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

51–60 of 71 posts

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

#51

Post by ColinWright »

wolfgke wrote: post #42
jasode wrote: post #37

Earlier quoted context omitted.

>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.

Not really. It's a generalization of the long-division algorithm. In that we remove lumps and then work on the remainder, ensuring that at each stage we've removed the largest lump possible given the number of digits we can have used so far. Then we recurse on the remainder.

In the square-root algorithm we remove the largest square possible at each stage, and work on the remainder. There's a hitch, because there's more detail to carry over at each stage. We do that with a linear factor of what we've already removed (that's the bit where we multiply the quotient so far by two) and then add the "epsilon" for the next lump to remove.

This can all be derived from first principles using the fact that (x+a)^2 = x^2 + 2ax + a^2. I did that when I was about 13.

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

#52

Post by djokkataja »

tedunangst wrote: post #50
j2kun wrote: post #27

Earlier quoted context omitted.

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.

I agree with your title suggestion. The author phrases the problem in an intentionally misleading way: "They examined the fuselages of all the damaged planes that had flown back after being hit to see where the most damage was. It turned out that the engines had an average of 1.11 bullet holes per square foot, the fuel system had 1.55, the fuselages 1.73, and the rest of the plane 1.8."

He says that they checked to see where the most damage was, but really they were checking to see where the least damage was. If the planes were able to fly back with terrible damage to everything except for one part, then adding armor to that part is a pretty obvious solution.

On the other hand, figuring out exactly how much armor to add is much more complicated, because you have the constraints of weight, production time, etc, plus you have to have some kind of estimate for the effectiveness of the armor in actual combat. I would probably go with "Most Real-World Math Problems Do Not Have a Unique Right Answer".

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

#53

Post by NAFV_P »

michael_nielsen wrote: post #34
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…

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 believe this is based on an old Greek method...

  #define DECK 11

  double z() {
    const double s=2.0, c=0.1;
    double arr[DECK], low=1.0, up=s, inc=c;
    int j, k;
    for(j=0; js ) {
          low=*(arr+k-1);
          up=*(arr+k);
          break;
        }
      }
      inc*=c;
    }
    if( s-low*low
This particular example only works to fifteen decimal places (roughly the limit on the precision of a double).

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

#54

Post by james1071 »

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…

Agree-have to feel comfortable with numbers to do anything at all.

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

#55

Post by dbecker »

j2kun wrote: post #35
dbecker wrote: post #23

Since uncountably many problems have unique right answers, I'd guess the cardinality of the set of problems with unique right answers is the same as the cardinality of the set of problems without. :)

Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.

Though, if we apply your requirement, the set of problems without a unique answer must be countable as well. So, the cardinality of problems with a unique answer is still equal to the cardinality of problems without one.

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

#56

Post by j2kun »

jordigh wrote: post #45
j2kun wrote: post #30

Earlier quoted context omitted.

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?

The computations just get more technical. So in research it's rare to "know" a computation in the sense that I think you mean because it's different almost every time you apply it. A common example is rounding linear programs to integer solutions for the purpose of approximation (or exact solutions, as the case may be). There is almost no way to tell in advance if it will work, and most problems require significantly different versions of this technique. There are many textbooks showcasing the many variations. So you might need days or weeks before you figure out if it will work on your problem, and then it's a "standard calculation" only in retrospect.

People really don't compute things "on the fly" in mathematics. When you hear stories about Gauss these are folk legends about people with extraordinary practiced abilities, and the majority of the mathematical world doesn't work like that. People do the grunt calculations offline so they can spend their time in talks/discussions doing actual work.

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

#57

Post by thret »

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…

sqrt(2) sounds like a doodle to me, something you do to distract yourself while considering another problem.

For some reason I developed the habit of sketching an Escher Cube* (with a cross in the middle) when I was in highschool, I still do it when my mind is blank.

*Perhaps it isn't Escher after all, I cannot find an image online. It is an impossible 3d cube where each corner crosses through the centre.

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

#58

Post by skierscott »

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?

Thanks for that -- fixed.

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

#59

Post by pbhjpbhj »

nileshtrivedi wrote: post #36
japhyr wrote: post #24

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 m…

You will love this blog: http://talkingmathwithkids.com

This too, http://jdh.hamkins.org/category/math-for-kids/, which featured on HN a couple of months ago.

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

#60

Post by j2kun »

tedunangst wrote: post #50
j2kun wrote: post #27

Earlier quoted context omitted.

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.

And, perhaps surprisingly, most mathematical work in the real world is spent figuring out which assumptions are useful to model the problem and which are not. You sound like you are claiming that modeling the problem is not part of the problem, but somehow "before" the problem is posed. It sounds like a fantasy to me.
Post reply on HN