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  How many grains of sand are there in the world?

  (Physics, Oxford)

  This is a venerable question that dates back to the time of Ancient Greece. There was an old Greek proverb which talked about the amount of sand in the world as being so large that the grains are beyond number. It was, if you like, the ancient world’s description of infinity. But as early as the third century BC, the great Greek scholar Archimedes sought to challenge the idea that the number of sand grains was infinite and inherently unknowable. In his famous book called The Sand Reckoner, Archimedes showed that though the number of grains of sand in the universe is, of course, uncountable, it’s not incalculable.

  Our modern (Indo-Arabic) numbering system provides a neat way of handling and writing very, very large numbers simply by adding a place for every step in magnitude of ten. We can even put them in shorthand by expressing them in powers of ten. But this was far beyond the numbering system of Ancient Greece, in which every number was simply a separate letter of the alphabet, giving just 27 possible numbers. Undaunted, Archimedes devised his own numbering system in The Sand Reckoner by taking the largest Greek number, the myriad (10,000) and creating multiples of myriads until he got to a myriad myriads (100 million), and then a myriad myriads times a myriad myriads (10,000 trillion). He then went on to estimate the number of grains of sand it would take to fill a universe which he assumed to be spherical, and with a radius of the distance then known between the earth and the sun. The answer he came up with was 1063.

  More recently, geologists Horn and Adam made a serious attempt to work out the number of grains of sand in the world, by making an estimate of the total volume of sand in all the world’s sedimentary rocks, in all the world’s deserts and on all the world’s beaches. They calculated that there are 147 million cubic kilometres of sand, and by making assumptions about average grain size and packing, they hit on a figure of 875 trillion trillion for the number of sand grains in the world. A little while ago mathematicians at Hawaii University estimated the length, width and depth of all the world’s beaches added together, and announced that there are 700,500 million billion grains of sand on all the world’s beaches!

  Although it’s impossible to know the number of sand grains in the world accurately, modern research techniques, from satellite surveys to seismic studies, allow us to refine such calculations to a degree where we at least have confidence that they are in the right order of magnitude. But as our estimates become more accurate so there is less and less point in making them. As Archimedes strove to demonstrate, if the number of grains is knowable it can’t be used as anything but a purely poetic image of the infinite. Moreover, as our knowledge of the universe expands both on the very largest scales and the very smallest scales, we have been introduced to all kinds of other unimaginably large numbers, from the number of stars in the universe (10 trillion trillion) to the number of atoms in your body (7,000 trillion trillion). The concept of infinity, however, remains as elusive as ever, and continues to intrigue mathematicians as they struggle to resolve the paradoxes of the finite and the not finite.

  Was Romeo impulsive?

  (Modern and Medieval Languages, Cambridge)

  Of course Romeo was impulsive. We see it in the speed with which he switches to Juliet from his old love Rosaline, so:

  That fair for which love groaned for and would die,

  With tender Juliet matched is now not fair.

  We see it as he risks sparking a deadly feud between the Capulets and Montagues for the instant gratification of his love for Juliet. We see it in the way he kills Tybalt in revenge, regardless of the consequences, actually calling for help to abandon sense, shouting:

  Away to heaven respective lenity,

  And fire-eyed fury be my conduct now!

  And we see it in the extremity and suddenness of his love for Juliet, which even she is startled by, saying:

  It is too rash, too unadvised, too sudden;

  Too like the lightning, which doth cease to be

  Ere one can say ‘It lightens’.

  And, of course, this is exactly as Shakespeare intended; both Romeo and Juliet are the epitome of rash, impulsive youth, the personification of thoughtless, unconsidered love – which is both immensely appealing and ultimately disastrous.

  But is it love that makes Romeo impulsive or is it because he is impulsive that he falls in love so readily? Love is so often described as a kind of madness, and falling in love can drive apparently sane men and women to the most extreme acts of folly. To many of the characters in the plays of Shakespeare and his near contemporaries, such as Lope de Vega and Calderón de la Barca, love and madness are flipsides of the same coin, a sweet madness. Even Freud admitted, ‘One is very crazy when in love’. For romantics, the madness of love was liberating because it freed one from the bonds of normal behaviour.

  Yet it’s quite clear that Romeo is born to be a lover. He is looking for love, for the rush of adrenaline it brings. He is in some ways addicted to love, and addicted to being impulsive for the same reason. It is of course in the character of so many tragic heroes from Oedipus to Othello to act rashly, not stopping to think, driven by their hearts and not their heads. If only they stopped to think, maybe, things wouldn’t have turned out so tragically.

  But that is half their appeal to an audience, for they thus appear brave and passionate, even rather foolhardy, and watching them is excitingly dangerous. As we watch in fascination, there is always a tug between saying, ‘No, don’t do it! That’s madness!’ and ‘Yes, throw caution to the winds and go for it!’ that keeps us on the edges of our seats, deeply involved in all their ups and downs. Their impulsiveness draws us in, in a way that more controlled, measured behaviour never would. They are risk-takers, intoxicating for an audience who take risks by proxy. But of course the risks they take make them prone to fall. That is what gives the audience an exhilarating emotional journey, but their failure reassures us that our more balanced approach to life is ultimately right, saving us from their tragic fate.

  The Greek philosophers had a word for this kind of behaviour, akrasia, which gives us the word ‘crazy’, and understanding it has preoccupied both philosophers and psychologists ever since. For the Greeks, it was behaviour which goes against sense – not madness, but a lack of self-control, of knowingly doing something you think is not good for you. Why on earth would you do that? Socrates believed that the only explanation was ignorance – if people really know what’s good for them they will behave accordingly. That’s a view echoed today by many modern economists such as the late Milton Friedman. But that’s not true of Romeo, who knowingly shuns the safe course. Thomas Aquinas took a more moral attitude, asserting that impulsive, excessive behaviour was a lack of the restraining modesty of virtue, a lack of shame, or as the Greeks described it, aidos – something Nietzsche later castigated as a fatal weakness and absence of vision.

  During the eighteenth century, philosophers began to look on impulsive behaviour in a more positive light, seeing it as coming from our untamed, natural side, and counterbalancing our more schooled, rational selves – the noble savage with simple instincts compared to the calculating elaborations of the civilised man. Freud took it much further, suggesting that impulsive behaviour came from the ‘id’, including basic subconscious drives such as sex that often conflict with the rational, conscious self of the ‘superego’, with the ‘ego’ in the middle trying to balance their demands.

  Although Freud’s ideas are no longer in vogue, many psychologists still think of a dual process in which human behaviour is torn between two different ways of thinking – a logical, language-based mode and a more irrational mode governed by associations. In this way of thinking, Romeo’s behaviour is tipped towards the impulsive by all the powerful associations that have formed in his young mind between excitement and danger, pleasure and love. Even his language is engaged not by logic, but by the associations of love with everything of beauty, so that Shakespeare puts on his tongue some o
f the most wonderful, heady words of romance ever said.

  But soft! What light through yonder window breaks?

  It is the East and Juliet is the sun!

  Arise fair sun and kill the envious moon.

  No wonder Juliet swoons.

  How would you describe a human to a person from Mars?

  (Medicine, Cambridge)

  Since the general consensus is that no one has encountered a person from Mars outside the realms of fiction – though of course followers of UFOs would demur – it’s probably reasonable to assume that the question is intended as a hypothetical one. Of course, if I were talking to a ‘real’ Martian, I’d be trying to find points of reference in my description that the Martian would understand. If the Martian were rather human-like, this would be pretty simple. If the Martian were basically a living rock and had never encountered any other life-form than rocks, this might be a little harder. And, of course, without a common language, my task would be even harder! In fact, until the nature of this hypothetical Martian is known, the range of possibilities and difficulties is infinite and the question pointless.

  I could go off on a flight of fancy, imagining my own kind of Martian and my own way of describing humans to this ill-informed alien, but my suspicion is that the questioner is trying to elucidate a description of humans rather than imagine our Martian friend. In other words, the Martian is irrelevant to the answer, and the idea is to describe humans to a completely unknown entity that we have no real contact with except via some interplanetary internet that translates perfectly but is, for some reason, unable to send pictures or anything other than a verbal description – a kind of weird obverse of the famous Turing test, in which a human questioner puts questions to a computer and tries to decide if the answers are coming from a computer or a living human.

  Even so, it’s hard to know where to begin without establishing some points of reference with the Martian. I could perhaps start with the simplest physical facts by describing the chemical basis of life on earth. Life on earth, I could say, depends on the remarkable ability of some large molecules of carbon to self-replicate and organise themselves into colonies. Life forms on earth are basically communities of self-replicating carbon molecules, each typically housed in its own sustaining package or cell. Over billions of years, these communities have diversified tremendously, as groups of cells develop their own specialised role. Now the variety of forms these communities take is enormous, although there are two major forms – plants, which are unable to move by themselves, but can obtain their sustenance from sunlight, and animals which can move but must actively seek food. Most animals have a head at one end containing a brain that controls their movement, a body containing the organs that they need to sustain themselves, and, usually, appendages to enable them to move.

  Humans are complex animals that developed quite recently in earth’s history – no more than 150,000 years ago. They have a head, a body and four limbs like many other animals, but unlike most others, they stand upright on just two of them, which means they can use the other two for grasping on to things, and their eyes, their visual apparatus, can look directly at what they are grasping. This gives them an enormous physical advantage over other animals in controlling their environment. But they have become unique in other ways. They dominate planet earth and manipulate their environment in a way that no other animal does.

  They have developed complex brains, the ability to communicate sophisticated messages through language, and a level of self-awareness far beyond other animals, and this has enabled them in many ways to transcend the purely physical. They need to satisfy basic material needs such as food and warmth to keep themselves alive. But for most humans, life is so much more than just staying alive.

  We humans can think about abstract ideas. We have imagination. By working together, and working separately, we can create a fantastic array of things to make life easier, from houses to live in to machines to move us around or to make tasks easier. We can create wonderfully elaborate things such as the varied sounds of music that have no other purpose but to entertain, stimulate or fill us with pleasure at their beauty. We have boundless curiosity which has allowed us to understand something of the nature of the world around us, building amazing machines to see far beyond our physical limits into the minute world of subatomic particles and the immensity of the universe, and formulating equally amazing theories to explain them. We have the ability to be both unutterably cruel to each other, and inspiringly caring. We have much in common, but each of us is slightly different, not only physically but also in our thoughts and needs, our hopes and fears. We are at the same time very simple creatures who have basic needs for material comfort, and complex beings who constantly worry about our place in the world. We sleep, we eat, we love, we talk, we work, we think, we analyse, we rant, we dance, we mope, we laugh, we cry.

  Our lives are short and ultimately quite similar, but each lives it differently, each has his or her own inner life, dreams and fears, hopes and disappointments. In fact, we treasure our frailties, saying they are ‘only human’ because, when they don’t cause us heartache, they impel us to learn, to explore, to adapt and to find ways of communicating … even with imaginary Martians.

  What do you like most about the brain?

  (Medicine, Cambridge)

  What a question! What I like is that someone can pose this question and that I can try, however ineffectively, to answer it. I like not what the brain is but what it can do. I like the fact that it enables me to think, to analyse and respond to experience, to formulate ideas and to know what pleasure is, and pain. I like the fact that it gives me my life.

  I can’t say I especially like it physically. In fact it’s a little bit disgusting to look at. But I like the fact that with it I can contemplate that this big dollop of paté contains a microscopic network of nerve cells that make it the most fantastically complex thing in the universe. Estimates vary wildly, since of course no one has counted them all, but there are some 100 billion separate cells in the human brain – which is, by strange coincidence, much the same number as there are galaxies in the universe. But this bald number, however awesome, doesn’t begin to capture the almost miraculous complexity of the brain, with each of those 100 billion cells making 1,000 or more separate connections with other cells – and unimaginably more alternative pathways for nerve signals to crackle, fizz and buzz along as they make us jump up or sit down, laugh and cry, love and hate, sing, shout, swear, eat, drink and do everything that makes us human.

  I like the way it’s continually defying expectations and proving a more wonderful mechanism than a few scientists would have us believe. There was a time not long ago, when scientists portrayed it as a dying machine, a computer that gradually lost its effectiveness as you went through life, as old nerve cells died off and were never replaced. Using techniques such as fMRI (functional Magnetic Resonance Imaging) scans, scientists have now discovered that the brain is not simply a failing machine that steadily loses its working parts. Instead, it’s a flexible, living organism with an amazing capacity for change and development throughout your life. It’s said to be ‘plastic’ – that is, it can go on programming and reprogramming itself almost limitlessly.

  I like the fact that it has become clear that your mental powers have very little to do with the number of brain cells that you have. It’s the connections between them that matter – and there is an almost infinite possible number of connections between brain cells, at any age. Indeed, your brain goes on making more and more of these connections as you get older, not fewer. I like too that while you may lose brain cells as you get older, some, at least, can actually grow again. And losing brain cells may actually be good for mental powers, not bad. When you’re a child, your brain evolves as it learns through repeated mass extinctions of little-used brain cells that allow those that prove their use to develop properly – a Darwinian ‘survival of the fittest’ process called apoptosis. Losing brain cells as you get older could actua
lly be the brain’s way of improving efficiency – cutting away the dead wood to allow the good to flourish.

  But perhaps the brain’s best victory over the reductive scientists is the recent discovery that its performance is in your control to a much, much greater extent than we ever realised. It has become clear that the future of your brain is at least partly in your own hands, or rather, your own thoughts. What I like best of all about the brain, though, is the fact that I have one, just like everyone else’s but unique to me.

  Why do so few Americans believe in evolution?

  (Human Sciences, Oxford)

  It’s one of the strange anomalies of the modern world that the USA, one of the most technologically advanced of all nations, should have so many uncompromising Christians who challenge some of science’s best-established theories.

  On a simple level, those who believe in the literal truth of the Bible, which describes how God made all the world’s creatures at a stroke, must inevitably disagree with the theory of evolution, which suggests that all the world’s living things evolved naturally over billions of years. But there is more to this question than that. Although there have been tussles, there are many Christians all around the world who are quite happy to accept the theory of evolution, without feeling their basic beliefs compromised.

  The USA, however, is different. Recent surveys suggest that more than half of all Americans entirely reject the notion of evolution. In fact, in a survey of the US plus 32 European countries, Turkey and Japan, it turned out that fewer people in America accepted evolution than in any other country but Turkey. And when it comes to acceptance of Darwin’s theory of natural selection, in which evolution progresses in an entirely automatic, mechanical (entirely God-free) way, the numbers shrink further – with only 14 per cent of Americans agreeing with it. Yet this idea has been accepted by scientists for not far short of a century.