Breaking logic with self-referential sentences

I don’t especially like people who talk a lot about themselves, but I have a soft spot for sentences that do. Case in point: the self-referential sentence “This sentence is false.” I really like that one. You may think you’ve seen this sentence before, but in fact you’ve seen other sentences, each made of the exact same words in the exact same order. I admit it’s hard to tell apart all those sentences, each separately subverting its own lonely self!

(If you think that two sentences that have the same words in the same order must really be the same sentence, consider the following two-sentence passage: The other sentence in this passage is lying! The other sentence in this passage is lying! The two sentences have the same words in the same order, but they’re saying very different things. And now that I’ve written down those two mutually-referential sentences, I have to say I don’t like them at all. They’re kind of shrill and unpleasant, and they remind me too much of current events.)

Going Negative, part 2

Last month, when I gave some ideas about how to justify the law of signs, my focus was on the kind of explanation that works when kids first encounter negative numbers. But in a way I wasn’t being 100% honest, and my use of some farfetched examples (like the balloon-stealing clown) was a tip-off. I think that the real justifications of the law of signs — not the most pedagogically appropriate ones, but the most historically honest ones — come from the body of material the students will encounter later in their studies, long after they’ve learned, enthusiastically or reluctantly, to calculate products in the standard way. These are justifications teachers seldom talk about with their students, but I think they matter.

So this month I’ll talk about those other rationales, and try to resolve any remaining qualms you may have about the law of signs that stem from a sense of symmetry-violation. I’ll also discuss the option of chucking negative numbers entirely. (Seems extreme, but as a parent of two children, I can sympathize with this way of brokering the conflict between yes-it’s-negative and no-it’s positive. “You know what, kids? If you can’t agree on a restaurant, we won’t go out to dinner at all.”)

Going Negative, part 1

Minus times minus equals plus. / The reason for this we will not discuss.
— W. H. Auden, recalling a popular verse from his school days

Ever tried mixing together your two least favorite foods?  I suspect you haven’t.  Nobody mixes two noxious ingredients and expects the results to be tasty.  So why should numbers behave differently in the numerical recipe called multiplication?  What mystical two-wrongs-make-a-right alchemy removes the taint of negativity and makes the product of two negative numbers positive? It just don’t make no sense!

The pioneering sixteenth-century algebraist Girolamo Cardano had qualms about this alchemy, and toyed with the idea of defining the product of two negative numbers to be negative.  In the intervening centuries, legions of schoolchildren have been tempted to follow that road.  But the mathematical community staunchly insists that it’s the wrong road.  Why?

Bertrand’s Ballot Problem

Early leads don’t always lead to much. Remember Republican presidential candidate Rudy Giuliani? Back in 2008, the outspoken New Yorker polled so well that some pundits predicted he’d win the nomination, but his lead fizzled out before the convention. The same thing has happened with many other would-be presidents from both sides of the aisle over the past couple of decades. So you might be inclined to discount a candidate’s early lead in polls. In fact, this reasonable inclination was part of why FiveThirtyEight analyst Harry Enten didn’t see Donald Trump’s nomination coming; Enten figured that Trump would wind up being just another Giuliani. Oops.

So, how closely do we expect early polling in an election process to correspond with the final outcome? A classic problem in probability theory strips this real-world question down to manageable size.

“The Man Who Knew Infinity”: what the film will teach you (and what it won’t)

During my years as a mathematician, not one film-maker has tried to teach me how to write better articles. So I’m not going to tell Matt Brown, the writer/director of “The Man Who Knew Infinity”, what he should have done differently in a movie that, as the fine print on the poster reminds us, is merely based on the life of Ramanujan. If I knew as much about movie-making as Matt Brown does, I probably would have made the same choices he did.

But I am going to tell you, fellow-members of the movie-going public, what characteristics of the math life are conveyed by the film, and what characteristics aren’t. I’m not saying that the film in and of itself is inaccurate, but it does recycle some tropes about mathematics that you’ve probably seen in other movies about mathematicians and that give an inaccurate picture of mathematics. Along the way, you’ll meet the surprising base-ten expansion of the infinite product .9 × .99 × .999 × .9999 × … and learn what it has to do with Ramanujan’s story. (I’m going to assume that you’ve read my blog essay Sri Ramanujan and the Secrets of Lakshmi from last month, or that you already know something about the life and work of Ramanujan.)

Sri Ramanujan and the Secrets of Lakshmi

Srinivasa Ramanujan

What can you say about a thirty-two-year-old mathematician who died?  That he loved numbers and equations.  That he had a mysteriously intimate understanding of infinite numerical processes (infinite sums, infinite products, infinite continued fractions, and the like). That to the mathematicians of England, his ideas seemed to spring from nowhere — while he himself said that his ideas came from a goddess.

The collaboration that took place in the years 1914–1919 between the Indian mathematician Srinivasa Ramanujan and the English mathematician G. H. Hardy — perhaps the most famous collaboration in the history of mathematics — is the subject of the 1991 book The Man Who Knew Infinity by Robert Kanigel.  It’s also the subject of a new film by the same name, which I’ll say more about next month.  Today I want to show you three “flowers from Ramanujan’s garden” (to steal a metaphor from Freeman Dyson), with some puzzles strewn along the way, and to explain why Kanigel’s choice of title makes sense. I’ll also speculate a bit about where mathematical ideas come from, and here the case of Ramanujan can be instructive, not because he was typical but because he was such an outlier; it’s hard to think of a parallel example of someone who came up with so many beautiful ideas but had so much trouble leading others to the wellsprings of his inspiration.  (His early death was certainly the greatest and most final source of this trouble, but it wasn’t the only one.)

Fermat’s Last Theorem: The Curious Incident of the Boasting Frenchman

The story as told by math popularizer Simon Singh in a Numberphile video goes as follows: There’s this 17th century French mathematician, Pierre de Fermat, sitting in his private library reading a book.  He excitedly records in the margin a new discovery he’s made — the assertion we now call Fermat’s Last Theorem, or FLT for short — but he writes that the margin is too small to contain his proof, and then, before he can communicate the details to anyone, “he drops dead.”

Pierre de Fermat.

I have two problems with this version of the FLT story, and the way it shows a character’s death preventing the revelation of a vital secret.  First, we’ve all “seen that movie” many, many times (see the TV Tropes entry on the His Name Is … trope, as well as the related tropes Conveniently Interrupted Document  and Lost in Transmission); it’s kind of hokey, isn’t it?  Second, there’s no evidence that Fermat wrote the passage right before he died.  We can’t go back and date the ink he used (or breathalyze the page for alcohol content, for that matter), as the book was lost after his annotations were transcribed, but we know from Fermat’s correspondence that he read the book fairly early in his career, in the 1630s. Most scholars agree that this particular remark was written two decades before he died. So Singh is putting an over-dramatic spin on things at the 02:15 mark. Still, he is right about Fermat dying without revealing a proof of his claim to anyone.

Andrew Wiles.

Thanks to modern-day mathematician Andrew Wiles, who this month receives the prestigious Abel Prize for his work on Fermat’s Last Theorem, we know that Fermat’s claim was correct.  But did Fermat have a proof?  That’s the question I want to explore today. And while we’re discussing the most famous proof Fermat never revealed, I’ll tell you about the proof method that Fermat did reveal — one that beautifully solves other problems of the same kind.