Most Math Problems Do Not Have a Unique Right Answer • Hacker News Board

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

#41

Post by skierscott » Sat, Aug 02, 2014, 5:16 PM UTC

hebbarp wrote: post #31

Most Math Problems Do Not Have a Unique Right Answer The title is both right and wrong. You are actually comparing school or college math with math applied in real world. School or college math works with few variables, for instance, and we consider most others remaining constant. Rarely have I seen school or college level students working with, say, derivatives of more than three variables. School or college math is…

> Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :)

I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely real numbers to describe one solution.

I've written a blog post on this topic[3]; that blog post works through all the underlying stuff before getting to these closed form solutions.

[1]:https://en.wikipedia.org/wiki/Nonlinear

[2]:https://en.wikipedia.org/wiki/List_of_nonlinear_partial_diff..., https://en.wikipedia.org/wiki/Closed_form_solution

[3]:http://scottsievert.github.io/blog/2014/07/31/common-mathema...

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

#42

Post by wolfgke » Sat, Aug 02, 2014, 5:37 PM UTC

jasode wrote: post #37

michael_nielsen wrote: post #34

Earlier quoted context omitted.

I'm a bit surprised by this. Most mathematicians know Newton's method. And square roots are trivial to calculate using Newton's method: (1) Goal: compute square root of x. (2) Make a guess, g. (3) Compute an updated guess, g' = (g + x/g) / 2. (4) Repeat step (3) until you achieve the desired accuracy. It's not incredibly quick, but it works and it's easy (though tedious) to carry out. Lots of other fun approaches her…

>I'm a bit surprised by this. Most mathematicians know Newton's method. The mathematician was probably talking about this procedure that does not require convergence: http://en.wikipedia.org/wiki/Methods_of_computing_square_roo... This is the method I also learned (at one time and then forgot). Apparently, this procedure was part of a typical math curriculum many decades ago. Like Latin and cursive handwriting, it's…

Looking over it, it just seems to be a special case of building nested intervals using the monotonicity and continuity of the sqrt function.

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

#43

Post by jordigh » Sat, Aug 02, 2014, 5:37 PM UTC

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…

Heh, I knew someone would come up with the Grothendieck prime example. That's why I said, "the computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise." I think Grothendieck certainly was a master of these and performed them regularly.

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

#44

Post by [deleted] » Sat, Aug 02, 2014, 5:37 PM UTC

[deleted]

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

#45

Post by jordigh » Sat, Aug 02, 2014, 5:41 PM UTC

j2kun wrote: post #30

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…

I think if you're already doing computations all the time as a matter of course, there's no need to specifically set time aside to do them. But would you agree that it's necessary to know how to do them? Perhaps when working in an unfamiliar area, you take some time to learn how to compute some things in this field?

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

#46

Post by skybrian » Sat, Aug 02, 2014, 5:41 PM UTC

skierscott wrote: post #41

hebbarp wrote: post #31

Most Math Problems Do Not Have a Unique Right Answer The title is both right and wrong. You are actually comparing school or college math with math applied in real world. School or college math works with few variables, for instance, and we consider most others remaining constant. Rarely have I seen school or college level students working with, say, derivatives of more than three variables. School or college math is…

> Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :) I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely re…

I'm confused by: "If f(x)=mx+b, f(x+y)=f(x)+f(y)+b". Shouldn't that be f(x)+f(y)-b?

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

#47

Post by ecesena » Sat, Aug 02, 2014, 5:58 PM UTC

skybrian wrote: post #46

skierscott wrote: post #41

Earlier quoted context omitted.

> Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :) I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely re…

I'm confused by: "If f(x)=mx+b, f(x+y)=f(x)+f(y)+b". Shouldn't that be f(x)+f(y)-b?

Yes, it's a misprint.

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

#48

Post by qnaal » Sat, Aug 02, 2014, 6:09 PM UTC

The idea that mathematics as a whole is intuitive (as opposed to specific facets of its machinations) is as pervasive as it is misleading.

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

#49

Post by ecesena » Sat, Aug 02, 2014, 6:09 PM UTC

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 want to add something here: great mathematicians compute too.

I totally agree. To me, there are 2 kinds of algorithms that one should get used to. 1. is computing fast enough (that doesn't mean super fast, but sometimes it means hours instead of months), and 2. is spotting out errors, both evident and subtle.

This said, I'm certainly not in the set of great mathematicians.

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

#50

Post by tedunangst » Sat, Aug 02, 2014, 6:11 PM UTC

j2kun wrote: post #27

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.

That sounds like different problems have different solutions. "Most math problems are (initially) poorly defined" would have been a better title, I guess.