Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Most Math Problems Do Not Have a Unique Right Answer

devlinsangle.blogspot.com

Post reply on HN
Print view View original

21–30 of 71 posts

Re: Most Math Problems Do Not Have a Unique Right Answer

#21

Post by ssivark »

jordigh wrote: post #2

I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…

I kinda agree with you, but there are counter-examples. For eg: Grothendieck.

> One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

( taken form http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf )

Btw, 57 is now jocularly referred to as the "Grothendieck prime".

Re: Most Math Problems Do Not Have a Unique Right Answer

#22

Post by watwut »

calhoun137 wrote: post #18
jasode wrote: post #4

Earlier quoted context omitted.

>Riemann's notes ..., there's a computation of sqrt(2) to 38 decimal places for no discernible purpose. A while back I read about a mathematician taking an informal survey of other mathematicians. One of the questions was if they knew how to manually calculate square roots. (Apparently, the mathematician was part of an older generation that was taught this procedure in class.) Most mathematicians did not know how and…

There is a funny story about Gauss from the book Men of Mathematics, in which Gauss asked two of his students to compute a certain infinite series to 30 decimal places. After many days and nights, the students had a disagreement over what number was in the 30th decimal place, so they asked for help. After looking at the calculations for a few seconds, Gauss accurately pointed at the correct answer and went back to wh…

That does not mean he did some tedious calculation fast. It is more likely that he understood some special property of those series to determine which number is incorrect.

Re: Most Math Problems Do Not Have a Unique Right Answer

#24

Post by japhyr »

I have a three year old son, and it is absolutely fascinating to watch his mathematical understanding develop. I've been a math teacher my entire adult life so I've had plenty of experience watching older students develop their understanding. It's entirely different watching your kid develop their understanding from scratch.

I recently looked for some kids' books that would focus on the more interesting problems in math, rather than just counting. I was happy to find a few books that have helped him see math as more than just counting. My favorite so far is The Boy Who Loved Math: The Improbable Life of Paul Erdos. [0] I knew of Erdos, but I didn't know much about him. I learned from reading this book, and my kid loves it as well. He is fascinated with aging, and he now sees it as normal that someone would spend their whole life focusing on numbers.

We are also starting to enjoy Bedtime Math: A Fun Excuse to Stay Up Late. [1] The idea is to give your kid some interesting math problems to think about at bedtime. We've found that it's a good way to help him think about things other than the dark, and strange noises while he's falling asleep.

It's fascinating to watch this development. A few nights ago: "Did you know that one of the oldest questions people have asked is, How many stars are there in the sky?"

"No, I didn't know that!"

"How many stars do you think there are in the sky?"

"Eight!"

[0] - http://www.amazon.com/The-Boy-Who-Loved-Math/dp/1596433078

[1] - http://www.amazon.com/gp/product/1250035856

Re: Most Math Problems Do Not Have a Unique Right Answer

#26

Post by kps »

JibberMeTimbers wrote: post #8

As for the armor plating question, did the plate the engine because of its size? The engines combined make up a smaller surface area than the other places but it was still shot up quite a bit. It would be less weight and not to mention it would be more 'mission critical'.

I've read this story before, but asserting it was the RAF rather than the USAAF, which triggered my urban-legend detector.

In fact, the mathematical work was done by Abraham Wald¹, a Jewish refugee from Romania, at Columbia University in 1943. A reprint of his memos is available², as is a more accessible article about his work³; the specific one is Part V Subdivision of the Plane⁴ Into Several Equi-Vulnerability Areas:

“Thus, for the observed data of this hypothetical example [italics added], the engine area is the most vulnerable in the sense that a hit there is most likely to down the plane. The fuselage has a relatively low vulnerability.”

I haven't seen evidence that his work was applied to actual aircraft design using real data.

¹ http://en.wikipedia.org/wiki/Abraham_Wald

² http://cna.org/sites/default/files/research/0204320000.pdf

³ http://people.ucsc.edu/~msmangel/Wald.pdf

⁴ the flying kind

Re: Most Math Problems Do Not Have a Unique Right Answer

#27

Post by j2kun »

tedunangst wrote: post #25

How is adding armor to the engine area not a unique right answer? I'll leave you to figure out why that is the best solution. If there is only a single best solution, isn't that then the unique right answer?

It depends on the assumptions you make. For example, are the planes you're measuring a uniformly random sample of all planes you're interested in? The answer is different depending on how you answer this question.

Re: Most Math Problems Do Not Have a Unique Right Answer

#28

Post by cscurmudgeon »

ssivark wrote: post #21
jordigh wrote: post #2

I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…

I kinda agree with you, but there are counter-examples. For eg: Grothendieck. > One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendie…

I think people are taking this point too literally. One could still do math exercises at the abstract level which don't involve any arithmetic.

Re: Most Math Problems Do Not Have a Unique Right Answer

#30

Post by j2kun »

jordigh wrote: post #2

I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…

There's a difference between the training you need to be a world-class mathematician and what the rest of the world thinks of as "mathematics."

One of the key differences (being a mathematician myself) is that the big leaps of progress often come at the high level, often talking with others, and ignoring computations. Then when you have three hours to sit down and calculate, you go back and make sure your high-level ideas pan out. And you worry that you're making mistakes the whole time :)

The point is that you don't just sit and compute for its own sake, nor is there a time crunch, nor is there even a single "right" way to get your computations to go through. Often you can choose one of many routes to bound some quantity, or many different proof techniques that involve very different computations.

Post reply on HN