The first of our nine paradoxes goes back two and a half millennia and, given how long we’ve had to mull it over, you will not be surprised to hear that it has been thoroughly understood and explained. And yet most people encountering it for the first time still scratch their heads in bafflement. It is known as the Paradox of Achilles (or the Problem of Achilles and the Tortoise) and is just one of a series of problems set by the Greek philosopher Zeno in the fifth century BC. As an example in pure logic it couldn’t be simpler. But don’t be fooled; in this chapter we will consider several of Zeno’s paradoxes and finish off by bringing his ideas right up to date with a version of one that can only be explained using quantum theory. Well, I never said I would go easy on you.
But let us begin with the most famous of Zeno’s paradoxes. A tortoise is given a head start in a race against the swift-running Achilles such that it reaches some point (let’s call it point A) along the route by the time Achilles sets off. Since Achilles runs much faster than the tortoise walks, he will very soon get to point A. However, by the time he gets there the tortoise has already moved on a short distance to a further point, which we will call point B. By the time Achilles reaches point B, the tortoise has moved on to point C, and so on. So while Achilles is clearly catching up with the tortoise, and the gap between them becomes a bit smaller at each stage, it seems he will never actually overtake it. Where are we going wrong?
When it comes to being clever, mastering logical conundrums and brainteasers, or just deep thinking in general, you can’t beat the ancient Greeks. In fact, so sharp were these philosophers of antiquity, so insightful in their logic, we tend to forget that they lived over two thousand years ago. Even today, when we wish to give examples of genius, along with the ever-popular Einstein we often invoke such familiar names as Socrates, Plato, and Aristotle as representing the ultimate in intellectual brilliance.
Zeno was born in Elea, an ancient Greek town in what is now southwestern Italy. Little is known about his life and work other than that he was a student of another Elean philosopher, Parmenides. Together with a third man from the same town, Melissus, they formed what is now called the Eleatic movement. Their philosophy was that you should not always trust your senses and sense experience in order to understand the world, but should rely ultimately on logic and mathematics. On the whole, this is a sensible approach; but, as we shall see shortly, it would lead Zeno down the wrong path.
From what little we know of his ideas it would seem that Zeno did not have many positive views of his own, but was instead committed to demolishing the arguments of others. Despite this, Aristotle himself, who lived a century after Zeno, regarded him as the founder of a type of debate called “dialectic.” This is a form of civilized discussion at which the ancient Greeks—particularly men like Plato and Aristotle—excelled, using logic and reason to resolve disagreements.
Only one short piece of Zeno’s original work survives today, so what we know about him derives from the writings of others, notably Plato and Aristotle. Around the age of forty Zeno traveled to Athens, where he met the young Socrates. Later in life he became active in Greek politics, and was eventually imprisoned and tortured to death for his part in a conspiracy against the local ruler in Elea. One story about him is that he bit off his own tongue and spat it out at his captors rather than betray his coconspirators. But he is most famous for a series of paradoxes that are brought to us by Aristotle in his great text the Physics. There are believed to have been around forty in total, but only a handful survive.
All Zeno’s paradoxes—the four most famous of which are known by the names Aristotle gave them: the Achilles, the Dichotomy, the Stadium, and the Arrow—are centered on the idea that nothing ever changes; that motion is just an illusion and that time itself does not really exist. Of course, if there was one thing the Greeks excelled at it was philosophizing, and grand statements like “all motion is an illusion” are just the sort of provocative abstraction they were famous for. Today we can demolish these paradoxes scientifically; but they are such fun that it is well worth revisiting them here. In this chapter I will consider them in turn and show how each one can be resolved with a little careful scientific analysis. Let’s begin with the one I’ve already outlined.
This is my favorite of Zeno’s paradoxes because at first glance it seems perfectly logical, and yet in fact it defies logic in an unexpected way. Achilles is the greatest warrior in Greek mythology, endowed with great strength, bravery, and military skill. Part human, part supernatural being—his parents were King Peleus of Thessaly and a sea nymph named Thetis—he figures prominently in Homer’s Iliad, which tells the story of the Trojan War. It was said that even as a young boy he could run fast enough to catch a deer and was strong enough to kill a lion. So Zeno was clearly going for extremes when he chose this mythical hero to race against the ponderous tortoise.
The paradox is based on the even older fable of the hare and the tortoise, attributed to another ancient Greek, by the name of Aesop, who lived about a century before Zeno. In the original fable, the tortoise is ridiculed by the hare and so challenges him to a race, which the tortoise duly wins thanks to the hare’s arrogance in thinking he can afford to stop for a nap halfway through, only to awaken too late to catch the tortoise.
In Zeno’s version, the fleet-footed Achilles takes on the role of the hare. Unlike the hare, he is completely focused on the task; but he does give the tortoise a head start, and this seems to be his undoing, as the tortoise appears always to win the race, however long it is, albeit probably by the ancient Greek equivalent of a photo finish. According to Zeno’s account, however fast the hero runs, and however slowly his opponent plods, Achilles will never overtake the tortoise. Surely this cannot be what happens in reality?
This was a serious conundrum to the Greek mathematicians, who had no real concept of what we call a converging infinite series, or indeed of the notion of infinity itself (ideas I will explain briefly to you in a moment). Aristotle, who was certainly no slouch when it came to thinking about such matters, regarded Zeno’s paradoxes as “fallacies.” The problem was that neither Aristotle nor anyone else in ancient Greece properly understood one of the most basic algebraic formulas in physics: speed equals distance divided by time. Today we can do much better.
The statement “will never overtake the tortoise” is of course wrong, since the ever-decreasing distances that are being considered with each stage (between points A and B, and between B and C, and so on), also involve ever-decreasing time intervals, and so even an infinite number of stages does not imply an infinite length of time. In fact, the stages all add up to a finite time: the time it takes for Achilles to reach the tortoise! The confusing thing about the paradox is that most people don’t appreciate that adding up an infinite sequence of numbers does not necessarily lead to an infinite result. Strange though it might sound, an infinite number of stages can be completed in a finite time, and the tortoise will be reached and overtaken easily enough, as logic insists. The solution relies on what mathematicians call a geometric series.
Consider the following example:
It is clear that you could keep adding smaller and smaller fractions for ever, with the total getting closer and closer to the value 2. Try this by drawing a line and then dividing it in half, then taking the right-hand side and chopping that in half, and so on until the fractions get so tiny that you can no longer mark them separately on the paper. If half of the line is one unit in length (it doesn’t matter if this is a centimeter, an inch, a meter or a mile), then by adding successive fractions, as in the series written above, we converge on the total length of two units.
Figure 2.1 A converging infinite series
Summing an infinite number of ever-decreasing lengths—adding lengths forever doesn’t mean the final answer is infinity since the lengths are getting shorter all the time.
A good way to apply this to the paradox is to consider not the points at which Achilles and the tortoise respectively have reached at any stage, but rather the ever-decreasing gap between them. Since they are both moving at a constant speed, this gap is decreasing at a constant rate too. For example, if Achilles gives the tortoise a 100-meter start and then proceeds to catch it up at a rate of 10 meters per second, how does this pan out according to Zeno? Well, the distance between them is halved after five seconds. The remaining distance is itself halved after two and a half seconds, then what is left is halved after one and a quarter seconds, and so on. We could, if we wished, keep adding the ever-smaller distances covered during these ever-decreasing time intervals, but the fact remains that if Achilles is catching the tortoise up at a rate of 10 meters per second then he will overtake it in 10 seconds, which is the time he needs to close that initial gap of 100 meters to zero. And this value of 10 seconds is just the number we would get if we added up 5 seconds + 2.5 seconds + 1.25 seconds + 0.625 seconds … and so on until the next number to add is so small we are happy to call it a day (at 9.9999 … seconds). After 10 seconds Achilles will then of course streak on ahead of the tortoise as expected (unless he decides to stop for a beer along the way—not something Zeno felt was necessary to clarify in his argument).
The next of Zeno’s paradoxes refutes the reality of motion itself and is a variation on the same theme as the Paradox of Achilles. It is very simple to state:
In order to reach your destination you must first cover half the distance, but in order to cover half the distance you must first cover a quarter of the distance, and in order to cover a quarter of the distance you must cover an eighth of the distance, and so on. If you can keep chopping the distances in half forever, then you never reach that very first distance marker, and so you never actually start your journey. What’s more, this never-ending sequence of ever-shorter distances is infinite. So to complete the journey requires you to complete an infinite number of tasks. So you could never finish it. If you cannot start the journey and would never finish it, then motion itself is impossible.
We learn about this paradox from Aristotle, who knows that it is nonsense but searches for the logical argument with which he can refute it conclusively. After all, it is quite obvious that there is such a thing as motion. However, Zeno was applying a form of argumentation called reductio ad absurdum, which is the reduction of an idea to absurdity by demonstrating the inevitably absurd conclusion to which it would logically lead. We must also remember that Zeno was no mathematician. He was arguing with reference only to pure logic, and that is often just not enough. Other Greek philosophers resorted to a more direct and pragmatic approach in refuting Zeno’s arguments about the illusion of motion. One of them was Diogenes the Cynic.
Our word “cynicism” has its origin in an idealistic philosophical movement of ancient Greece. The Greek Cynics seem to have been a nicer bunch than the modern connotations of their name suggest: they rejected wealth, power, fame, even possessions, and chose instead to lead a simple life free of all the traditional human vices. They believed that all humans were equal and that the world belonged in equal measure to everyone. Probably the most famous of the Cynics was Diogenes, who lived during the time of Plato in the fourth century BC. This philosopher is responsible for some of the most wonderful quotes you will find, such as “Blushing is the color of virtue,” “Dogs and philosophers do the greatest good and get the fewest rewards,” “He who is most content with the least has the most,” and “I know nothing, except the fact of my ignorance.”
Diogenes took the teachings of Cynicism to their logical extremes. He seems to have made a virtue of poverty and spent years living in a tub in an Athens marketplace. He became famous for being, well, cynical about everything, particularly much of the philosophical teaching of the time, even that of eminences such as Socrates and Plato. So you can imagine what he thought of Zeno and his paradoxes. On hearing about Zeno’s Dichotomy Paradox regarding the illusion of motion, he simply stood up and walked off, thus demonstrating the absurdity of Zeno’s conclusions.
While we may applaud Diogenes for his practical approach, we still need to investigate a little more carefully where Zeno’s logic is breaking down. And that turns out not to be so difficult—after all, we’ve had over two thousand years to figure it out. In any case, while you may feel that sheer common sense is sufficient to dismiss Zeno’s paradox, I do not. I have spent most of my life working and, more importantly, thinking as a physicist, and I am not satisfied with mere commonsensical, philosophical, or logical arguments that refute the Dichotomy. I need watertight physics—which, for me, does a far more convincing job.
What we need to do is to convert Zeno’s argument about distance into one about time. Assume you are already moving at a constant speed at the moment in time when you are at the starting point of the journey to be covered. The notion of speed, which Zeno would not have understood very well, means covering a certain distance in a finite time. The shorter the distance you must cover, the shorter the time interval needed to cover it, but whenever you divide the first number by the second they always give the same answer: your speed. By considering shorter and shorter distances that must be covered before you begin your journey, you are also considering shorter and shorter intervals of time. But time marches on regardless of how we might wish to split it up artificially into these ever-decreasing periods. Thinking about time, rather than space, as a static line that can be subdivided indefinitely is fine (and we often think of time in this way when solving problems in physics), but the crucial point is that the way we perceive time is not as a static line in the same way as we can view lines in space. We cannot take ourselves outside of time’s stream. Time marches on regardless—and so we move.
If we consider the situation from the point of view of someone not already moving, but starting from rest, there is just one more bit of physics we need to think about. This is something we all learn about at school (and most of us, no doubt, promptly forget). It is referred to as Newton’s second law, which states that to make an object begin to move, a force needs to be applied to it. This will cause it to accelerate—to alter its state from being at rest to being in motion. But once it is moving, the same argument applies: namely, that, as time goes by, the distances covered are based on the moving object’s speed, which need not be constant. The Dichotomy argument is then an abstract irrelevance that has nothing to say about true motion in the physical world.
I should make one final remark before moving on. Albert Einstein’s theory of relativity teaches us that maybe we should not dismiss the Dichotomy Paradox so confidently. According to Einstein, time can be regarded in a similar way to space—indeed, he refers to time as the fourth axis, or fourth dimension, of what is called space–time. This suggests that maybe the flow of time is just an illusion after all—and, if it is, then so is motion. But I would argue that, despite the success of relativity theory, this conclusion takes us away from physics and into the murky waters of metaphysics—abstract ideas that don’t have the solid backing of empirical science.
I am not suggesting that Einstein’s theory of relativity is wrong; of course not. It is just that Einstein’s ideas only really manifest themselves when things start to move very fast—close to the speed of light. At normal everyday speeds we are quite within our rights to ignore such “relativistic” effects and think of time and space in the familiar commonsense way we are used to. After all, if we push Zeno’s argument to its logical extreme, then it is in fact wrong to say that time and space are infinitely divisible into ever-smaller but still discrete intervals and distances. At some point things get so small that quantum physics comes into effect, when time and space themselves become fuzzy and indefinable and it no longer makes sense to chop them up into smaller pieces. Indeed, motion itself is a little illusory in the quantum domain of atoms and subatomic particles. But that is not what Zeno had in mind.
While fun to explore and discuss in this context, neither quantum physics nor relativity theory is needed to explain away Zeno’s Dichotomy. Using such ideas from modern physics to argue that all motion is an illusion is to miss the point and brings us dangerously close to turning physics into mysticism. So let’s not make things more complicated than they need to be. There will be plenty of time for such craziness later on in the book, believe me.
And so we move swiftly on. A related paradox of Zeno’s that plays with the notion of speed is known as the Moving Rows Paradox. It is somewhat obscure and reaches us via Aristotle, who called it the Stadium Paradox. I will try to describe it as succinctly as possible.
Imagine there are three trains, each consisting of one engine and two carriages. The first train is standing still at a station. The second and third trains don’t stop at the station but are moving at an equal and constant speed in opposite directions to each other, B from the west and C from the east.
At a given moment in time the trains are positioned as in Figure 2.2(a). Then, one second later, they are aligned as in Figure 2.2(b). The problem, according to Zeno, concerns the motion of train B: in one second, it has moved the length of one carriage past train A, but at the same time it has moved the length of two carriages past train C. The paradox is that train B has moved a distance and twice that distance at the same time. Zeno seems to have been aware that these are just relative distances, and so attempted to cast the paradox in terms of time instead. Dividing each of the two distances by the constant speed of train B, we arrive at two different time periods, one double the duration of the other. But both paradoxically seem to describe how long it takes to get from the situation in the upper diagram to the aligned one in the lower diagram!
Figure 2.2 The Moving Rows Paradox
(a) Train A is stationary. Train B travels from left to right and train C from right to left at the same speed as B.
(b) One second later, the trains are all aligned.
Resolving this apparent paradox is easy, since we can see what is going wrong in the reasoning. There is such a thing as relative speed, of course, so that we cannot say that B is traveling at the same speed relative to the moving C as it is relative to the stationary A. Did Zeno know this, and was he simply making a more subtle point about the illusory nature of motion? It is not clear; but, as every schoolboy and schoolgirl should be able to appreciate, there is really no paradox here at all. B passes C at twice the relative speed that it passes A, and so of course it will pass two carriages of C in the same time it takes to pass one carriage of A.
Like the Dichotomy, this is another paradox centered on the idea that true motion is just an illusion. It is stated thus by Aristotle: “If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.”
Eh? Well, let me try to put it more clearly.
An arrow in flight has, at any given instant in time, a certain, fixed position—as we would see in a snapshot. But if we only see it at this instant, it will be indistinguishable from a truly motionless arrow in the same position. So how can we ever say an arrow is in motion? Indeed, since time is made up of a sequence of consecutive instants, in each of which the arrow is motionless, it never moves.
The paradox, of course, is that we know there is such a thing as motion. Of course the arrow moves. So where is the logical error in Zeno’s statement?
Time can be considered to be made up of a sequence of infinitesimally short “moments,” which we can think of the smallest possible, indivisible, intervals of time. As a physicist, I can see the problem with Zeno’s argument. If these indivisible instances are not exactly of zero duration (true snapshots) then the arrow will be in a slightly different position at the start of each one from its position at the end, and it cannot therefore be said to be at rest. On the other hand, if such instances are truly of zero duration, then it doesn’t matter how many of them sit together side by side, they will never add up to a nonzero interval of time: we can add zero to itself as many times as we like and the answer is still zero. So Zeno’s argument that a finite duration of time is just made up of a sequence of such consecutive instants is wrong.
It would take advances in mathematics as well as physics for this paradox to be finally laid to rest. More specifically, it was an understanding of calculus, the field of mathematics developed by Isaac Newton and others in the seventeenth century, which describes how to add up tiny quantities in order to describe the notion of change correctly, that would lay Zeno’s naïve ideas to rest.
Well, almost. In 1977, two physicists at the University of Texas published a surprising research paper that suggested Zeno’s Arrow Paradox might have been laid to rest too prematurely. Their names were Baidyanaith Misra and George Sudarshan, and the title of their paper was “The Zeno’s Paradox in Quantum Theory.” Physicists around the world were intrigued. Some thought the work silly, while others rushed to try to test the idea. But before I go any further, let me say as little as I feel I can get away with at this early stage of the book about the weird and wonderful set of ideas that is quantum mechanics.
Quantum mechanics is the theory that describes the workings of the microscopic world—by which I do not mean the tiny world only visible under a microscope, but rather the far, far tinier world of atoms and molecules and the subatomic particles (the electrons, protons, and neutrons) of which they are made up. Indeed, quantum mechanics is the most powerful, important, and fundamental mathematical set of ideas in the whole of science. It is remarkable for two seemingly contradictory reasons (almost a paradox in itself!). On the one hand, it is so fundamental to our understanding of the workings of our world that it lies at the very heart of most of the technological advances made in the past half-century. On the other hand, no one seems to know exactly what it means.
I must make it clear from the outset that the theory of quantum mechanics is not in itself weird or illogical; on the contrary, it is a beautifully accurate and logical construction that describes nature superbly well. Without it we would not be able to understand the basics of modern chemistry, or electronics, or materials science; we would not have invented the silicon chip or the laser; there would be no television sets, computers, microwaves, CD and DVD players, or mobile phones, let alone much else that we take for granted in our technological age.
Quantum mechanics accurately predicts and explains the behavior of the very building blocks of matter with extraordinary accuracy. It has led us to a very precise and almost complete understanding of how the subatomic world behaves, and how the myriad of particles interact with each other and connect up to form the world we see around us, and of which we are of course a part. After all, we are ultimately just a collection of trillions of atoms obeying the rules of quantum mechanics and organized in a highly complex way.
These strange mathematical rules were discovered in the 1920S. They turn out to be very different from the rules that govern the more mundane everyday world we are familiar with: the world of objects we see around us. Near the end of the book I will explore just how strange some of these rules are when we consider the Paradox of Schrödinger’s Cat. For now, I wish to focus on one particularly strange feature of the quantum world, namely that an atom will behave differently if left to its own devices from how it behaves when it is being “observed”—by which I mean when it is being monitored in some way: poked or prodded, knocked or zapped. This feature of the quantum world is still not fully understood, partly because it is only now becoming clear what exactly constitutes “observation” in this sense. This is known as the “measurement problem” and is still an active area of scientific research today.
The quantum world is ruled by chance and probability. It is a place where nothing is as it seems. If left alone, a radioactive atom will emit a particle, but we are unable to predict when this will take place. All we can ever do is work out a number called the half-life. This is the time it takes for half of a large number of identical atoms to “decay” radioactively. The larger the number, the more accurate we can be about this half-life, but we can never predict in advance which atom in the sample will go next. It is very much like the statistics of tossing a coin. We know that if we toss a coin again and again, then half the time it will end up heads and the other half tails. The more times we toss it, the more accurate this statistical prediction will be. But we can never predict whether the very next toss of the coin will be heads or tails.
The quantum world is probabilistic in nature not because quantum mechanics as a theory is incomplete or approximate, but rather because the atom itself does not “know” when this random event will take place. This is an example of what is called “indeterminism,” or unpredictability.
Misra and Sudarshan’s paper, which was published in the Journal of Mathematical Physics, describes the astonishing situation whereby a radioactive atom, if observed closely and continuously, will never decay! The idea can be summed up perfectly by the adage “the watched pot never boils,” which was first used, as far as I can tell, by the Victorian writer Elizabeth Gaskell in her 1848 novel Mary Barton—although it is the sort of saying that probably dates back much further. The notion has its origins, of course, in Zeno’s Arrow Paradox and our inability to detect motion by considering a snapshot of a moving object in an instant of time.
But how and why might this happen in reality? Clearly the saying about the watched pot is nothing more than a simple lesson in patience: you cannot make a kettle boil any more quickly by staring at it. However, Misra and Sudarshan seemed to be suggesting that when it comes to atoms you really do influence how they behave by watching them. What is more, this interference is unavoidable—the act of looking will inevitably alter the state of the thing you are looking at.
Their idea goes to the very heart of how quantum mechanics describes the microscopic world: as a fuzzy, ghostly reality in which all sorts of weird goings-on seem to take place routinely when it’s left alone—an idea we will revisit in Chapter 9—none of which we are ever able to detect actually happening. So an atom that would, if left to its own devices, spontaneously emit a particle at any moment will somehow remain too shy to do so if it is being spied upon, so that we can never actually catch it in the act. It is as though the atom has been endowed with some kind of awareness, which is crazy. But then the quantum world is crazy. One of the founding fathers of quantum theory was the Danish physicist Niels Bohr, who in 1920 set up a research institute in Copenhagen to which he attracted the greatest scientific geniuses of the time—men such as Werner Heisenberg, Wolfgang Pauli, and Erwin Schrödinger—to try to unlock the secrets of the tiniest building blocks of nature. One of Bohr’s most famous sayings was that “if you are not astonished by the conclusions of quantum mechanics then you have not understood it.”
Misra and Sudarshan’s paper was entitled “The Zeno’s Paradox in Quantum Theory” because of its origins in the Arrow Paradox. However, it is fair to say now that, while its conclusion remains somewhat controversial, it is for most quantum physicists no longer a paradox. In the literature today it is referred to more commonly as the “Quantum Zeno Effect,” and has been found to apply far more widely than in the situation described by Misra and Sudarshan. A quantum physicist will happily tell you that the effect can be explained by “the constant collapse of the wave function into the initial undecayed state,” which is the sort of incomprehensible geeky gobbledygook one should expect from such people—I should know, I am one of them. But I don’t think I will pursue this line of thought in any more detail here, just in case you are nervously wondering what you’ve let yourself in for.
This recent discovery that the Quantum Zeno Effect is pretty much ubiquitous is down to a better understanding among quantum physicists of how an atom responds to its surroundings. A big breakthrough was made when scientists at one of the world’s most prestigious laboratories, the National Institute of Standards and Technology in Colorado, confirmed the Quantum Zeno Effect in a famous experiment in 1990. The experiment took place within the wonderfully named Division of Time and Frequency, which is best known for setting the standards for the most accurate measurement of time. Indeed, scientists here have recently built the world’s most accurate atomic clock, precise to within one second every three and a half billion years—that’s getting close to the age of the Earth itself.
One of the physicists working on these incredibly high-precision clocks is Wayne Itano. It was his group who designed the experiment to test whether the Quantum Zeno Effect could really be detected. It involved trapping several thousand atoms in a magnetic field and then zapping them delicately with lasers, forcing them to give up their secrets. Sure enough, the researchers found clear evidence of the Quantum Zeno Effect: under constant watching, the atoms behaved very differently from what the scientists had expected.
One final twist: there is now evidence for the opposite effect, something called the “Anti-Zeno Effect,” which is the quantum equivalent of staring at a kettle and making it come to the boil more quickly. While still somewhat speculative, such research goes to the heart of some of the most profound and possibly important areas of science in the twenty-first century, such as working toward building what is a called a quantum computer. This is a device that makes direct use of some of the strange behavior of the quantum world in order to carry out its calculations far more efficiently.
I am not sure what Zeno of Elea would have made of this revival of his paradoxes, or of his name being attached to a remarkable phenomenon in physics nearly two and a half thousand years later. Here, the paradox has nothing to do with tricks of logic, but everything to do with the even stranger tricks that nature seems able to play down at the tiny scale of atoms—tricks that we are only beginning to understand.
Zeno’s paradoxes have taken us from the very birth of physics to cutting-edge ideas in the twenty-first century. All the other paradoxes in this book arose somewhere in between. In resolving them we will have to travel to the furthest reaches of the Universe and explore the essence of space and time themselves. Hold on tight.