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Why Discrete Math Is Important

artofproblemsolving.com

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Re: Why Discrete Math Is Important

#21

Post by gh02t »

kccqzy wrote: post #7
craigching wrote: post #4

I have to admit, being a hybrid math/csci student, I never understood the place of discrete math in mathematics or computer science. It always seemed like a mish-mash of different topics I'd studied in algebra->geometry->calc (including mv calc, linear algebra, diff eq, and series and sequences)->real analysis. This article is a bit too brief to properly place it (at least I still don't see it), could someone provide…

Discrete math in my school is mostly about combinatorics, but also graph theory, trees, and things like that. It doesn’t really fit into the standard math sequence IMO. The standard math sequence is essentially single-variable calculus -> multi-variable calculus -> linear algebra and differential equations -> real analysis. Note that all of the above fields essentially operate on continuous things like the real numbe…

That's not quite true, almost any respectable math program is gonna include a significant amount of material on finite set theory and discrete algebra (though maybe not in the non math major escalator, that usually stops at differential equations). That said, I think there is definitely a place for a "discrete math" course, which focuses on teaching things in a more computer science relevant manner.

I also think numerical analysis is a much better choice than real analysis for CS majors, but that's a different topic...

Re: Why Discrete Math Is Important

#23

Post by fjsolwmv »

psandersen wrote: post #19
luckyt wrote: post #17

Earlier quoted context omitted.

I disagree that CS deals only with discrete math. Machine learning makes heavy use of linear algebra, with a decent amount of calculus to model statistical phenomena. All of this uses continuous variables, not discrete.

A computer still deals with a finite amount of bits (e.g. 32-bit floating point numbers) so its a discrete system that is large enough to approximate a continuous system. It may not matter much in practice except for numerical precision issues, but it is useful to understand the foundations and occasionally throw away abstractions for performance/other requirements.

That's silly. Floating point math is an approximation of continuous math. It's only discrete in a narrow technical sense.

Re: Why Discrete Math Is Important

#24

Post by fjsolwmv »

glockenspielen wrote: post #20
craigching wrote: post #15

Earlier quoted context omitted.

So all responses thus far have been great, I think the thing I struggled with in HS and into university is trying to place everything and see what the current math leads to. So, responding here, but upvoting those that responded to me at this time because all provided me with insight. Thanks!

Head down the Prof Wildberger rabbit hole if you'd like to explore why we might be able to cast aside chunks of conventional math including real numbers and infinite sets. https://m.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ https://njwildberger.com

Might be? Of course we can. Obviously infinite objects are irrelevant to modeling the real world. They are just a shorthand for adding a bunch of annoying qualifiers to every mathematical statement.

Re: Why Discrete Math Is Important

#25

Post by fjsolwmv »

gh02t wrote: post #21
kccqzy wrote: post #7

Earlier quoted context omitted.

Discrete math in my school is mostly about combinatorics, but also graph theory, trees, and things like that. It doesn’t really fit into the standard math sequence IMO. The standard math sequence is essentially single-variable calculus -> multi-variable calculus -> linear algebra and differential equations -> real analysis. Note that all of the above fields essentially operate on continuous things like the real numbe…

That's not quite true, almost any respectable math program is gonna include a significant amount of material on finite set theory and discrete algebra (though maybe not in the non math major escalator, that usually stops at differential equations). That said, I think there is definitely a place for a "discrete math" course, which focuses on teaching things in a more computer science relevant manner. I also think nume…

Of course applied math is much more relevant to CS than pure math. CS is an application of math.

Re: Why Discrete Math Is Important

#28

Post by kqr »

saagarjha wrote: post #13
craigching wrote: post #4

I have to admit, being a hybrid math/csci student, I never understood the place of discrete math in mathematics or computer science. It always seemed like a mish-mash of different topics I'd studied in algebra->geometry->calc (including mv calc, linear algebra, diff eq, and series and sequences)->real analysis. This article is a bit too brief to properly place it (at least I still don't see it), could someone provide…

A common use case is prove correctness or running time of algorithms.

Prove running time as in complexity analysis is more analysis than algebra though, right?

Re: Why Discrete Math Is Important

#29

Post by enriquto »

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 (which in a discrete setting amounts to matrix associativity) is a self-evident fact. Most of applied math is thus reduced to a few lines of octave code.

Only when you lose discreteness or compactness things start to get nasty. But this is just a flaw in our current definition of real numbers.

Re: Why Discrete Math Is Important

#30

Post by bordercases »

simonbyrne wrote: post #16
kccqzy wrote: post #9

Earlier quoted context omitted.

I loved Knuth’s concrete mathematics too. But I don’t think it is an attempt at reformulating calculus into the discrete math framework. Instead in many places in concrete math, knowledge of calculus is assumed, especially in later chapters about generating functions etc.

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.

UPenn's Calculus I+II courses with Robert Ghrist uses this sort of notation right at the beginning (it takes the Talyor Polynomial as the natural starting point, rather than derivatives, with knocking off terms of the summation involves factoring them out into the O-notation block).
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