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

Why Discrete Math Is Important

artofproblemsolving.com

Post reply on HN
Print view View original

71–80 of 127 posts

Re: Why Discrete Math Is Important

#71

Post by lmm »

vog wrote: post #69
lmm wrote: post #55

Earlier quoted context omitted.

Unknown. There's at least one approach to quantum gravity based on the idea that it is (causal sets), but we haven't proven it one way or another.

I'd rather ask: Is space discrete? We know there are discrete particles, but AFAIR they can potentially move to any position in the continuum, i.e. for each particle, x, y and z are in the domain of reals (or some huge real interval), not some discrete subset of it. Or, am I mistaken?

It's the same question (since time and space are not fully distinct), and the answer's equally unknown.

Re: Why Discrete Math Is Important

#72

Post by BeetleB »

gosubpl wrote: post #49
enriquto wrote: post #29

Discrete math is important because the universe is discrete. Continuous math is an approximation that sometimes, but not always, is rather convenient. Once I wrapped my mind around this, I started to understand something. Manifolds are just graphs with many vertices. Fourier analysis studies the eigen-decomposition of the laplacian on a graph, and is used to solve heat, wave and dispersion equations. Stokes theorem (…

Nice thing about "continuous" math is that we have so many "standardised" tools in its toolbox, contrasted with "ad-hoc-edness" of discrete math. Hence interesting is solving discrete problems with "continuous" tools - like e.g. http://ac.cs.princeton.edu/home/

>Hence interesting is solving discrete problems with "continuous" tools - like e.g. http://ac.cs.princeton.edu/home/

I'm often interested in the opposite: Solving continuous problems by going to the discrete domain.

I'm not a mathematician, but I did enjoy taking math courses and dabbling a little.

My personal highlight was when I was struggling for months to solve a continuous variable problem, but then one day I decided to "pixelate" it and converted it to a discrete problem. I solved the discrete problem, and got the answer to the continuous problem by taking the limit of the solution.

The problem was: If you choose n numbers at random (uniformly) in the interval (0,1), what is the expected value of the maximum?

I showed this problem to a number of people (including math professors) who struggled with it. Finally, one day, a colleague solved the problem in 5 minutes and 3 lines using continuous math. (It's not a clever solution either - surprising so many people missed it).

Still, I feel content with my discrete proof (which was about 2 pages). Since then I've often thought I should collect interesting continuous problems solved this way and put them on a web site, but never did. :-(

Re: Why Discrete Math Is Important

#73

Post by Verdex_3 »

onychomys wrote: post #65

Even if we don't teach a single day of number theory, I think we can all agree that modern society would be better if everybody had to have a semester of basic probability or statistics as part of their education.

Not really. You have to find a way to make the math real to your students or else it becomes just another exercise of "what set of words do I need to say in order to make the teacher happy". At least in my experience, most learning seems to be either incidental OR some sort of vestigial residue of the social component of making the system happy.

Re: Why Discrete Math Is Important

#74

Post by contravariant »

woopwoop wrote: post #64
enriquto wrote: post #29

Discrete math is important because the universe is discrete. Continuous math is an approximation that sometimes, but not always, is rather convenient. Once I wrapped my mind around this, I started to understand something. Manifolds are just graphs with many vertices. Fourier analysis studies the eigen-decomposition of the laplacian on a graph, and is used to solve heat, wave and dispersion equations. Stokes theorem (…

>Manifolds are just graphs with many vertices. Okay, I'll bite. How? What is the definition of the tangent space? Dimension?

Yeah you'll want hypergraphs (or even better simplicial complexes) in general but you can make sense of manifolds as discrete objects. In fact I'm fairly sure you can triangulate any (smooth?) manifold.

The tangent space can be defined in terms of derivations, as soon as you define what a smooth function on the 'discrete' manifold should look like (you may have to define the derivative at a face, rather than a vertex, or have the function take values on faces rather than vertices).

Re: Why Discrete Math Is Important

#75

Post by svat »

jonsen wrote: post #37
simonbyrne wrote: post #16

Earlier quoted context omitted.

He did elsewhere suggest teaching calculus by Big O notation: http://www.ams.org/notices/199806/commentary.pdf I would be excited to see someone try that.

I found his original and not abbreviated version https://www-cs-staff.stanford.edu/~knuth/calc Now, how do I TeXify it on an iPad?

I've TeXed it a while ago; scroll down here: https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...

Re: Why Discrete Math Is Important

#76

Post by crimsonalucard »

enriquto wrote: post #29

Discrete math is important because the universe is discrete. Continuous math is an approximation that sometimes, but not always, is rather convenient. Once I wrapped my mind around this, I started to understand something. Manifolds are just graphs with many vertices. Fourier analysis studies the eigen-decomposition of the laplacian on a graph, and is used to solve heat, wave and dispersion equations. Stokes theorem (…

>Discrete math is important because the universe is discrete. Continuous math is an approximation that sometimes, but not always, is rather convenient.

This is not true. There are "things" in the universe that are discrete (for example: matter). But whether the universe itself is discrete is something we don't know.

What is the smallest discrete measurable length in the universe? We don't know.

Re: Why Discrete Math Is Important

#78

Post by e12e »

onychomys wrote: post #65

Even if we don't teach a single day of number theory, I think we can all agree that modern society would be better if everybody had to have a semester of basic probability or statistics as part of their education.

You propose we stop teaching children to count? That two comes after one, is one more than one etc? I don't think we can get away from all discreet math... ;-)

Re: Why Discrete Math Is Important

#79

Post by baldfat »

killjoywashere wrote: post #77

> Discrete math shows up on most middle and high school math contests. That seems a terribly weak reason for anything to be important.

When funding for your school is based off of test scores in a horrible way this is what we get.

My daughter is in 6th Grade and she has no Science or Social Studies this year. Reason: She has her Math and Science testing this year.

When did science become the enemy of math?

Re: Why Discrete Math Is Important

#80

Post by jumpman500 »

Verdex_3 wrote: post #73
onychomys wrote: post #65

Even if we don't teach a single day of number theory, I think we can all agree that modern society would be better if everybody had to have a semester of basic probability or statistics as part of their education.

Not really. You have to find a way to make the math real to your students or else it becomes just another exercise of "what set of words do I need to say in order to make the teacher happy". At least in my experience, most learning seems to be either incidental OR some sort of vestigial residue of the social component of making the system happy.

What? You can make easily make basic statistic and probability about real problems and interesting. Talk about sports. Talk about risks of the stock market and financial planning. Talk about politics/polling. Talk about gambling/poker. Just takes an interesting teacher to make any subject interesting.
Post reply on HN