Once Upon a Prime, page 23
All his fiction and poetry has a reductio ad absurdum flavor to it that in fact is common to both mathematics and children’s games of make-believe. If, for example, we assume that you can grow and shrink at will (or by eating cake and drinking potion), then it would be possible to swim in a lake of your own tears, as Alice does shortly after falling down the rabbit hole. You observe the internal logic of the game—precisely as mathematicians do. We agree on the ground rules of our mathematical playground, and then we explore.
Mathematically, pushing assumptions to their logical limits in the hope that they’ll break is a common proof technique. The trick is to assume the opposite of what you think is true. Already it’s a bit Through the Looking-Glass. This was how we proved way back in Chapter 1 that there are infinitely many prime numbers. We assumed there weren’t, in which case there would be a finite list containing all the prime numbers, and then we deduced from that the existence of a prime number not on the list, which would be impossible. This is a true mathematical reductio ad absurdum, a trick we call “proof by contradiction.” In a similar vein, Alice’s encounter with the Mock Turtle features, as well as some very silly puns, a sequence that Alice tries to take to its logical conclusion. The Mock Turtle is recounting his school days, when they studied “all the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
“And how many hours a day did you do lessons?” said Alice.…
“Ten hours the first day,” said the Mock Turtle: “nine the next, and so on.”
“What a curious plan!” exclaimed Alice.
“That’s the reason they’re called lessons,” the Gryphon remarked: “because they lessen from day to day.”
This was quite a new idea to Alice, and she thought it over a little before she made her next remark. “Then the eleventh day must have been a holiday?”
“Of course it was,” said the Mock Turtle.
“And how did you manage on the twelfth?” Alice went on eagerly.
“That’s enough about lessons,” the Gryphon interrupted in a very decided tone.
And no wonder, because in this plan, they would have had to learn for fewer than zero hours on every subsequent day.
There is a good deal of absurd arithmetic in Lewis Carroll’s verse too. In his poem “The Hunting of the Snark” (subtitled “An Agony in 8 Fits”), ten crew members, all of whose names begin with b, set out on a voyage to track down the snark, though ultimately they fail because it turns out that all along the snark was a boojum. At one point the Beaver is struggling to work out how to add two to one to get three. The Butcher steps in to help. He “explained all the while in a popular style, which the Beaver could well understand”:
Taking Three as the subject to reason about—
A convenient number to state—
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.
The result we proceed to divide, as you see
By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true.
At first glance this looks, to use a technical term, like a load of nonsense. But in fact it is a clever little mathematical trick, a series of precise logical steps that leads inexorably (if ridiculously) to the right answer. The Butcher is trying to show that 3 is the answer to the fiendish sum 2 + 1. So he starts with 3 and then does a bunch of arithmetical calculations that if you follow them through carefully, actually bring you back precisely to 3 again. But even better, this works whatever number you start with. If I start with my favorite number, 4, I’ll end up with 4. Try it: We add 7 and 10 to our number. So we have 4 + 17 = 21. Then multiply this by 1000 - 8, which is 992. Then we divide by 992. So far, then, we have found which is just 4 + 17. The final instruction is to subtract 17, which brings us back to 4. Wherever you start, the answer must indeed be exactly and perfectly true.
There’s one specific number that Lewis Carroll seems to have been a little obsessed with: 42. It crops up all over the place in his writing. In Alice’s Adventures in Wonderland (which happens to have forty-two illustrations), the King of Hearts, infuriated by the rapidly enlarging Alice’s disruption of court proceedings, reads out from his notebook, “Rule Forty-two. All persons more than a mile high to leave the court.” When Alice follows the White Rabbit into his hole, she tumbles down a very deep well, and keeps falling down and down. She wonders if she will fall right through the earth. It’s a curious mathematical fact that falling through a tunnel leading between any two points on the earth’s surface takes a constant amount of time (being a pure mathematician, I ignore prosaic things like friction and air resistance). Guess how long it would take Alice to fall all the way through the earth to the other side? You got it: forty-two minutes.
There are other possible 42s hidden in Through the Looking-Glass. While Alice’s Adventures in Wonderland is full of playing cards, in Through the Looking-Glass the theme is chess. The entire book has the structure of a chess game, white versus red, with Alice moving through a board laid out over the fields. Alice meets several of the pieces during the course of her adventure, which it’s possible, says Lewis Carroll, to play as a real game of chess with Alice as a pawn who crosses the board to become a queen. During one conversation, Alice tells the White Queen she is 7 and a half years old exactly, or 7 years, 6 months; 7 times 6 is, of course, 42. The Queen’s age is much greater: 101 years, 5 months, and a day. How many days is that? The answer depends on where your leap years go, but the highest attainable total is 37,044. Is this a randomly chosen number? Perhaps. But the Red Queen and the White Queen, being from the same chess set, are presumably the same age. So their combined age is 74,088 days. What of that? Well, it’s exactly I struggle to believe that this is a coincidence.
I’ve never read a convincing explanation as to why Lewis Carroll was so fixated on 42. In spite of his enthusiasm for logic, I suspect he just took a shine to it. But there is one school of thought that says there might be a religious interpretation. In the preface to “The Hunting of the Snark,” for example, we hear how the crew’s rules had entangled them in a logical impasse:
Rule 42 of the Code, “No one shall speak to the Man at the Helm,” had been completed by the Bellman himself with the words “and the Man at the Helm shall speak to no one.” So remonstrance was impossible [and] during these bewildering intervals the ship usually sailed backwards.
Some people believe that the number 42 is a reference to an important religious document, the Forty-Two Articles of Thomas Cranmer, which laid out important doctrinal rules of the Church of England. Lewis Carroll was an Anglican priest, so he would certainly have been familiar with this document. Article 42, by the way, is “All men shall not be saved at the length.” Make of that what you will.
The number 42 has become much better known in the last forty-two years (or thereabouts: the TV series appeared in 1981) for its role in Douglas Adams’s book The Hitchhiker’s Guide to the Galaxy. He may have been inspired by Lewis Carroll—after all, the episodes of the original radio series, on which the TV series and books were based, were named Fit the First, Fit the Second, and so on, just like the parts of “The Hunting of the Snark.” In The Hitchhiker’s Guide, an alien civilization of hyperbeings creates a huge computer called Deep Thought that takes seven and a half million years (a million Alice ages) to determine the answer to “Life, the Universe, and Everything.” After those eons have passed, Deep Thought reveals that the ultimate answer is 42. The problem then becomes to work out, in a sort of existential version of Jeopardy!, what is the question?
Let’s address a final arithmetical mystery. This one combines numbers with Carroll’s favorite pastime: setting up a mathematical chain of events that leads down a logical rabbit hole. When Alice arrives in Wonderland, she starts to doubt her own sanity because everything is so confusing. She decides to see if she still knows reliable things like multiplication tables. “Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate!” Poor Alice—but what does she mean, she’ll never get to twenty? The prosaic interpretation is that traditionally we learn our times tables only up to twelve, and, following her pattern, if and then and Since we stop at twelve, we won’t reach
But there’s a much more mathematically interesting interpretation, and that’s trying to find a scenario in which 4 times 5 really is 12. This isn’t as ridiculous as it seems when you remember that clocks follow an arithmetic in which 6 plus 8 is 2. What I mean is, if you add eight hours to six o’clock, you don’t arrive at fourteen o’clock (unless you are in the military and have to use the twenty-four-hour clock), but two o’clock. In some situations, then, it’s legitimate to say that
Another way to get surprising answers to sums is to work in an unexpected base. In our usual base 10, we write numbers in terms of powers of 10 (units, 10s, 100s, 1,000s, and so on), so that 1101 in base 10 means one thousand one hundred and one. But in binary or “base 2” arithmetic, we work in powers of 2 (units, 2s, 4s, 8s, and so on). This time, 1101 means 8 plus 4 plus 1: in other words, 1101 equals 13. And we can write sums that look crazy but are correct: 1 + 1 = 10, or 10 + 1 + 1 = 100. Computer programmers occasionally use a base 16 system (hexadecimal). In hexadecimal, 14, say, would mean 1 lot of 16 plus 4 units. In other words, 20. So in hexadecimal, we can correctly write that The game now becomes, in what number base is it true that ? It turns out the answer is base 18, because in base 18, 12 means one lot of 18 plus 2, which indeed is twenty. What about ? Here we need base 21, because is 24, and 13 in base 21 means one lot of 21 plus 3, which is the required 24. This pattern continues nicely if we add 3 each time to the base. We get
(base 24)
(base 27)
(base 30)
The pattern continues up to 4 times 12, which indeed does equal 19 in base 39 (one lot of 39 plus 9). But frabjous day! Callooh! Callay! We can never get to 20 like this. Four times 13 is 52, and to fit the pattern the next base should be 42 (there it is again). But in base 42, writing “20” would represent two 42’s, which is 84. So we really will never get to 20. I especially like that the pattern breaks not only at base 42, but also when the total reaches 52—which is the number of playing cards in a pack. It’s a nice reference to the later appearance of cards, like the Queen of Hearts, as characters in the story.
The examples I’ve shown you are glimpses of the common thread that runs through all Lewis Carroll’s writing, both mathematical and nonmathematical: a drive to understand the power and possibilities of logic. As well as his children’s books, he invented games and puzzles, many for children, that were aimed at teaching the laws of logical inference, starting with the most basic syllogisms (All men are mortal; Socrates is a man; therefore Socrates is mortal) and moving to sequences of deductions with as many as a dozen or more sentences strung together.
Carroll’s fiction, including the Alice books, is for me simply another facet of his lifelong exploration into how far we can get by setting the scene and then following the logic. The discussions of words and their meanings in the Alice books is one telltale sign of these mathematical undercurrents. For a mathematician, there is resonance in Humpty Dumpty’s remark “When I use a word it means just what I choose it to mean—neither more nor less.” In mathematics we have to be absolutely clear about the meanings of the words we are using, and must not load them with unspoken qualities. It’s not mere pedantry—any ambiguities risk tying us in logical knots and can even mean our deductions are false. It doesn’t matter what names we give to our new concepts, but we have to be careful to get the definitions right. As I mentioned earlier, all sorts of things go wrong if our definition of prime numbers allows 1 to be prime. Just like Humpty Dumpty, a mathematician’s words must mean no more, or less, than what they say.
Lewis Carroll, along with other Victorian mathematicians like John Venn (of Venn diagram fame), was interested in taking this precision further and codifying the very processes of logic itself. This “symbolic logic” allows you not just to look at whether individual statements are true or false, but to make deductions about the truth or falsity of statements made by connecting them with words like “and,” “or,” or “implies.” Even these simple words can trip us up if we aren’t careful. The word “or,” for instance, can mean different things according to context. Don’t believe me? Would you like a cup of tea or a cup of coffee? In that sentence, we all know “both” isn’t encompassed in “or.” On the other hand, a job advertisement saying that applicants should be fluent in Spanish or Portuguese would presumably not exclude people fluent in both languages. In normal speech we can tell from context which meaning is intended. If you are trying to create a set of rules of logic that covers all possibilities, you don’t have that luxury.
The “symbolic” bit of “symbolic logic” comes from the fact that we use symbols for words like “or” and “and,” and with them we construct a sort of algebra of logic. The aim is to be able to extract all possible logical conclusions from a collection of statements. One example given by Carroll is these two sentences: “No son of mine is dishonest” and “All honest men are treated with respect.” Now, we don’t concern ourselves with determining whether these sentences are true or false. Our job is to say what can be deduced, assuming they are true. Carroll describes how this is just one instance of a more general archetype, of the form “No x is not y, and every y is z.” If both these are true, then it must follow that “no x is not z.” Carroll represents these ideas both with diagrams and with symbols. In symbolic form, using his notation, we have the rather forbidding (here, means “and,” and means “therefore”). Once we know this general formula, we can apply it in our special case, where x is “my sons,” y is “honest,” and z is “treated with respect.” And hey presto, we can deduce that “no son of mine fails to be treated with respect.” Carroll assures us that it gets easier with practice!
This example comes from a book that Lewis Carroll wrote, aimed at popularizing symbolic logic for the general public. In the introduction, he extols its virtues thus:
Mental recreation is a thing that we all of us need for our mental health.… Once master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest.… It will give you … the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you!
Lewis Carroll’s contributions to the academic study of symbolic logic were valuable and important. As befits his character, he was adorably enthusiastic too about bringing the joys of that subject to the masses. In spite of his valiant efforts, though, I’m sorry to say that it did not really catch on as a fun family pastime.
I can’t resist ending this chapter with a story that may be apocryphal but is so good it deserves to be true. Believing unlikely things is, after all, just a matter of practice, says the White Queen: “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.” Legend has it, anyway, that Queen Victoria was so delighted with Alice that she asked to be sent the very next book Mr. Carroll produced. History does not relate her reaction on receipt of An Elementary Treatise on Determinants, with Their Application to Simultaneous Linear Equations and Algebraical Geometry. One suspects she was not amused.
10
Moriarty Was a Mathematician
The Role of the Mathematical Genius in Literature
In the bestselling Millennium series, there’s a scene at the start of the second book in which the hero, Lisbeth Salander, comes up with a short proof of Fermat’s Last Theorem. This may be the most famous misnomer in mathematical history. The mathematician Pierre de Fermat, undoubted genius though he was, made a great many claims that he didn’t back up with proof. This “theorem” of his was one such statement. In mathematics, we call these things conjectures. Most of Fermat’s conjectures were resolved either by Fermat himself or by other mathematicians within a few years. But this particular one was not—hence the “last.” What made it even more enticing was the accompanying marginal note—“I have a truly marvelous proof, but this margin is not big enough to contain it.” Hundreds of mathematicians tried to find this proof, but as the decades stretched into centuries, nobody succeeded. Even partial progress involved major new mathematical advances, far beyond anything Fermat could have had in mind. Eventually, in one of the great achievements of the last half century, the problem was cracked, and a proof found, by Andrew Wiles in 1993, using brilliant, beautiful, incredibly sophisticated mathematical machinery.
Anyway—we are asked to believe that Lisbeth Salander, genius hacker with no mathematical training, has proved what I propose should be called not “Fermat’s Last Theorem” but “Fermat’s midlife boast.” This gambit, of course, is shorthand for Salander’s being a maverick genius, with perhaps a side order of being good only at logic, not human emotion. The author might just as well have written, “Insert evidence of extreme cleverness here.”1
In this final chapter of the book, I’m going to show you some of the ways that people who do mathematics have been portrayed in literature. All too often, as mentioned briefly in Chapter 8, we see the trope of the emotionless, uncaring, obsessive, even insane mathematician. This stereotyped portrayal does mathematics a disservice, perpetuating the idea that only “freak” geniuses can be mathematicians when really everybody can delight in the fascination of mathematics. More sympathetic portrayals are out there, and I’ll show you some of these too, from Aldous Huxley’s heartrending “Young Archimedes” to Alice Munro’s Too Much Happiness, a mesmerizing fictionalized account of the life and death of the Russian mathematician Sofya Kovalevskaya.
