Really enjoyed this book! ChatGPT suggested it for me and I bought it at Mr. K’s in Asheville for $5.
It’s interesting that I was drawn to this book because math was my hardest, and least favorite, subject in school.
It was also the source of tense struggle between my father and I as he attempted, God bless him, to help me complete my homework many nights at the kitchen table. Oh the pain! Oh the teenage obstinance!
Some memories from the book:
- There’s something I’m drawn to about thinkers and doers who managed to do memorable things (Erdos’ math prolifics in this case) while growing up under less-than-free and less-than-ideal societies.
- In this case, post WWI and WWII fascist Hungary for Erdos.
- He had a rather pure pursuit of mathematics which is highlighted by his rejection of lots of other things.
- This seems like a common quality of prolific and great thinkers/doers: They are almost completely focused on it.
- I.e., Erdos owned hardly anything, was painful to be around it seemed, completely dependent on others for basic needs so that he could focus on Ramsey theory problems e.g.
- I’m not saying this is aspirational, but one does need to appreciate that if you choose (or are chosen?) to do Great Things, you probably will do many fewer things, especially at the margins, and may be quite aloof or unthinking to those around you.
- Ramsey theory: “complete disorder is impossible. The appearance of disorder is really a matter of scale. Any mathematical ‘object’ can be found if sought in a large enough universe.”
- “‘In the TV series Cosmos, Carl Sagan appealed to Ramsey theory without knowing that’s what he was doing,’ Graham said. Sagan said people often look up and see, say, eight stars that are almost in a straight line. Since the stars are lined up, the temptation is to think they are artificially put there, as beacons for an interstellar trade route, perhaps. Well, Sagan said, if you look at a large enough group of stars, you can see almost anything you want. That’s Ramsey theory in action.’”
- Math is an infinite quest, according to Erdos, unlike physics, and there’s this big difference is applied mathematics and, I guess, theoretical mathematics. Super interesting! You could work on something, theorize it, and then many years later (thousands?!) someone could use apply that to something you could’ve never imagined. Incredible.
Discoveries in pure mathematics, which can seem unconnected to the real world, often turn out to have very practical applications, far from the minds of those who made them. That was the case with Stuart’s work. Historically, many of the brightest minds in mathematics have prided themselves on doing math that has no applications. Math for math’s sake was the rallying cry. They feared real-world relevance might distract from the pristine order and beauty that mathematics laid bare. When Euclid was investigating prime numbers, he was proud that they contributed nothing practical to Greek life. G. H. Hardy, too, reveled in his uselessness. “I have never done anything ‘useful,’” he once said, not as an apology, but in defiance. “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Hardy was a committed pacifist, who proudly claimed that his area of expertise, number theory, would never be used by the military. But he was wrong. In the past two decades, prime numbers, so thoroughly useless for 2,300 years, have found a home in the Pentagon as the basis of the military’s most secure codes.
“This is the remarkable paradox of mathematics,” observed commentator John Tierney. “No matter how determinedly its practitioners ignore the world, they consistently produce the best tools for understanding it. The Greeks decide to study, for no good reason, a curve called an ellipse, and 2,000 years later astronomers discover that it describes the way the planets move around the sun. Again, for no good reason, in 1854 a German mathematician, Bernhard Riemann, wonders what would happen if he discards one of the hallowed postulates of Euclid’s plane geometry. He builds a seemingly ridiculous assumption that it’s not possible to draw two lines parallel to each other. His non-Euclidean geometry replaces Euclid’s plane with a bizarre abstraction called curved space, and then, 60 years later, Einstein announces that this is the shape of the universe.”
Unlike Hardy, Erdős was more accepting of the applications of mathematics, even though such applications did not drive his own work. And he was realistic that in mathematics, as in the other sciences, “everything that can be used for good things can also be used for bad things. After all, the same differential equations which govern the spread of poison gases also govern how pollutants spread. So one can spread poison gases deliberately, but also one can prevent spreading of pollution.”
But when the interests of Erdős’s colleagues drifted away from pure mathematics, he made no secret of his disapproval. “When I wasn’t sure whether to stay a mathematician or go to the Technical University and become an engineer,” Vászonyi recalled, “Erdős warned me: ‘I’ll hide, and when you enter the Technical University, I will shoot you.’ That settled the matter.” When probability theorist Mark Kac had a paper published in the Journal of Applied Physics based on his work during the war at MIT’s Radiation Laboratory, Erdős sent him a one-sentence postcard: “I am praying for your soul.” Erdős was “reminding me,” Kac said, “that I might be straying from the path of true virtue, which, as a matter of fact, I was.”
Today, the distinction between pure and applied mathematics is more muddled than ever.
- The Monty Hall dilemma story was fascinating. Maybe the best part of the whole book for me.
- Basically, a woman in the 90s, Marilyn vos Savant, had the highest IQ in the world and a column in Parade magazine “read by millions every Sunday.”
- One day, she offers this brain teaser where you’re a contestant on a TV show. There are three doors and behind the three are two goats and a car. “You choose, say door 1, and the host, who knows where the car is, opens another door, behind which is a goat. He now gives you the choice of sticking with door 1 or switching to the other door? What should you do?”
- You then get an option: do you want to stay on your choice or switch?
- Here’s where the math comes in and everyone (very credentialed types) gets super mad:
- von Savant says that if you stick with your first choice, you odds of winning the car are 1/3 but if you switch doors your odds double to 2/3.
- How can this be?! cries the phD. The odds should be 50/50: there are two doors left: 1/2 chance of picking the car.
- “Imagine, she said, that just after the host opened the door, revealing a goat, a UFO lands on the game-show stage and a little green woman emerges. Without knowing what door you originally chose, she is asked to choose one of the two unopened doors. The odds that she’ll randomly choose the car are fifty-fifty. ‘But that’s because she lacks the advantage the original contestant had–the help of the host… If the prize is behind No. 2, the host shows you No. 3; and if the prize is behind No. 3, the host shows you No. 2. So when you switch, you win if the prize is behind No. 2 or No. 3. YOU WIN EITHER WAY! But if you don’t switch, you win only if the prize is behind door No 1.’”
- The phD cries:
- “‘You are utterly incorrect about the game-show question,’” wrote E. Ray Bobo, a Ph.D. at Georgetown, ‘”and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively toward the solution to a deplorable situation. How many irate mathematicians are needed to get you to change your mind?’”
- This line from von Savant is why this whole story is interesting to me.. It’s not the brain teaser, which is interesting, is her social/philosophical observation about why everyone got so mad that’s interesting:
- “When reality clashes so violently with intuition, people are shaken.“
This was a good book.