation of calculus – and although maybe someone else would have thought of it if they hadn’t, it was they who had the insight to work out how to analyse something as fundamental, yet as hard to pin down, as how things change. It turned our view of the natural and physical world from one that was essentially static to one that was dynamic. By the late nineteenth century, Gilbert and Sullivan’s Modern Major General could proudly boast in The Pirates of Penzance about the heights of his achievement:
With many cheerful facts about the square of the hypotenuse.
I’m very good at integral and differential calculus.
Thank goodness someone is.
[1] The problem was initially overcome using ‘infinitesimals’ – numbers smaller than any given number, yet not zero. If an infinitesimal is added to itself, no matter how many times, it always remains less than a given number. This system made the calculations possible, but was vague and imprecise, which is why most calculus since the nineteenth century has used limits instead. For some mathematical operations, though, infinitesimals have been proven to work better.
[2] Leibniz was one of the great figures of German science and philosophy, and his interests were wide-ranging. Besides creating calculus, he devised the binary number system and made significant contributions to everything from geology to the study of logic. But his reputation suffered a terrible blow when he was satirised by Voltaire in Candide as the absurdly optimistic Dr Pangloss, who even in the face of catastrophic miseries and suffering insists that ‘all is for the best’. Only in recent years has his brilliance been fully recognised again.
[3] If you find this hard to follow, it’s not surprising. When the great mathematician Johann Bernoulli was trying to get his head around Leibniz’s ideas, he wrote: ‘But just as much as it is easy to find the differential [derivative] of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.’
#24 Arable Farming
In 2008, amid talk of a world food crisis, with global shortages of basic foods and rising prices, the UN met in Rome to work out how to feed the world. The participants agreed that it was wholly unacceptable that 862 million people worldwide were malnourished, given the resources available. They also resolved that global food production must be doubled by 2030.
There are two points here that are especially remarkable. The first is that there is enough food to feed the world’s entire population well right now – if only it were more equally distributed. The second is that experts consider it entirely realistic to talk of doubling world food production over the next two decades.
This is an incredible testament to farming. Farmers are already producing enough to feed more than 6 billion people, not to mention a vast quantity of livestock (70 per cent of the world’s maize crop is fed to animals). By 2030, if the UN’s goals are actually achieved, they will be producing enough to feed 8.5 billion.
There is no other way besides farming that even a tiny fraction of the food needed to feed so many could ever be found. Arable farming is a simple but brilliant idea. Where hunter-gatherers must be constantly on the move in search of elusive sources of wild food, arable farming provides food on tap simply by the farmer collecting the seeds or tubers of particular wild food plants such as grasses, then planting them together near at hand, so enough grow where one has easy access to them. The variety of foods may be restricted, but the supply is reassuringly predictable.
Over history, by planting selected seeds and improving techniques, farmers have boosted crop plants’ ability to yield food far above that of their wild counterparts. The result is that farmers can now typically grow 120 bushels of wheat on a single acre – which is enough to keep 120 people in bread throughout the year (a bushel of wheat gives flour for about 100 loaves). Some farmers get well over 400 bushels.
Other methods of getting food can’t come even remotely near this. The few remaining hunter-gatherer tribes around the world, such as the Yora and Awa Guaja of the Amazon, need vast areas to find food even for a small tribal group. And of course, there is now very little wild food left. So now farming is literally a matter of life and death. If farms stopped producing food, most of us would be dead within a very short time.
Of course, part of the reason there are so many people in the world today is because farming has made more food available. The beginning of arable farming 10,000 or so years ago triggered the world’s first population boom.
A second boom – the massive urbanisation of the Industrial Revolution – is linked to the eighteenth-century revolution that replaced ancient subsistence farming with commercial farms, using new techniques to produce food to sell to city folk at a profit.
A third boom, that has seen the global population double since the 1960s, has been enabled by a massive increase in global food production through new intensive farming techniques involving specialised crop strains, fertilisers and pesticides, including the so-called Green Revolution.[1]
The traditional view was that farming began in response to a shortage of food for foraging and hunting in the dry conditions after the Ice Age about 10–8,000 years ago. But most experts now agree that this doesn’t account for the variations in the times when farming first appeared around the world – 11,000 years ago in the Middle East, 8,000 years ago in China and 10,000 years ago in South America. If anything, crop farming may have developed in times of plenty.
What’s more, recent findings have pushed the first crops well back before the end of the Ice Age. Previously, the oldest known arable farms were in the Middle East, where people were growing things such as emmer and einkorn wheat (the forerunners of modern wheat strains), barley, rye, vetch and lentils up to 11,500 years ago. Then in 2005, researchers working near the Sea of Galilee in Israel discovered a site labelled Ohalo II. Here, at least 23,000 years ago, people were collecting seeds of wild grasses such as emmer wheat and barley, and grinding them into flour to make bread.
In fact, it seems that the origins of farming are very much older than once thought and emerged gradually rather than all together. People domesticated dogs as hunting companions and may have tended herds of sheep and goat tens of thousands of years ago. They may also have nurtured wild fruit and nut trees. The idea of planting selected seeds of wild plants such as grasses in order to grow crops probably developed from practices like this separately around the world.
When arable farming did begin, the effect on the human way of life was dramatic. No longer would people move around hunting and foraging. To plant, tend and harvest their crops, people had to settle permanently in one place and build stable houses and villages. The need to organise labour, planting, harvesting and storage, and the availability of surpluses for trading, soon led to the development of the first towns, the first governments, building, writing and everything we associate with the development of civilisation.
Interestingly, though, not everyone agrees that this makes farming a great idea. Recently, Cambridge academic Jay Stock described the invention of arable farming as ‘the worst mistake in human history’. The idea that primitive man was superior is an old thread, typified by Rousseau’s picture of the ‘noble savage’ uncorrupted by the complexities of the civilised world, and sprouting up in 1960s hippy anthropology. But the idea that farming in particular was a retrograde step was put forward by Jared Diamond in his 1997 book Guns, Germs and Steel. We may think that farming has brought us health, wealth and long life, Diamond contended, but it has in fact brought us misery. Farming not only replaced a very varied and healthy diet of wild food with a poor diet of a few staples; it also led to the development of soldiers, warfare, class divisions and all the bad aspects of civilisation, because of the way it allowed the food to be stockpiled, and freed some, but not others, from the necessity of looking for it.
Moreover, a sedentary lifesty
le and the proximity of animals and shared water resources led to the spread of infectious diseases. Jay Stock made his damning comment after he studied the bones of hunter-gatherers living in Egypt just before farming developed and found that these people were significantly taller and healthier than the first farmers were. Yet there is plenty to suggest that a hunter-gatherer lifestyle, too, was far from idyllic. And, although the first farmers were unhealthy and stunted, things have improved. A huge number of people today are taller and live much longer, healthier lives than their hunter-gatherer ancestors.
And, of course, it is not simply about health. The food surpluses and division of wealth may indeed have brought us warfare, taxes, robbery, murder and a whole host of other problems. But they have also brought us wonderful civilisations, beautiful buildings, ingenious craftsmanship, amazing technology and pretty much every significant scientific and artistic achievement of humanity. In fact, it has, you could say, delivered us the whole of human history, since history only began once farming allowed people to settle long enough in one place to record it. Of course, there is indescribably bad in human history along with the incontestably glorious, but one cannot wish it away. It is in the way we live and practise farming rather than in the idea of farming itself that any faults lie.
Farming continues to give us the indispensable gift of food. Thanks to the efforts of farmers, 5 billion people are able to eat well every day. Those who lack food lack it because they are unable to farm or gain access to the fruits of farming.
Yet delivering all this food is taking an enormous toll on the environment. Over more than one third of the world’s land surface, the natural ecosystem has been replaced by the artificial landscape of farms, with their meadows and enclosures, their ploughed lands and crops. And farming acreage is still growing. Every year an area of the Amazon rainforest the size of Scotland is chopped or burned down to create new farmland. Modern intensive farming can damage the natural environment in other ways too, from poisoning the land and the oceans with pesticides and fertilisers to adding carbon to the atmosphere with energy intensive practices.
The World's Greatest Idea Page 16
