How wonderful that we have met with a paradox. Now we have some hope of making progress.
– Niels Bohr
Please accept my resignation. I don’t want to belong to any club that will accept me as a member.
– Groucho Marx
The word ‘paradox’ comes from the Greek para (‘beyond’) and doxa (‘opinion’ or ‘belief’). Literally, then, it means anything that’s hard to believe or runs counter to our intuition or common sense. We’ll often say, in everyday conversation, that something is paradoxical just because it seems almost unbelievable. For example, the fact, mentioned in Chapter 3, that in a room of 23 people there’s a 50-50 chance of two people having the same birthday is sometimes called the ‘birthday paradox’, even though it’s an easily proven statistical fact and only surprising because it jars with our expectations. Among academics in maths and logic, the word has a narrower and more precise meaning: it refers to a statement or situation that gives rise to a self-contradiction. One such paradox, as we’ll see, led to an important breakthrough in a fundamental area of maths. Others, to do with the nature of self, free will, and time, have opened up fruitful discussions in philosophy and science.
The fourteenth-century French priest and philosopher Jean Buridan played an important role in encouraging the Copernican revolution – the idea that the Sun is at the centre of the solar system – in Europe. But his name is better known through its association with a paradox of medieval logic. Buridan imagined an ass that is standing exactly midway between two piles of hay that are the same in every respect – size, quality, and appearance. The ass is hungry but also unremittingly rational, so that it has no reason to favour one pile over the other. Thus conflicted it simply stays where it is, having no basis on which to make a decision, until it starves. With one pile of food it would have lived, yet with two identical piles it dies. How, if only pure reason is involved, can this make sense?
Buridan’s ass is in a predicament similar to that of a perfectly round ball balanced at the top of a steep, rounded hill. As long as no unbalanced forces act on it, there’s nothing to cause it to roll down the side. Its state is unstable, in that the slightest nudge will set it in motion. But without a nudge of any kind it will remain forever in place. Like many thought experiments, Buridan’s ass makes a number of assumptions that are never fully realised in practice. For one thing, it assumes complete symmetry: that the decision to choose one pile of hay or the other involves an identical series of states and steps. Yet this would never be the case in reality. The ass might habitually favour its right side over its left or, perhaps through a trick of the light, gain the impression that one pile looks slightly more appetising than the other. Any of a dozen different reasons might tip the balance in favour of one of the food piles. In a practical example from digital electronics, a logic gate may hang indefinitely midway between the values 0 and 1 (analogous to the bales of hay), until some random flicker of noise in the circuit causes it to flip into one of the stable states. Buridan’s ass has been used in discussions of free will, since it’s argued that a creature with free will, however rational, would never choose not to eat simply because there was no reason to favour one food source over another.
Another paradox that bears on the problem of free will was devised as recently as 1960 by William Newcomb, a theoretical physicist at the Lawrence Livermore Laboratory and great-grandson of the brother of the famous nineteenth-century astronomer Simon Newcomb. In Newcomb’s paradox a superior being, with predictive powers that have never been known to fail, has put $1,000 in a box labelled A and either nothing or $1 million in a box labelled B. The being presents you with a choice: (1) open box B only, or (2) open both box A and B. But here’s the catch: the being has put money in box B only if it predicted you’ll choose option (1). It put nothing in box B if it predicted you’ll do anything other than choose that option. The question is, what should you do to maximise your winnings? In fact, there’s no known consensus on what to do or even whether the problem is well-defined. You might argue that, since your choice now can’t alter the contents of the boxes, you may as well open them both and take whatever’s there. This seems reasonable until you recall that the being has never been known to predict wrongly. In other words, in some way, your mental state is correlated with the contents of the box: your choice is linked to the probability that there’s money in box B. These arguments and many others have been put forward in favour of either choice. But there’s no generally agreed-upon ‘right’ answer, despite the concerted attentions of philosophers and mathematicians for more than half a century.
Newcomb came up with his paradox while thinking about a slightly older one known as the surprise hanging, which seems to have begun circulating by word of mouth sometime in the 1940s. It concerns a man who’s been condemned to hang. A judge, with a reputation for reliability, tells the prisoner on Saturday that he’ll be hanged on one of the next seven days but that he won’t know (and won’t be able to know in any way) which day until he’s told on the morning of the execution. Back in his cell, the prisoner thinks about his predicament for a while and then reasons that the judge has made a mistake. The hanging can’t be left until the following Saturday because the prisoner would certainly know, if this day dawned, that it was his last. But if Saturday is eliminated, the hanging can’t take place on Friday either, because if the prisoner survived Thursday he would know that the hanging was scheduled for the next day. By the same argument, Thursday can be crossed off, then Wednesday, and so forth, all the way back to Sunday. But with every other day ruled out for a possible surprise hanging, the hangman can’t arrive on Sunday without the prisoner knowing in advance. Thus, the condemned man reasons, the sentence can’t be carried out as the judge decreed. But then Wednesday morning comes around and, with it, the hangman – unexpectedly! The judge was right after all and something was awry with the prisoner’s seemingly impeccable logic. But what?
More than half a century of attack by legions of logicians and mathematicians has failed to produce a resolution that’s universally accepted. The paradox seems to stem from the fact that whereas the judge knows beyond doubt that his words are true (the hanging will occur on a day unknown in advance to the prisoner), the prisoner doesn’t have this same degree of certainty. Even if the prisoner is alive on Saturday morning, can he be certain that the hangman will not arrive?
Sometimes our use of language, and especially a lack of precision when making statements or asking questions, can lead to perplexing problems. Berry’s paradox is named after George Berry, a part-time employee of the Bodleian Library at Oxford University, who, in 1906, drew attention to statements of the form: ‘The smallest number not nameable in under ten words.’ At first sight, there doesn’t seem anything particularly mysterious about this sentence. After all, there are only finitely many sentences that have less than ten words, and fewer still that specify unique numbers; so there’s clearly a finite number of numbers nameable in under ten words, and hence a smallest number N that is not. The trouble is, the Berry sentence itself is a specification for that number in only nine words! In this case, the number N would then be nameable in nine words, contradicting its definition of being the smallest number not nameable in under ten words. You could try picking a different number as N, but the paradox will still hold. What Berry’s paradox shows is that the concept of nameability is inherently ambiguous and a dangerous one to be used without qualification.
A paradox of a different kind deals with the notion of identity. We generally take identity for granted; for example, it seems obvious that the person known as Agnijo an hour ago is still the same person now. However, paradoxes can call our intuitive ideas about identity into question. One such paradox involves a thought experiment known as the Ship of Theseus. Legendary King Theseus, famous for his association with the story of the Minotaur, fought many successful naval battles so that the people of Athens, it’s said, honoured him by preserving his ship in port. Over time, however, the planks and other parts of the all-wooden vessel gradually rotted and had to be replaced, one by one. The question is: at what point, if any, does the ship stop being the Ship of Theseus and become instead a replica or a different entity in its own right? After one plank is replaced, or half of all the wood, or some other amount? Does the answer depend on the speed of replacement? If the old planks were then reassembled to form another ship, which one, if any, is the true Ship of Theseus? In modern times, the same kind of conundrum has been called the ‘Sugababes principle’. The Sugababes, an English band, formed in 1998 with Siobhán Donaghy, Mutya Buena, and Keisha Buchanan. Members gradually joined and left until in 2009 the members were Heidi Range, Amelle Berrabah, and Jade Ewen – all three original members had left. In 2011 Donaghy, Buena, and Buchanan formed a new band. Which one has more of a claim to being the ‘true’ Sugababes?
Such questions may seem unimportant in the case of an inanimate object – although archaeologists and conservationists may debate to what extent ancient buildings and artefacts that have been repaired and rebuilt can be said to be original or legitimate continuations of the original. But thought experiments along the lines of the Ship of Theseus take on a new dimension when applied to ourselves and, in particular, to the subject of personal identity. The time is fast approaching when it will be possible to replace almost every body part with an organic transplant (donated or lab-grown) or a prosthesis. If a large part of our body is replaced by various means, over time, are we still the same person in the end? The tendency might be to say ‘yes’ unless the substitution involves significant portions of the brain, because the brain is generally regarded as being key to who we are.
Of course, everyone would agree that if a person loses an arm in an accident and receives a prosthetic arm instead, they remain the same in every way that matters. Also, it’s true that the atoms, molecules, and cells that make up our bodies are changing to some extent every moment. In the time it takes you to read this sentence, about 50 million of your cells will have died and been replaced. If it’s a like-for-like swap and it happens over time, or we get a transplant or a prosthesis, we don’t worry about there being a threat to our identity. We also recognise that people age without becoming someone new. But what if the replacement happened all at once? What if every particle in your body, down to the atomic level, were suddenly swapped out for an identical copy?
Teleportation, in which particles (or, to be more precise, properties of particles) are made to vanish in one place and reappear instantly some distance away, is already possible with photons. It’ll probably be a long time before such ‘quantum teleportation’ can be achieved with larger objects. But let’s suppose that human teleportation becomes possible. You step onto a teleportation pad in, say, London, the position and state of every atom in your body is scanned in exquisite detail, and a moment later this information is used to reconstitute your body from a new collection of identical atoms in Sydney. The reconstruction is so swift and accurate that, aside from a moment of slight disorientation, you don’t notice that your old body in London has been dissolved, the component atoms recycled into the environment, and your new body fashioned a split-second later from identical atoms in identical states halfway around the planet. As far as you’re concerned you’ve just been whisked across more than 10,000 miles in the blink of an eye and can start your Australian adventure without the customary jet lag and tiredness that follows a day’s plane journey. You were even thinking exactly the same thought at the instant you were reconstructed as when your old body was dissolved at the London end. Two weeks later, it’s time to go home and you go through the reverse process of having your atoms disassembled in Sydney and an exact copy fashioned a microsecond later in the UK. You step off the pad, bronzed and relaxed, ready to head home. But at that moment you get a call on your mobile phone from a technician in Australia. There’s been a problem at the Sydney end and the ‘old’ you didn’t get dissolved. Instead he/she is complaining to the staff there that nothing happened, the teleportation failed, and they should either do the whole thing again or offer a refund. So now, it seems, there are two ‘yous’, identical in every respect down to the exact thought and memory at the moment the teleportation took place. Which is the real ‘you’? And how can you be in two places at once? What happens to your consciousness in such a situation? And what would it feel like for a single consciousness to be replicated in this way?
The technological barriers to human teleportation are immense and it’s by no means certain they’ll ever be surmounted. However, the feasibility of uploading the contents of our minds into a computer so that we achieve a kind of mental immortality is already being discussed. The ultimate goal would be not only to store all of our memories but also to recreate our consciousness, our active experience of self and the world around us, in an inorganic medium. The issue of what it would mean and feel like to be reconstituted in this way then becomes of central importance. If one copy of your consciousness could be made, then two or more could also be created – backups perhaps in case the main one was lost or damaged. Such possibilities will raise interesting personal and ethical dilemmas over the coming decades. They’ll also forge a direct link between mathematics and the mind. The way the uploading is carried out and the technology of the computational support system required will be the product of intense and complex mathematical analysis, as well as advancements in science and engineering. The outcome, if it happens, will be a new form in which human-level consciousness can exist and be sustained indefinitely. At that point, the ultimate expression of objective universality, drained of emotion and opinion – mathematics – will meet the essence of subjectivity, the feeling of ‘what it is like to be’.
Time is another great mystery around which the paradoxical swirls. The twins paradox is a thought experiment in which one twin (A) travels into space at nearly the speed of light and returns, after a long interstellar journey, to find that he’s aged much less than his twin (B) who remained on Earth. The slowing down of time, or time dilation, for an object moving at very high speed is a proven effect of Einstein’s special theory of relativity. The riddle posed by the twins paradox is why twin B doesn’t also age at a slower rate because he could be considered to be moving equally fast, in the opposite direction, if we switch to a frame of reference in which twin A is at rest. The fact is, though, that the roles played by A and B, despite appearances, aren’t symmetric. Twin A had to accelerate to reach a high speed, whereas twin B, back on Earth, underwent no acceleration. It’s this shifting of twin A out of Earth’s frame of reference that causes him to age at a different rate than his stay-at-home brother.
Travelling very fast is a proven way of jumping into the future, assuming we can develop the technology for super high-speed travel, though unfortunately it’s a one-way trip. We don’t know of any trick to hop back into the past, except perhaps by means of something exotic (and disturbingly unpredictable) like jumping into a wormhole – a hypothetical tunnel in space and time. But that hasn’t stopped people speculating about what might happen if we could make journeys backwards through time. One difficulty that might arise is that we end up changing something in the past that renders our future existence problematic. In Back to the Future, Marty McFly is hurled back to 1955 in a plutonium-powered DeLorean, only to encounter his mother-to-be as a hormonal teenager; he wisely avoids her amorous advances. We might go back and accidentally kill our grandfather when he was still a boy. That would mean we couldn’t be born and so couldn’t go on to become a time traveller who went into the past and caused the early demise of their grandfather. This ‘grandfather paradox’ is a classic argument against the possibility of going back in time. On the other hand, it’s been suggested that if we did go back we’d cause a split in the time line so that whatever we did in the past, as a result of our time machine exploits, would only happen along a new branch, completely separate from the original, thereby sidestepping any logical conflicts or endless loops.
Such conflicts and loops are not so easily avoided in other cases, however. Suppose these three sentences are written on a card:
(1) This sentence contains five words.
(2) This sentence contains eight words.
(3) Exactly one sentence on this card is true.
Is sentence (3) true or false? Obviously sentence (1) is true and (2) is false. If (3) is also true then two sentences are true, which immediately makes (3) false. But if (3) is false then it isn’t true to say that exactly one sentence on the card is true. However, in that case the only true statement is (1), which means (3) must be true. A statement can’t be both true and false at the same time. Can it be neither?
This little conundrum is similar to one credited to the sixth-century Greek seer and philosopher-poet Epimenides, who purportedly said: ‘All Cretans [people from the island of Crete] are liars’. Because Epimenides himself was a Cretan his statement implies that he also is a liar, so that, at first glance, what he says appears to be paradoxical. In fact, though, it isn’t, even if we assume that every Cretan either always lies or always tells the truth. Where some people make a mistake is that they know that if Epimenides is truthful then all Cretans, including himself, are liars (which is a contradiction) but assume that if Epimenides is lying then all Cretans, including himself, are truthful. This is false, because if Epimenides is lying this implies only that at least one Cretan is truthful, not necessarily all Cretans.
Epimenides’ statement, however, can easily be turned into a genuine paradox. The so-called liar paradox, credited to Eubulides of Miletus, of the fourth century bc, can be put as crisply as: ‘This statement is a lie’. It then follows that if it’s true then it’s false, and if it’s false then it’s true.
Different versions of the basic Eubulides liar paradox have popped up over the centuries. Jean Buridan used it in an argument for the existence of God. Just over a hundred years ago, the English mathematician Philip Jourdain offered a version in which two statements are written on opposite sides of a single card. On one side appears: ‘The sentence on the other side of this card is true’. On the other is a baffling counter to this: ‘The sentence on the other side of this card is false’.
No one has come up with an easy or single resolution of the liar paradox. Common reactions to it are to dismiss it out of hand as being a pointless game of words or to say that the sentence(s) involved, although grammatically correct, are devoid of real content. Both are attempts to stop the paradox dead in its tracks, but don’t stand up to scrutiny. The first just refuses to acknowledge that there’s any substantive problem. The second denies any meaning to the statement(s) on the grounds that they lead to a paradox. On the face of it, the statement ‘This statement is a lie’ is very similar to the one that declares: ‘This sentence is not in French’. How can the first be meaningless if the second makes perfect sense?
Apart from being interesting talking points, such brain-twisters don’t seem to serve much real purpose. But there’s one paradox, leading to a self-contradiction, which has had a pivotal influence on the development of one of the most fundamental areas of modern mathematics. The paradox in question is best understood in a form called the barber paradox. In this, there’s a barber who claims to shave everyone who doesn’t shave themselves. As a result he faces a dilemma: does he shave himself? If he does, he isn’t shaved by the barber so he doesn’t shave himself. If he doesn’t, he is shaved by the barber so he does shave himself. A more abstract form of this paradox appeared in a letter from the English philosopher and logician Bertrand Russell to the German philosopher and logician Gottlob Frege in 1902. The timing couldn’t have been worse from Frege’s point of view. Frege was just about to send the second volume of his monumental work Die Grundlagen der Arithmetik (The Foundations of Arithmetic) to the publisher. In his letter, Russell drew attention to a peculiar mathematical object: the set of all sets that don’t contain themselves. He then asked: does this set contain itself? If so, then it isn’t contained within the set of all sets that don’t contain themselves, which means it doesn’t contain itself. If not, it is contained within the set of all sets that don’t contain themselves, which means it does contain itself. Such a monstrosity, Frege realised in horror, could not be accommodated within the set theory he’d spent many years formulating and which now, it seemed, was broken and discredited before it had even seen the light of day.
Russell’s paradox, as it became known, exposed a fatal inconsistency of the ‘naïve’ set theory that Frege had developed. ‘Naïve’, in this context, refers to early forms of set theory that aren’t based on axioms and that assume there’s such a thing as a ‘universal set’ – a set that contains all objects in the mathematical universe. On reading Russell’s letter, Frege immediately grasped its implication. In reply to Russell he said:
Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic … It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.
The existence of this one paradox at the heart of Frege’s cherished theory meant that, effectively, all the statements the theory could generate were both true and false at the same time. It’s a simple fact that any system of logic, if found to harbour a paradox, is rendered useless.
The emergence of Russell’s paradox, at the dawn of the twentieth century, shook logic and mathematics to their very core. With a paradox lurking in their midst, no proof they could generate was ultimately trustworthy, no theory rooted in them was well grounded. Operationally, it’s true, maths could continue as it had always done. For everyday purposes no one was about to deny that 2 + 2 = 4 was true and that 2 + 2 = 5 was just as obviously false. But the disturbing fact remained that there was no way to prove these things, or anything else in maths for that matter, starting from what had been assumed to be firm mathematical bedrock – set theory, developed by the likes of Georg Cantor and Richard Dedekind (both of whom we’ll learn more about in Chapter 10 on infinity), David Hilbert (whom we first met in Chapter 1, and then again in Chapter 5 in connection with Turing machines), and Frege, in the form in which it existed in late Victorian times. The crumbling of naïve set theory began with a paradox to do with transfinite ordinals, known as the Burali-Forti paradox, although it was Cantor who first grasped its disturbing implications in about 1896. Then came Russell with his coup de grâce and it was clear that mathematicians must either abandon their faith in proof or find an alternative to naïve set theory. The former was unthinkable. So some way was needed to rebuild set theory from the ground up but in a way that excluded, from the very start, anything with a whiff of the paradoxical.
The answer lay in the development of what are called formal systems. In contrast to naïve set theory, which grew out of common sense assumptions and rules based on natural language, the new approach started by defining a specific set of axioms. An axiom is a statement or premise that’s put forward in precise terms and taken to be true at the outset. Different systems and authors are free to adopt different sets of axioms. But after the axioms have been declared in a formal system the only statements that can be said to be true or false within the system have to be constructed from those starting assumptions. The key to the success of formal systems is that by carefully choosing axioms in the first place anything as unwelcome and destructive as the liar paradox can be prevented from arising.
What is sometimes called a paradox may not actually be a paradox but merely a true statement that seems counterintuitive or a false statement that seems obvious. A classic example in mathematics is the so-called Banach–Tarski paradox, which states that you can take a ball, cut it up into finitely many pieces, and rearrange them to make two balls, each of the same volume as before. This seems crazy and, indeed, it’s important to understand that this isn’t a claim about what can actually be done with a real ball, a sharp knife, and some dabs of glue. Nor is there any chance of some entrepreneur being able to slice up a gold ingot and assemble in its place two new ones like the original. The Banach–Tarski paradox tells us nothing new about the physics of the world around us but a great deal about how ‘volume’, ‘space’, and other familiar-sounding things can assume unfamiliar guises in the abstract world of mathematics.
Polish mathematicians Stefan Banach and Alfred Tarski announced their startling conclusion in 1924, having built on earlier work by the Italian mathematician Giuseppe Vitali, who proved that it’s possible to chop up the unit interval (the line segment from 0 to 1) into countably many pieces, slide these bits around, and fit them together to make an interval of length 2. The Banach–Tarski paradox, which mathematicians often refer to as the Banach–Tarski decomposition because it’s really a proof not a paradox, highlights the fact that among the infinite set of points that make up a mathematical ball, the concepts of volume and measure can’t be defined for all possible subsets. What this boils down to is that quantities that can be measured in any familiar sense aren’t necessarily preserved when a ball is broken down into subsets and then those subsets reassembled in a different way using just translations (slides) and rotations (turns). These unmeasurable subsets are extremely complex, lacking reasonable boundaries and volume in the ordinary sense, and simply aren’t attainable in the real world of matter and energy. In any case, the Banach–Tarski paradox doesn’t give a prescription for how to produce the subsets: it only proves their existence.
Paradoxes can come in many different forms. Some of them are merely errors in our reasoning; others raise interesting questions about what we may take for granted. Still others can threaten to destroy an entire field of mathematics, but provide an opportunity for rebuilding it on more solid foundations.