Learning Strategies

Catching up on posts about recent reading:

I highly recommend Case of Lies, by Perri O’Shaughnessy, especially if you are interested in math or linguistics. If you’re not, it’s still a solid mystery, well above average for the genre even though there are a lot of holes in the plot.

“It seemed to be about two different things, math and murder,” complains one reviewer. Well, yes — but he forgot to add, “...not that there’s anything wrong with that.” Math and murder: sounds like an excellent combination to me, especially when most of the math is primarily number theory and cryptography. The rest of it includes probability, combinatorics, and the history of math, taking off from the famous Bringing Down the House: The Inside Story of Six M.I.T. Students Who Took Vegas for Millions, by Ben Mezrich. There are a few errors, but not enough to spoil the effect. Let’s look at a few carefully chosen quotations from the book — chosen from many possibilities to highlight the math and a bit of linguistics, not to be in any sense representative of the entire novel:

He told Elliott about Sanskrit. Linguistics wasn’t about languages, it was about logic. Pop showed him how to diagram Sanskrit grammar so the little x’s and y’s added up to a sentence... Subject plus verb plus direct object equals a sentence. In English, anyway. English moved like a number line, marching to the right. But there were other languages that put the direct object first, or even the verb. X still described the subject, y still described the verb. Math described language; wow!

 

To him these four numbers were as real as rocks, more real, alive in some sense. But what were they? What was a number? Where did numbers come from? Had humans invented them or discovered them? Where did they go? He thought they followed a line toward some far infinity where a little breeze sprang up and supported them.

 

“You can’t divide by zero, Elliott. It’s a rule.”
“Why is it a rule?”

“Because the rest of arithmetic won’t work otherwise. You just have to accept it.”

“I thought math was supposed to be logical.”

“It is.”

“Then how come multiplying by nothing wipes out a number?”

“Talk to me after class.”

 

“I know what an exponent is,” he boasted. “I know what a square root is. What’s the square root of minus one?”

“This is way beyond third-grade arithmetic,” Mr. Pell said. “Who told you to ask me these questions?”...

“Nobody. My pop. He’s a Sanskrit scholar. What’s the square root of minus one?”

“You know what? I bet your father already told you the answer, told you it’s an imaginary number with its own number line.”

“Egg-zackly. So if you can set up a brand-new number line for negative square roots, why can’t you set up a new number line for one divided by zero?”

 

“Then somebody said, ‘Let’s try that triangle out with a side that measures a single unit, a one,’ ” Pop said. “They tried it out. And a devil sprang out! Because one squared plus one squared equals two. Therefore the hypotenuse was the square root of two.” He leaned toward Elliott and said in a chilling theatrical whisper, “And that number couldn’t exist.”

“Wow!” Elliott said.

“That thing, that square root of two, couldn’t be described as an integer or a ratio. It completely contradicted the beautiful universe the Pythagoreans had constructed. Now they had a choice — to accept this ugly thing into their system and work with it, or to try to suppress the fact that it existed. To lie about it, because the Pythagorean religion could not encompass something as ill-formed, as unlocatable as this.”

“So what did they do?” Elliott’s mother asked.

“They swore the whole brotherhood to strict secrecy. This secret made a mockery of their beliefs. Now their religion was based on a lie.”

 

He read everything he could about the attempts to find a formula to predict the primes. The geniuses of mathematics, the smartest people who ever lived, had tried to understand the primes, and been defeated. Some had lived long, quiet lives, but many who flirted with the primes had fallen while very young: Gauss, who left math forever in this twenties; Ramanujan, the vegetarian Brahmin who died at thirty-two; Gödel, who starved himself to death; Nash, teeting on the edge of the void most of his life; Grothendieck, still alive, cloistered in a hut in the Pyrenees, obsessed with the devil; Turing, who killed himself at forty-one by eating a cyanide-laced apple.

And the greatest of them all, in Elliott’s mind at least, Bernhard Riemann, who died in Italy at thirty-nine. Because of pleurisy, the books said, but Elliott figured he had died because the heat in him had died. Riemann had simply gone as far as he could. He had found a possible order in the primes and given the world a direction in the Riemann Hypothesis. It made sense to die then.

 

“...As soon as I had that, I could factor really large non-prime numbers, too, as a simple corollary. Are you understanding any of this?”

“I’m not sure. But you sound autthoritative. That’s half the battle.”

“Maybe for lawyers. You only have to be convincing. Not in math. In math you have to be right.”

 

“Well, really big numbers can’t be factored — nobody can find the primes they’re made of — even with today’s computers. So a company called XYC invented a method of encoding financial and other information using that fact, so information couldn’t be hacked as it traveled from one Web site to another. The code lets you type in your credit-card number for certain eyes only.”

Not exactly standard material for a mystery!

Labels: books, linguistics, math