Do You Think You're Clever? Page 7

  A mathematician might come at the description from a different angle. This would be tricky because the shape is variable, and it’s a complex shape. It’s only very roughly approximate to a sphere. You could say it’s an oblate spheroid, since it’s slightly flattened at the ends. But that would not allow for the flattening around the sides on many apples, too. And of course, this completely ignores the toroidal indenting at top and bottom. It might be simpler to sum up its shape as apploid. We might even be able to come up with an equation to describe an apploid, by assuming an imaginary, perfectly symmetrical apple. Of course, real apples are never perfectly symmetrical, but our mathematical apploid could be a good model of the average apple.

  A botanist might have yet another description: An apple is a pome, a special kind of fruit that grows on Rosaceae trees of the Maloid family, which also includes pears, quinces, medlars and rowans. Pomes are distinctive because they are ‘accessory’ fruits, in which the flesh is not made from the carpels (the fertilised ovary) but grows around them. In an apple, the carpels are the five-pointed star of the core alone, each with its seeds or ‘pips’. The shrivelled remains of sepals, style and stamen often form a little complementary five-pointed star at the base of the apple. The apples we eat are cultivars, varieties developed from Malus domestica, the cultivated tree taken from the wild Malus sieversii, which grows in the mountains of Central Asia.

  An organic chemist might describe an apple as an arrangement of cellulose in cells containing water, sugars such as fructose and glucose, mild acids such as malic acid, Vitamin C, minerals and amino acids.

  A mythologist could describe an apple in many ways – as the forbidden fruit, the fruit of the tree of knowledge, for instance. It was said to be an apple that the Biblical Eve coaxed Adam to share with her, so that they both learned they were naked and were expelled from the Garden of Eden. Golden apples were the fruit of the Greek Tree of Life in the Garden of the Hesperides, which Hercules had to pluck as one of his Twelve Labours. They were the gifts of Aphrodite, the goddess of love, too. Indeed, pretty much every culture has its own special apple stories.

  A greengrocer might describe an apple as ‘a lovely eater, really sweet and crisp – delicious’ or ‘a great cooker, big and juicy’. A cook might add that it’s very versatile fruit, which can be used in a huge variety of dishes, including apple pie and apple crumble – not to mention as an accompaniment to pork. A wine merchant could say it’s the basic ingredient of cider or calvados. To a harassed parent, an apple might be a healthy way to keep the kids quiet in between meals. For disappointed medieval audiences, a (rotten) apple might be a critical theatre review. William Tell’s son might describe an apple as the difference between life and death, or ‘dad’s psychopathic moment’. To Isaac Newton, it was a matter of some gravity. And for Salvador Dali it could be a nun’s posterior or a skull’s eye socket …

  And finally, listen to the very bad but aptly named romantic poet, Bramley Laxton:

  ‘On the Apple’

  (Written at Egremont Russet)

  Oh apple, glorious autumn’s bounty,

  Richest fruit of shortening days,

  Come when summer’s brightness dwindles

  Softly into moistening haze.

  In lovely luscious clusters drooping

  Your gold and russet globes aglow

  Ripe and ready now for plucking –

  Can I reach one? Can I? Oh!

  Yes, I have you now and swiftly

  Bite on flesh so crisp and sweet.

  Here’s a moment’s taste of heaven:

  Before the winter, autumn’s treat.

  Oh orchard pome, gold and delicious,

  Let me ask you one thing more:

  Please apple fair, you’re so appealing,

  Come back next year for your encore!

  The stage: a platform for opinions or just entertainment – what are your thoughts?

  (Education Studies, Cambridge)

  The stage today can be pretty much what the performer wants it to be, yet it’s rarely either opinion or entertainment alone. There is quite a difference, for instance, between even the most undiluted piece of ‘opinion’ theatre and someone giving a political speech in parliament or haranguing passers-by at Speaker’s Corner, even though all can be highly theatrical.

  A speech seeks to communicate the speaker’s ideas directly to listeners. A stage performance seeks to engage the audience’s imagination, and help them understand the ideas by living through them in their imagination. This is why, generally, stage performances use characters to tell a story or create a picture, rather than talk to the audience directly. When they do appear to address the audience directly, it’s in character or within the imaginary world created for the performance.

  It’s this creation of an imaginary world that marks out the stage from the direct address of a public speaker. The stage is the arena for the creation of this world, a defined area in which the vision is created – whether it’s an actual stage, or simply a subtle imaginary barrier set up by the performers between them and their audience as they wander among them in ‘promenade’ performances. The stage in essence, then, is a playground for the imagination. Stand-up comedians, singers and musicians all perform upon stages, and all to a lesser or greater extent engage the imagination. What makes the stage in the sense of theatre different is that a story is enacted and a picture embodied.

  All stage performances must entertain in the broadest sense. ‘Entertain’ is a word that came originally from the French entretenir, meaning ‘hold together’ – and that’s what theatre seeks to achieve: binding performers and audience together in a communal act of imagination. But of course, by ‘just’ entertainment, people generally mean something rather less – something that merely makes the audience laugh or smile but rarely makes them think. By contrast, then, ‘opinion’ theatre would make them think, but maybe not laugh or smile. The juxtaposition, though, is false. Even the crudest jokers are expressing an opinion – and their audience have to think, however shallowly and briefly, in responding. And if the audience doesn’t fall asleep, they are entertained in the sense of engagement by even the most undiluted opinion piece.

  What really matters is the quality of thought and the richness and worth of the ideas involved, and this may be where modern theatre often loses its way, setting up false opposites between the brash, flashy world of ‘sheer entertainment’ in West End musicals and the intensely serious world of ‘challenging’, ‘relevant’ drama. The irony, then, is that often both end up simply poor – being neither great entertainment nor particularly interesting opinion. (Opinion without knowledge or wisdom is facile and attention-seeking rather than instructive and engaging – and ultimately no more profound or valuable than ‘sheer entertainment’, besides being much less enjoyable.)

  Back in the earliest days of theatre in Ancient Greece, Aristotle in his Poetics highlighted the difference between tragedy and comedy, and until recently these were seen as the two kinds of theatre, with countless theatre companies carrying as their trademark the opposite weeping mask of tragedy and the smiling mask of comedy. The essence of comedy was a happy ending, and the audience was intended to enjoy the triumph of a lowly, often mischievous character or outcast. The essence of tragedy was the downfall of a good or noble person through some fatal error, and the audience was intended to feel pity and fear that led them to what Aristotle called ‘catharsis’ – a purging of these negative emotions. Yet most of the best tragedies are leavened with laughter, because humour helps engage an audience; and many of the best comedies have a darker or sadder side that likewise helps engage the audience’s sympathies.

  Aristotle traced the origins of both tragedy and comedy to religious rituals, and if the overtly religious element has long since vanished, there remains an element of ritual. Without really thinking about it, as an audience we willingly fall hushed at the beginning of the performance and sit rapt in silence in the dark – a condition that has distant echo
es of the ritual trances of primitive religions in which people voyaged in their minds into visions and other worlds.

  The Ancient Greeks called dramatists didaskalos, which is typically translated as teachers, but a better word might be guides. Dramatists are guides through these visions, and they may be opinionated or entertaining, reflective or passionate, thoughtful or impetuous – in fact, anything they may wish to be. Above all, though, they must, like all the best guides, know their way …

  I am an oil baron in the desert and I need to deliver oil to four different towns which happen to lie in a straight line. I must visit each town in turn, returning to my oil tank between each visit. Where should I position my tank to drive the shortest possible distance? Roads are no problem, since I have a sheikh friend who will build me as many roads as I like for free.

  (Mathematics, Oxford)

  This appears be intuitively simple. A first guess might be that the tank should be located somewhere on the line between the four towns, at the mid-point between one of the two central towns. Then, if the towns are 1 km apart, this would give two 1-km round trips and two 3-km round trips, a total of 8 km. However, intuition is not always right, and proving that answers like these are correct has led mathematicians into some of the most complex and important of all mathematical problems. Known as optimisation, these are about mathematically finding the best solution to a problem, whether it’s the shortest route or the optimum structure.

  The shortest distance problem has always fascinated mathematicians, not only because it’s an interesting geometrical conundrum, but also because it has clear practical applications. The point we are looking for to locate our oil baron’s tank is known as the geometric median, which is the point which minimises the distance to a set of sample points.

  Back in the seventeenth century, the great French mathematician Pierre de Fermat posed a similar problem to Evangelista Torricelli, the Italian inventor of the barometer. Fermat’s problem was to find the shortest distance from a single point to three sample points. Torricelli ingeniously solved the problem, and the point is now known as the Fermat or the Torricelli point. (German economist and pioneer of globalisation theory Alfred Weber also gets in on the act because of his 1909 thesis on the location of industries, so this question is sometimes called the Fermat-Weber problem.)

  Fermat’s work gave a solution for three points; the problem of finding the geometric median for four points was actually solved by the brilliant Italian priest and mathematician Giovanni Fagnano dei Toschi around 1750. Fagnano realised that when the fixed points form a convex quadrangle, the geometric median is the intersection point of both diagonals; otherwise it’s the fixed point in the triangle formed by the three other fixed points.

  Mathematicians now know that there is only one geometric median when the sample points are not in a line. In the special case, when the points are in a line or ‘colinear’ as in the oil baron’s problems, the geometric median is the median. The median is the number that divides the population in half. In the case of the oil baron, it is indeed as we guessed – the midpoint between the two middle towns.

  Interestingly, the idea of optimising routes is not simply a geometrical problem. It leads into network theory, a branch of mathematics that has attracted extraordinary interest in recent years. The more people study it, the more relevant it seems as an organising principle to explain how the world really works. Networks crop up everywhere. People network socially. The internet is a network. Transport links form networks. Ecosystems form networks. Computers depend on them. What is really exciting people is how lessons learned in one discipline, such as biology, are feeding into others, such as economics.

  The recent banking crisis, for instance, has provoked a number of economic theorists such as Domenico Delli Gatti and Joseph Stieglitz to try to tease out the problems using network theory, showing how a few banks in a network emerge as connecting hubs with more than their share of links. The problem with this is that when such hubs collapse, they can take down the whole network with them. So the network economists are looking to ecological food webs for remedies, because these have evolved ways of coping with crises better. Species with many links, for instance, are often connected to species with few. An insect may pollinate a wide variety of plants, but each plant may be pollinated by only one or a few insects. That way a catastrophe for even the multi-connected insect affects only a limited part of the network.

  As director of network research at the University of Notre Dame, Albert-László Barabási writes: ‘The diversity of networks in business and the economy is mind-boggling. There are policy networks, ownership networks, collaboration networks, organizational networks, network marketing – you name it. It would be impossible to integrate these diverse interactions into a single all-encompassing web. Yet no matter what organizational level we look at, the same robust and universal laws that govern nature’s webs seem to greet us.’

  Think of a painting of a tree. Is the tree real?

  (Modern and Medieval Languages, Cambridge)

  The simple and immediate answer to this is that the image is real, but the tree clearly is not, even though it may be a real picture of a real tree. Yet philosophers have always had problems with the notion of reality and our knowledge of it. In commonsense, everyday terms most of us have no problem telling what is real and what is not. Yet when you start to think about trying to define just what reality is, it becomes much more elusive.

  Is the test of reality, for instance, a commonsense test – something we see with our own eyes? That seems valid at first, especially since the evidence of our eyes is often backed up with evidence from other senses. But can you be sure it’s real? Could it disappear as soon as you stop looking? Could it all be a dream? Or a hologram? How can you tell that you are not just a brain in a vat fed a virtual reality by some mad scientist? And might even memory be an illusion, too – and the world burst into existence a few minutes ago, with all our memories intact? Descartes asserted that the only thing you can be sure of is that you are thinking, an idea summed up in his famous phrase, ‘Cogito ergo sum’ – ‘I think therefore I am’.

  One of the problems with visual evidence is that very often our eyes see things differently at different times or are tricked altogether. The sky looks black at night but blue by day. A dress that appears white in sunlight may appear yellow by candlelight. Railway lines seem to converge in the distance. Things far away look much smaller. On hot days, illusory puddles shimmer on the road. The Greek philosopher Plato talked about how what we think of as reality may actually just be like shadows from a fire playing on the cave wall where we have been bound captive all our lives – when reality is the real objects revealed in bright sunlight. Plato proposed that beyond the shifting, imperfect world we perceive is another realm of perfect, unchanging Forms, an inner reality as brilliant as the sunlit world outside the cave that gives the shifting shadows their substance. Although philosophers have largely abandoned this Platonic vision, it has remained a source of fascination for fiction writers ever since.

  The English philosopher John Locke got round the problem of illusions a different way. Human understanding, Locke suggested, is like ‘a closet wholly shut from light, with only some little openings left, to let in external visible resemblances, or ideas of things without’. We don’t actually see real things at all, even though they may be there, but instead create or ‘represent’ them in our minds from purely visual sensations. In other words, seeing a tree is like seeing a picture of a tree in our minds and seeing someone sit down beneath the tree is like seeing a film of it. If so, our painting of a tree may actually be said to be no less real than a ‘real’ tree, because it’s simply a representation of the image of the tree the artist sees in his or her mind.

  However, one of the problems of this ‘representative realism’ is that you then have to ask who’s actually doing the watching. Descartes saw the mind effectively as a stage (later called the Cartesian theatre) on which ideas and percepti
ons are viewed by an inner observer or ‘homunculus’. But just who or what is the homunculus, and who’s watching inside the head of homunculus? It begins to sound like some absurd Russian doll.

  That’s why some philosophers, such as Berkeley, insisted that it was pointless to ask whether something had a reality outside our perception. There is no way we can be sure the world has any other reality. Berkeley insisted that ‘esse est percipi’ – to exist is to be perceived. In other words, reality exists only as long as it is perceived. If I blink, reality disappears momentarily. But this, though quite logical, is solipsistic, and so counter-intuitive that we’d probably regard someone who genuinely believed it as slightly unhinged.

  Maybe the best guess is that we do live in a world of real things, and that these real things are what cause our sense experiences, an idea known as causal realism. It sounds self-evident, but the crucial thing is that the real object causes the experience, so there is a direct link between the real thing and our sensory experience. It’s an assumption which cannot be proved by logic – we cannot know we are not just brains in a vat – but it’s the most fruitful way of treating reality, and there is no evidence to contradict it.

  We are reasonably confident that when we look at a tree we are looking at a real thing, and this is borne out by others also seeing the tree – although they might see it differently. Similarly, we are reasonably confident in our sensory experience of a painting to say that we know the painting itself is real. Equally, we can be confident enough in our senses to say that the tree in the painting is not real, but simply a representation of a tree created by the artist. The tree may be a representation of a real tree, or it may be the representation of an imaginary tree, created in the artist’s mind, maybe as a result of seeing real trees or being told about them. The artist chooses how to represent the tree – whether it’s ‘realistic’ like a pre-Raphaelite painting, which the viewer instantly connects visually to a real tree, or entirely abstract like a tree by Picasso.