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Why Discrete Math Is Important

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Re: Why Discrete Math Is Important

#81

Post by baldfat »

bootsz wrote: post #61

> Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring That was me. I grew up believing I hated math. Struggled all the way through middle & high school to AP calc and just found it incredibly boring and tedious. Ended up opting out of doing engineering/science in undergrad because I just couldn't stand doing all the math. Long story short, years late…

> even calculus dull and uninspiring

Calculus is totally based on the teacher. I had an awesome Calculus teacher (He actually was a Physicians Assistant and had degrees from Yale and Harvard but volunteered at my small Christian School). He taught me first class why calculus was awesome by challenging use that everything else in math was fake numbers. Showed us the difference between 1/3 and 0.33333 and studying the speed of two trains word problem was always wrong. He than stated that with Calculus you could see the world as we see it. We than used functions all semester long that would eventually get "exact" and it was an awesome ride. To bad we had 3 students and the other 2 were total math geniuses and got perfect math scores on their SATs. They always made me feel like an idiot.

Re: Why Discrete Math Is Important

#82

Post by dbcurtis »

crimsonalucard wrote: post #76
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.

Would you concede that the Planck length is a good approximation of the smallest measurable length in the universe?

Re: Why Discrete Math Is Important

#83

Post by whatshisface »

bootsz wrote: post #61

> Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring That was me. I grew up believing I hated math. Struggled all the way through middle & high school to AP calc and just found it incredibly boring and tedious. Ended up opting out of doing engineering/science in undergrad because I just couldn't stand doing all the math. Long story short, years late…

What you're describing is the basic shift between what lower-ed science/math is like and what "real" (college) science/math is like. The problem is that everything they teach in highschool and below needs to have an escape hatch for "what if they have anti-ADHD* but no clue what's going on?" That's why you were able to solve for v without obtaining any knowledge about the universe, and why taking discrete math did what they claimed geometry would do.

* It's kind of a stupid way to put it, but by anti-ADHD I mean sufficiently controlled behavior combined with the ability to focus on any rote task until it is completely learned (basically, whatever traits or life conditions you need to have the opposite of ADHD). No matter what, you're not allowed to fail that group.

Re: Why Discrete Math Is Important

#84

Post by dbcurtis »

Attention parents of "mathy" kids: A bit off topic, but I just want to put in a testimonial for AoPS online math classes. My daughter used it as the spine of her middle/high-school math education. Great program. Check it out.

Re: Why Discrete Math Is Important

#85

Post by woopwoop »

contravariant wrote: post #74
woopwoop wrote: post #64

Earlier quoted context omitted.

>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…

Sure. I think the notion of a manifold wouldn't be very interesting if it didn't have some kind of discrete analogue. But, for me at least, smooth manifolds are much easier to think about than simplicial complexes or PL manifolds.

Re: Why Discrete Math Is Important

#86

Post by daveslash »

randcraw wrote: post #58
jonsen wrote: post #48

Earlier quoted context omitted.

Discontinuous, probably.

Or concrete, as suggested by Knuth, Graham, and Patashnik in their book, "Concrete Mathematics: A Foundation for Computer Science". https://www.amazon.com/Concrete-Mathematics-Foundation-Compu...

The "Concrete Math" book title is a play on words - combining continuous and discrete. From the preface: "When DEK taught Concrete Mathematics for the first time.... [h]e announced that, contrary to the expectations of some of his colleagues, he was _not_ going to teach the Theory of Aggregates, not Stone's Embedding Theorem, not even the Stone-Cech compactification. (Several students from the civil engineering department got up and quietly left the room). [Edit]: Typo/Spelling fix.

Re: Why Discrete Math Is Important

#87

Post by JoeAltmaier »

dbcurtis wrote: post #82
crimsonalucard wrote: post #76

Earlier quoted context omitted.

>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.

Would you concede that the Planck length is a good approximation of the smallest measurable length in the universe?

The change in radius of a black hole when 1 bit of information is added is much smaller than 1 Planck length. The Planck length is just a convenient unit of measure in physics. It has no relevance to limits of space or time at all.

Here's an article on the subject:

https://www.quora.com/Is-there-anything-smaller-than-a-Planc...

Re: Why Discrete Math Is Important

#88

Post by leetcrew »

whatshisface wrote: post #83
bootsz wrote: post #61

> Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring That was me. I grew up believing I hated math. Struggled all the way through middle & high school to AP calc and just found it incredibly boring and tedious. Ended up opting out of doing engineering/science in undergrad because I just couldn't stand doing all the math. Long story short, years late…

What you're describing is the basic shift between what lower-ed science/math is like and what "real" (college) science/math is like. The problem is that everything they teach in highschool and below needs to have an escape hatch for "what if they have anti-ADHD* but no clue what's going on?" That's why you were able to solve for v without obtaining any knowledge about the universe, and why taking discrete math did wh…

> anti-ADHD

awkward way to put it (although i dunno how else to describe them), but i think everyone knows the students you are talking about. these people are the reason why highschool math/science is hell, and you can't get away from them by taken honors or AP courses.

Re: Why Discrete Math Is Important

#89

Post by dubya »

baldfat wrote: post #79
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?

Math contests are not related to standardized testing. The contests are entirely extra-curricular, and probably mostly benefit those kids who have exhausted their school's standard curriculum.

Re: Why Discrete Math Is Important

#90

Post by dubya »

BeetleB wrote: post #70
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 (…

Hello Doron Zeilberger. Did not know you read HN.

Apart from his interesting ideas about infinity, Zeilberger has things to say about math contests. See http://sites.math.rutgers.edu/~zeilberg/Opinion71.html I'd love to find a modern equivalent of the "gilyonot lematematika" he mentions.
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