Paradox

These are old fond paradoxes, to make fools laugh i' the alehouse.

William Shakespeare, Othello, Act II, Scene I

Our word paradox comes from two Greek words: para, meaning "contrary to," and doxa, meaning "opinion."7 It describes a situation in which, alongside one opinion or interpretation, there is another, mutually exclusive opinion. The word has taken on a variety of subtly different meanings, but at the core of each usage is the idea of a contradiction. Paradox is more than mere inconsistency, though. If you say "it is raining, it is not raining," then you have contradicted yourself, but paradox is more than this. A paradox arises when you begin with a set of self-evident premises and then, from these premises, deduce a conclusion that undermines them. If you have a cast-iron argument that proves it must certainly be raining outside, and then you look out of the window and see that it is not raining, then you have a paradox to resolve.

A weak paradox or fallacy can often be clarified with a little thought. The contradiction usually arises because of a simple mistake in a chain of logic leading from premises to conclusion.8 In a strong paradox, however, the source of a contradiction is not immediately apparent; centuries may pass before matters are resolved. A strong paradox has the power to challenge our most cherished theories and beliefs. Indeed, as the mathematician Anatol Rapoport once remarked: "Paradoxes have played a dramatic part in intellectual history, often foreshadowing revolutionary developments in science, mathematics and logic. Whenever, in any discipline, we discover a problem that cannot be solved within the conceptual framework that supposedly should apply, we experience shock. The shock may compel us to discard the old framework and adopt a new one."9

figure 2 A visual paradox. These impossible figures are Penrose triangles. They appear to show a three-dimensional triangular solid, but these triangles are impossible to construct. Each vertex of a Penrose triangle is in fact a perspective view of a right angle. Artists like Escher delight in presenting visual paradoxes.

Paradoxes abound in logic and mathematics and physics, and there is a type for every taste and interest.

A Few Logical Paradoxes

An old paradox, contemplated by philosophers since the middle of the 4th century BC and still much discussed, is the liar paradox. Its most ancient attribution is to Eubulides of Miletus, who asked: "A man says that he is lying; is what he says true or false?" However one analyzes the sentence, there is a contradiction. The same paradox appears in the New Testament. St. Paul, referring to Cretans, wrote: "One of themselves, even a prophet of their own, said the Cretans are always liars."10 It is not clear whether St. Paul was aware of the problem in his sentence, but when self-reference is allowed paradox seems almost inevitable.

One of the most important tools of reasoning we possess is the sorites. In logicians' parlance, a sorites is a chain of linked syllogisms: the predicate of one statement becomes the subject of the following statement. The following is a typical example:

all ravens are birds;

all birds are animals;

all animals require water to survive.

Following the chain, we reach a logical conclusion: all ravens need water.

Sorites are important because they allow us to make conclusions without covering every eventuality in an experiment. (So we do not need to deprive ravens of water to know they may die of thirst.) But sometimes the conclusion of a sorites can be absurd: we have a sorites paradox. For example, if we accept that adding one grain of sand to another grain of sand does not make a heap of sand, and given that a single grain does not itself constitute a heap, then we must conclude that no amount of sand can make a heap. And yet we see heaps of sand. The source of such paradoxes lies in the intentional vagueness of a word like "heap"; politicians, of course, routinely take advantage of these linguistic tricks.11

As well as sorites, when reasoning we all routinely employ induction — the drawing of generalizations from specific cases. For example, whenever we see something drop, it falls down: using induction we propose a general law, namely that when things drop they always fall down and never up. Induction is such a useful technique that anything casting doubt on it is troubling. Consider Hempel's raven paradox.12 Suppose that an ornithologist, after years of field observation, has observed hundreds of black ravens. The evidence is enough for her to suggest the hypothesis that "all ravens are black." This is the standard process of scientific induction. Every time the ornithologist sees a black raven it is a small piece of evidence in favor of her hypothesis. Now, the statement that "all ravens are black" is logically equivalent to the statement that "all non-black things are non-ravens." If the ornithologist sees a piece of white chalk, then the observation is a small piece of evidence in favor of the hypothesis that "all non-black things are non-ravens" — but therefore it must be evidence for her claim that ravens are black. Why should an observation regarding chalk be evidence for a hypothesis regarding birds? Does it mean that ornithologists can do valuable work whilst sat indoors watching television, without bothering to watch a bird in the bush?

Another paradox in logic is that of the unexpected hanging, wherein a judge tells a condemned man: "You will hang one day next week but, to spare you mental agony, the day that the sentence will be carried out will come as a surprise." The prisoner reasons that the hangman cannot wait until Friday to carry out the judge's order: so long a delay means everyone will know the execution takes place that day — the execution will not come as a surprise. So Friday is out. But if Friday is ruled out, Thursday is ruled out by the same logic. Ditto Wednesday, Tuesday and Monday. The prisoner, mightily relieved, reasons that the sentence cannot possibly take place. Nevertheless, he is completely surprised as he is led to the gallows on Thursday! This argument — which also goes under the name of the "surprise examination paradox" and the "prediction paradox" — has generated a huge literature.13

A Few Scientific Paradoxes

Although it is often fun, and occasionally useful, to ponder liars, ravens and hanged men, arguments involving logical paradoxes too frequently — for my taste at least — degenerate into a discussion over the precise meaning and usage of words. Such discussions may be fine if one is a philosopher. But for my money the really fascinating paradoxes are those that can be found in science.

Consider one of the oldest of all paradoxes: Zeno's paradox of Achilles and the tortoise.14 Achilles and the tortoise take part in a 100-m sprint. Since Achilles runs 10 times faster than the tortoise, he gives the animal a head start of 10 m. The two sprinters set off at the same instant; so when Achilles has covered the first 10 m, the tortoise has moved on by 1 m. In the time it takes Achilles to cover 1 m, the tortoise has moved on by 10 cm; in the time it takes Achilles to cover that 10 cm, the tortoise has moved on by a further 1 cm. And so on ad infinitum. Our senses tell us a fast runner will always overtake a slow runner, but Zeno said Achilles cannot catch the tortoise. There is a contradiction between logic and experience: there is a paradox. It took 2000 years to resolve the paradox — but the mathematical machinery for doing so found a host of other uses.15

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10 m 20 m 30 m 40 m 50 m 60 m 70 m 80 m 90 m figure 3 When the race begins, Achilles is 10 m behind the tortoise. By the time Achilles has run 10 m, the tortoise has crawled a distance of 1m. By the time Achilles has run a further 1 m, the tortoise has crawled a further 10 cm. Following this logic, it seems Achilles can never catch up____

10 m 20 m 30 m 40 m 50 m 60 m 70 m 80 m 90 m figure 3 When the race begins, Achilles is 10 m behind the tortoise. By the time Achilles has run 10 m, the tortoise has crawled a distance of 1m. By the time Achilles has run a further 1 m, the tortoise has crawled a further 10 cm. Following this logic, it seems Achilles can never catch up____

The twin paradox, which involves the special relativistic phenomenon of time dilation, is one of the most famous in physics. Suppose one twin stays at home while the other twin travels to a distant star at close to the speed of light. To the stay-at-home twin, his sibling's clock runs slow: his twin ages more slowly than he does. Although this phenomenon may be contrary to common sense, it is an experimentally verified fact. But surely relativity tells us that the traveling twin can consider himself to be at rest? From his point of view, the clock of the earthbound twin runs slow; the stay-at-home twin should be the one who ages slowly. So what happens when the traveler returns? They cannot both be right: it is impossible for both twins to be younger than each other! The resolution of this paradox is easy: the confusion arises from a simple misapplication of relativity. The twins' situations are not interchangeable: the traveling twin accelerates to light speed, decelerates at the half-way point of his journey, and does it all again on the trip back. Both twins agree that the stay-at-home twin undergoes no such acceleration. So the traveler ages more slowly than the earthbound twin; he returns to find his brother aged, or even dead. An extraterrestrial visitor to Earth would observe the same phenomenon when it returned to its home planet: its stay-at-home siblings (if aliens have siblings) would be older or long-since dead. It is a sad fact of interstellar travel, and it is contrary to our experience, but it is not a paradox.16

One of the most important of scientific paradoxes is that named after Heinrich Olbers.17 He considered a question asked by countless children — "Why is the night sky dark?" — and showed that the darkness of night is deeply mysterious. His reasoning was based upon two premises. First, that the Universe is infinite in extent. Second, that the stars are scattered randomly throughout the Universe. (Olbers did not know of the existence of galaxies — they were not recognized as stellar groupings until some 75 years after his death — but this does not affect his reasoning. His argument works in exactly the same way for galaxies as it does for stars.) From these premises we reach an uncomfortable conclusion: in whichever direction you look, your line-of-sight must eventually end on a star — the night sky should therefore be bright.

Olbers' Paradox

Suppose all stars have the same intrinsic brightness. (The following argument is simpler under this assumption, but the conclusion in no way depends upon it.) Now consider a thin shell of stars (call it shell A) with Earth at its center, and another thin shell of stars (shell B), also centered on Earth, with a radius twice that of shell A. In other words, shell B is twice as distant from us as shell A.

A star in shell B will appear to be 1 as bright as a star in shell A. (This is the inverse-square law: if the distance to a light source increases by a factor of 2, the apparent brightness of the light source decreases by a factor of 2 x 2 = 4.) On the other hand, the surface area of shell B is 4 times larger than that of shell A, so it contains 4 times as many stars. Four times as many stars, each of which is 1 as bright: the total brightness of shell B is exactly the same as the total brightness of shell A! But this works for any two shells of stars. The contribution to the brightness of the night sky from a distant shell of stars is the same as from a nearby shell. If the Universe is infinite in extent, then the night sky should be infinitely bright.

This argument is not quite correct: the light from an extremely distant star will be intercepted by an intervening star. Nevertheless, in an infinite Universe with a uniform distribution of stars any line of sight will eventually run into a star. Far from being dark, the entire night sky should be as dazzling as the Sun. The night sky should blind us with its brightness!

figure 4 If the stars are uniformly distributed throughout space, then shell B will contain 4 times as many stars as shell A (A is at a distance r and B is at distance 2r). But the stars in shell A will appear 4 times as bright as the stars in shell B. So the total brightness of the shells will be the same. Since there is an infinite number of such shells, the night sky should be infinitely bright. Even allowing for the stars in nearby shells that block the light from distant stars, the night sky should be blindingly bright.

How can we resolve the paradox? The first explanation you are likely to think of is that clouds of gas or dust obscure the light from distant stars. The Universe does indeed contain dust clouds and gaseous regions, but they cannot shade us from Olbers' paradox: if the clouds absorb light, they

will heat up until they are at the same average temperature as the stars themselves. It turns out that the paradox is explained by one of the most dramatic discoveries made by astronomers: the Universe has a finite age. Since the Universe is only about 13 billion years old, the part that we can see is only about 13 billion light years in size. For the night sky to be as bright as the surface of the Sun, the observable Universe would have to be almost 1 million times bigger than it is. (That the Universe is expanding also helps to explain the paradox: light from distant objects is redshifted by the expansion, and so distant objects are less bright than one would expect from the inverse-square law. The principal explanation, though, comes from the finite age of the Universe.)

It is fascinating that in pondering such a simple question — "Why is the night sky dark?" — one could infer that the Universe is expanding and that it (or at least the stars and galaxies it contains) has a finite age. Perhaps the simple question that Fermi asked — "Where is everybody?" — leads to an even more important conclusion.

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