The Science of Discworld III - Darwin's Watch tsod-3

by Terry Pratchett

  "s quite simple,' said Xeno. `Look, let's say this olive stone is an arrow and this, and this -' he cast around aimlessly -'and this stunned seagull is the tortoise, right? Now, when you fire the arrow it goes from here to the seag- the tortoise, am I right?'

  `I suppose so, but-'

  `But, by this time, the seagu- the tortoise has moved on a bit, hasn't he? Am I right?'

  'I suppose so,' said Teppic, helplessly. Xeno gave him a look of triumph.

  `So the arrow has to go a bit further, doesn't it, to where the tortoise is now. Meanwhile the tortoise has flow- moved on, not much, I'll grant you, but it doesn't have to be much. Am I right? So the arrow has a bit further to go, but the point is that by the time it gets to where the tortoise is now the tortoise isn't there. So if the tortoise keeps moving, the arrow will never hit it. It'll keep getting closer and closer, but it'll never hit it. QED.'

  Zeno has a similar set-up, though he garbles it into two paradoxes. The first, called the Dichotomy, states that motion is impossible, because before you can get anywhere, you have to get halfway, and before you can get there, you have to get halfway to that, and so on for ever ... so you have to do infinitely many things to get started, which is silly. The second, Achilles and the Tortoise, is pretty much the paradox enunciated by Xeno, but with the hare replaced by the Greek hero Achilles. Achilles runs faster than the tortoise - face it, anyone can run faster than a tortoise - but he starts a bit behind, and can never catch up because whenever he reaches the place where the tortoise was, it's moved on a bit. Like the ambiguous puzuma, by the time you get to it, it's not there. The third paradox says that a moving arrow isn't moving. Time must be divided into successive instants, and at each instant the arrow occupies a definite position, so it must be at rest. If it's always at rest, it can't move. The fourth of Zeno's paradoxes, the Moving Rows (or Stadium), is more technical to describe, but it boils down to this. Suppose three bodies are level with each other, and in the smallest instant of time one moves the smallest possible distance to the right, while the other moves the smallest possible distance to the left. Then those two bodies have moved apart by twice the smallest distance, taking the smallest instant of time to do that. So when they were just the smallest distance apart, halfway to their final destinations, time must have changed by half the smallest possible instant of time. Which would be smaller, which is crazy.

  There is a serious intent to Zeno's paradoxes, and a reason why there are four of them. The Greek philosophers of Roundworld antiquity were arguing whether space and time were discrete, made up of indivisible tiny units, or continuous - infinitely divisible. Zeno's four paradoxes neatly dispose of all four combinations of continuous/discrete for space with continuous/discrete for time, neatly stuffing everyone else's theories, which is how you make your mark in philosophical circles. For instance, the Moving Rows paradox shows that having both space and time discrete is contradictory.

  Zeno's paradoxes still show up today in some areas of theoretical physics and mathematics, although Achilles and the Tortoise can be dealt with by agreeing that if space and time are both continuous, then infinitely many things can (indeed must) happen in a finite time. The Arrow paradox can be resolved by noting that in the general mathematical treatment of classical mechanics, known as Hamiltonian mechanics after the great (and drunken) Irish mathematician Sir William Rowan Hamilton, the state of a body is given by two quantities, not one. As well as position it also has momentum, a disguised version of velocity. The two are related by the body's motion, but they are conceptually distinct. All you see is position; momentum is observable only through its effect on the subsequent positions. A body in a given position with zero momentum is not moving at that instant, and so will not go anywhere, whereas one in the same position with non-zero momentum - which appears identical - is moving, even though instantaneously it stays in the same place.

  Got that?

  Anyway, we were talking about Thief of Time, and thanks to Xeno we've not yet got past page 21. The main point is that Discworld time is malleable, so the laws of narrative imperative sometimes need a little help to make sure that the narrative does what the imperative says it should.

  Tick.

  Lady Myria Lejean is an Auditor of reality, who has temporarily assumed human form. Discworld is relentlessly animistic; virtually everything is conscious on some level, including basic physics. The Auditors police the laws of nature; they would very likely fine you for exceeding the speed of light. They normally take the form of small grey robes with a cowl - and nothing inside. They are the ultimate bureaucrats. Lejean points out to Jeremy that the perfect clock must be able to measure Xeno's smallest unit of time. `It must exist, mustn't it? Consider the present. It must have a length, because one end of it is connected to the past and the other is connected to the future, and if it didn't have a length then the present. couldn't exist at all. There would be no time for it to be the present in.'

  Her views correspond rather closely to current theories of the psychology of the perception of time. Our brains perceive an `instant' as an extended, though brief, period of time. This is analogous to the way discrete rods and cones in the retina seem to perceive individual points, but actually sample a small region of space. The brain accepts coarse-grained inputs and smooths them out.

  Lejean is explaining Xeno to Jeremy because she has a hidden agenda: if Jeremy succeeds in making the perfect clock, then time will stop. This will make the Auditors' task as clerks of the universe much simpler, because humans are always moving things around, which makes it difficult to keep track of their locations in time and space.

  Tick.

  Near the Discworld Hub, in a high, green valley, lies the monastery of Oi Dong, where live the fighting monks of the order of Wen, otherwise known as History Monks. They have taken upon themselves the task of ensuring that the right history happens in the right order. The monks know what is right because they guard the History Books, which are not records of what did happen, but instructions for what should.

  A youngster named Ludd, a foundling brought up by the Thieves' Guild, where he was an exceptionally talented student, has been recruited to the ranks of the History Monks and given the name Lobsang. The monks' main technological aids are procrastinators, huge spinning machines that store and move time. With a procrastinator, you can borrow time and pay it back later. Lobsang wouldn't dream of living on borrowed time, though - but if it wasn't nailed down, he would almost certainly steal it. He can steal anything, and usually does. And, thanks to the procrastinators, time is not nailed down.

  If you haven't got the joke by now, take another look at the title.

  Lejean's plan works; Jeremy builds his clock.

  Time stops, which is what the Auditors wanted. Not only on Discworld: temporal stasis expands across the universe at the speed of light. Soon, everything will stop. The History Monks are powerless, for they, too, have stopped. Only Susan Sto Helit, Death's granddaughter, can get time started again. And Ronnie Soak, who used to be Kaos, the Fifth Horseman of the Apocralypse, but left because of artistic disputes before they became famous ... Fortunately, the Auditors like obeying rules, and DO NOT FEED THE ELEPHANT really perplexes them when there is no elephant to feed. Fatally, they also have a love-hate relationship with chocolate. They are living on stolen time.

  A procrastinator is a sort of time machine, but it moves time itself, instead of moving people through time. Moreover, it's fact, not fiction, as is all of Discworld to those who live there. On Roundworld, the first fictional time machine, as opposed to dreams or narrative timeslip, seems to have been invented by Edward Mitchell, an editor for the New York Sun newspaper. In 1881 he published an anonymous story, `The Clock That Went Backward', in his paper. The most celebrated time-travel gadget appears in Herbert George Wells's novel The Time Machine of 1895, and this set a standard for all that followed. The novel tells of a Victorian inventor who builds a time machine and travels into the far future. There he finds that humanity has
speciated into two distinct types - the nasty Morlocks, who live deep inside caverns, and the ethereal Eloi, who are preyed on by the Morlocks and are too indolent to do anything about it. Several movies, all fairly ghastly, have been based on the book.

  The novel had inauspicious beginnings. Wells studied biology, mathematics, physics, geology, drawing, and astrophysics at the Normal School of Science, which became the Royal College of Science and eventually merged with Imperial College of Science and Technology. While a student there, he began the work that led up to The Time Machine. His first time-travel story `The Chronic Argonauts' appeared in 1888 in the Science Schools Journal, which Wells helped to found. The protagonist voyages into the past and commits a murder. The story offers no rationale for time travel and is more of a mad-scientist tale in the tradition of Mary Shelley's Frankenstein, but nowhere near as well written. Wells later destroyed every copy of it he could locate, because it embarrassed him so much. It lacked even the paradoxical element of the 1891 Tourmalin's Time Cheques by Thomas Anstey Guthrie, which introduced many of the standard time-travel paradoxes.

  Over the following three years, Wells produced two more versions of his time-travel story, now lost, but along the way the storyline mutated into a far-future vision of the human race. The next version appeared in 1894 in the National Observer magazine, as three connected tales with the title `The Time Machine'. This version has many features in common with the final novel, but before publication was complete, the editor of the magazine moved to the New Review. There he commissioned the same series again, but this time Wells made substantial changes. The manuscripts include many scenes that were never printed: the hero journeys into the past, running into a prehistoric hippopotamus[19] and meeting the Puritans in 1645. The published magazine version is very similar to the one that appeared in book form in 1895. In this version the Time Traveller moves only into the future, where he finds out what will happen to the human race, which splits into the languid Eloi and the horrid Morlocks - both equally distasteful.

  Where did Wells get the idea? The standard SF writer's reply to this question is that `you make it up', but we have some fairly specific information in this case. In a foreword to the 1932 edition, Wells says that he was motivated by `student discussions in the laboratories and debating society of the Royal College of Science in the eighties'. According to Wells's son, the idea came from a paper on the fourth dimension read by another student. In the introduction to the novel, the Time Traveller (he is never named, but in the early version he is Dr Nebo-gipfel, so perhaps it's just as well) invokes the fourth dimension to explain why such a machine is possible:

  `But wait a moment. Can an instantaneous cube exist?' `Don't follow you,' said Filby.

  `Can a cube that does not last for any time at all, have a real existence?'

  Filby became pensive.

  `Clearly,' the Time Traveller proceeded, `any real body must have extension in four directions: it must have Length, Breadth, Thickness, and - Duration ...

  .. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives ...

  .. But some philosophical people have been asking why three dimensions particularly - why not another direction at right angles to the three? - and have even tried to construct a Four-Dimensional geometry. Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago.'

  The notion of time as a fourth dimension was becoming common scientific currency in the late Victorian era. The mathematicians had started it, by wondering what a dimension was, and deciding that it need not be a direction in real space. A dimension was just a quantity that could be varied, and the number of dimensions was the largest number of such quantities that could all be varied independently. Thus the Discworld thaum, the basic particle of magic, is actually composed of resons, which come in at least five flavours: up, down, sideways, sex appeal, and peppermint. The thaum is therefore at least five-dimensional, assuming that up and down are independent, which is likely because it's quantum.

  In the 1700s the foundling mathematician Jean le Rond D'Alembert (his middle name is that of the church where he was abandoned as a baby) suggested thinking of time as a fourth dimension in an article in the Reasoned Encyclopaedia or Dictionary of Sciences, Arts, and Crafts. Another mathematician, Joseph-Louis Lagrange, used time as a fourth dimension in his Analytical Mechanics of 1788, and his Theory of Analytic Functions of 1797 explicitly states: `We may regard mechanics as a geometry of four dimensions.'

  It took a while for the idea to sink in, but by Victorian times mathematicians were routinely combining space and time into a single entity. They didn't (yet) call it spacetime, but they could see that it had four dimensions: three of space plus one of time. Journalists and the lay public soon began to refer to time as the fourth dimension, because they couldn't think of another one, and to talk as if scientists had been looking for it for ages and had just found it. Newcomb wrote about the science of four-dimensional space from 1877, and spoke about it to the New York Mathematical Society in 1893.

  Wells's mention of Newcomb suggests a link to one of the more colourful members of Victorian society, the writer Charles Howard Hinton. Hinton's primary claim to fame is his enthusiastic promotion of `the' fourth dimension. He was a talented mathematician with a genuine flair for four-dimensional geometry, and in 1880 he published `What is the Fourth Dimension?' in the Dublin University Magazine, which was reprinted in the Cheltenham Ladies' Gazette a year later. In 1884 it reappeared as a pamphlet with the subtitle `Ghosts Explained'. Hinton, something of a mystic, related the fourth dimension to pseudoscientific topics ranging from ghosts to the afterlife. A ghost can easily appear from, and disappear along, a fourth dimension, for instance, just as a coin can appear on, and disappear from, a tabletop, by moving along `the' third dimension.

  Charles Hinton was influenced by the unorthodox views of his surgeon father James, a collaborator of Havelock Ellis, who outraged Victorian society with his studies of human sexual behaviour. Hinton the elder advocated free love and polygamy, and eventually headed a cult. Hinton the younger also had an eventful private life: in 1886 he fled to Japan, having been convicted of bigamy at the Old Bailey. In 1893 he left Japan to become a mathematics instructor at Princeton University, where he invented a baseball-pitching machine that used gunpowder to propel the balls, like a cannon. After several accidents the device was abandoned and Hinton lost his job, but his continuing efforts to promote the fourth dimension were more successful. He wrote about it in popular magazines like Harper 's Weekly, McClure's, and Science. He died suddenly of a cerebral haemorrhage in 1907, at the annual dinner of the Society of Philanthropic Enquiry, having just completed a toast to female philosophers.

  It was probably Hinton who put Wells on to the narrative possibilities of time as the fourth dimension. The evidence is indirect but compelling. Newcomb definitely knew Hinton: he once got Hinton a job. We don't know whether Wells ever met Hinton, but there is circumstantial evidence of a close connection. For example, the term `scientific romance' was coined by Hinton in titles of his collected speculative essays in 1884 and 1886, and Wells later used the same phrase to describe his own stories. Moreover, Wells was a regular reader of Nature, which reviewed Hinton's first series of Scientific Romances (favourably) in 1885 and summarised some of his ideas on the fourth dimension.

  In all likelihood, Hinton was also partially responsible for another Victorian transdimensional saga, Edwin A. Abbott's Flatland of 1884. The tale is about A. Square, who lives in the Euclidean plane, a twodimensional society of triangles, hexagons and circles, and doesn't believe in the third dimension until a passing sphere drops him in it. By analogy, Victorians who didn't believe in the fourth dimension we
re equally blinkered. A subtext is a satire on Victorian treatment of women and the poor. Many of Abbott's ingredients closely resemble elements found in Hinton's stories[20].

  Most of the physics of time travel is general relativity, with a dash of quantum mechanics. As far as the wizards of Unseen University are concerned, all this stuff is `quantum' - a universal intellectual getout-of-jail card - so you can use it to explain virtually anything, however bizarre. Indeed, the more bizarre, the better. You're about to get a solid dose of quantum in Chapter 8. Here we'll set things up by providing a quick primer on Einstein's theories of relativity: special and general.

  As we explained in The Science of Discworld, `relativity' is a silly name. It should have been 'absolutivity'. The whole point of special relativity is not that `everything is relative', but that one thing - the speed of light - is unexpectedly absolute. Shine a torch from a moving car, says Einstein: the extra speed of the car will have no effect on the speed of the light. This contrasts dramatically with old-fashioned Newtonian physics, where the light from a moving torch would go faster, acquiring the speed of the car in addition to its own inherent speed. If you throw a ball from a moving car, that's what happens. If you throw light, it should do the same, but it doesn't. Despite the shock to human intuition, experiments show that Roundworld really does behave relativistically. We don't notice because the difference between Newtonian and Einsteinian physics becomes noticeable only when speeds get close to that of light.

  Special relativity was inevitable; scientists were bound to think of it. Its seeds were already sown in 1873 when James Clerk Maxwell wrote down his equations for electromagnetism. Those equations make sense in a `moving frame' - when observations are made by a moving observer - only if the speed of light is absolute. Several mathematicians, among them Henri Poincare and Hermann Minkowski, realised this and anticipated Einstein on a mathematical level, but it was Einstein who first took the ideas seriously as physics. As he pointed out in 1905, the physical consequences are bizarre. Objects shrink as they approach the speed of light, time slows to a crawl, and mass becomes infinite. Nothing (well, no thing) can travel faster than light, and mass can turn into energy.