
Why Discrete Math Is Important - jaoo
https://artofproblemsolving.com/articles/discrete-math
======
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.

~~~
gosubpl
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/](http://ac.cs.princeton.edu/home/)

~~~
enriquto
You can also see it in the opposite sense.

The continuous models are an ad-hoc, purely mental, construction. When you
have to solve a PDE, you actually build a discrete model (using finite
elements), and solve the discrete thing. Except in very simple toy problems,
you can never "solve" anything using only continuous tools.

~~~
gugagore
Spectral methods, or any methods where you have chosen a basis of continuous
functions and are solving for weights produces solutions in the continuous
domain. That's not a discrete model.

~~~
jacobolus
The grandparent poster might like Chebfun,
[http://www.chebfun.org](http://www.chebfun.org)

------
bootsz
> _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 later ended up going back to school for CS and took
discrete math as one of my first courses, and remember being blown away by how
cool it was. All this time thinking I hated math!

Hard to say exactly what the difference is. Partially I think my brain just
groks discrete concepts more easily.

But also the class had a heavy emphasis on proofs, which I think was really
important. At a certain level this type of problem-solving can start to
resemble philosophy. Chugging through a proof, figuring out just the right way
to construct it and slapping a triumphant "Q.E.D." at the end is an empowering
experience, especially the first time. There's a world of difference between
"you throw a ball, solve for its velocity at time x" and "prove that there
must be a ball" (I'm embellishing of course). It's a difference between
obtaining an answer for a specific instance of a situation, and shedding light
on some fundamental/universal property of the world. To me that feels profound
in a sense, which makes it exciting.

Proofs don't belong solely to the domain of discrete math, of course, so this
probably isn't as much a testament to the subject as it is to the general
problem-solving approach. It would be nice if students could get exposed to
this a bit earlier, I think there are many folks like myself who would realize
that they can love math too.

~~~
whatshisface
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.

~~~
leetcrew
> 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.

------
onychomys
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.

~~~
Verdex_3
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.

~~~
jumpman500
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.

~~~
Verdex_3
Like, I want to believe you that such a thing is easy. But I'm not really
convinced by the assertion that it is followed by a non-descriptive blurb. I
mean I get what you're saying, "Find what they care about and try to apply
statistics to it." However, I don't believe that this process is easy.

Sports is a good example of why I don't think this is easy. What exactly is
the point of statistics in sports? Predicting what teams are going to win or
what strategies are superior. Most children in school are not interested in
sports because they're into strategy or because they're predicting who's going
to win. They're interested in it for social reasons. Which team do their
parents or friends want to win. Telling everyone at thanksgiving that the
family team is going to lose the game will probably not go well for them. And
as far as strategy goes ... I thought the statistical analysis of the extra
point kick in football indicates that you should never do it. Teams rarely
make use of this at the professional level. Also didn't they make a movie
about how nobody pays attention to the math in baseball (Moneyball?). Anyway,
if adults who have money and fame on the line can't be bothered to care about
statistics in sports, then I don't see how children are going to be much
different.

Of course that's where you come in. You say it's easy. Personally, I would
love to see _why_ you think this because it looks hard to me. I look forward
to a more in depth response from you.

~~~
jumpman500
Well I'm commenting on the internet Verdex_3. I'm not going to take the time
to write out a bunch of interesting statistical examples for you. I wasn't
trying to say it's easy to make statistics interesting. Was only trying to
refute what I thought that you were asserting; statistics and probability are
not interesting to teach. Designing enjoyable, interesting, challenging and
fair curriculum is a really hard task to do for any subject...

If you don't believe math can be made interesting I would checkout websites
like FiveThirtyEight or youtube channels like numberphile. It's definitely
possible to describe statistics or math problems in interesting ways. It's
hard to make any class interesting, but any teacher can do it if they work at
it hard enough.

Also your moneyball comment makes no sense. All teams operate like the 2002
Oakland A's now, if anything the the movie shows the triump of statistics over
bad heuristics.

------
craigching
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 some proper context for discrete
mathematics that fits into the mold of the standard maths sequence?

~~~
JonathonW
It's hard to place a particular "context" for discrete math; it really is a
hodgepodge of topics, loosely linked because they deal with discrete
structures (integers, graphs, logic statements) rather than continuous ones
(real/complex numbers).

It's particularly relevant to computer science because, in CS, we're dealing
with discrete structures almost exclusively. The rise of computers and of CS
is both what led to the current interest in discrete math subjects as a
research field and what led to the development of university curricula in the
topic.

So, really, discrete math (as a university course) exists mostly to teach some
CS-relevant topics that don't necessarily get much dedicated time in the
"standard" algebra->geometry->calc progression, because they're more concerned
with continuous phenomena. It's sort of a parallel and independent track from
the "standard" math sequence.

~~~
craigching
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!

~~~
glockenspielen
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://m.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ)
[https://njwildberger.com](https://njwildberger.com)

~~~
fjsolwmv
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.

~~~
yters
It's a good thing we don't actually need the square root of two!

------
saagarjha
> Prominent math competitions such as MATHCOUNTS (at the middle school level)
> and the American Mathematics Competitions (at the high school level) feature
> discrete math questions as a significant portion of their contests. On
> harder high school contests, such as the AIME, the quantity of discrete math
> is even larger.

As someone who participated in these contests, this isn't the entire story.
Competitions such as these all require numerical answers, and as such skew
_extremely_ heavily towards counting and probability (as in, there's no other
discrete math topics but these two). It's only when you get into proof based
contents that the real meat of discrete math, namely recurrence, cardinality,
graphs, etc. start showing up.

------
pimmen
The vast majority of people who learn calculus in school will never model a
changing system in their life again.

The vast majority who didn't take statistics courses in college will still try
to use the limited understanding they have of statistics to assess statistical
claims or draw conclusions from reported figures. The vast majority of people
who never took discrete mathematics courses will still face problems of
figuring out the difference of combinations and permutations at some points in
their life.

I love calculus and I'm very happy I know it but I would be lying if I said it
even approaches the importance of discrete mathematics and statistics in
today's world.

------
comstock
I love discrete math, it seems so much cleaner in general. I wish there were
more reformulations of calculus, other numerical methods into discrete maths.

I think Knuth’s concrete mathematics might have been an attempt at this, but
I’ve never found time to dig into it in depth. Perhaps I should try again...

~~~
andars
In some sense, don't the modern formulations of real analysis, etc. already
start from as close to discrete maths as you can get (set theory)?

Sets -> Naturals -> Rationals -> Reals

I don't understand how you could reformulate study of continuous structures
into _discrete_ math in any sense other than the above.

~~~
ginnungagap
Every mathematical object (ok, this is false but that's not the point here)
can be constructed in ZFC (the standard axiomatic framework for set theory) so
you can construct the real numbers in terms of sets (if you want more precise
informations on this construction look up Dedekind cuts).

However this is irrelevant to, say, analysis, you could define the real
numbers as the unique (up to isomorphism) complete, ordered, archimedean field
and do analysis just as well, so I'd say that you are right in some sense and
some formulation, but it's a bit of a stretch to consider analysis as starting
from discrete maths.

I also don't see how set theory fits into discrete maths, apart from the
basics it seems pretty far from the common structures studied in discrete
maths.

~~~
EtDybNuvCu
Pick the Grothendieck-Tarski axiom instead, and use category theory to build
ZFC via topos. This path is "big" enough to handle all the interesting sets;
it can't deal with proper classes, but proper classes are kind of metaphysical
anyway.

[0]
[https://en.wikipedia.org/wiki/Tarski–Grothendieck_set_theory](https://en.wikipedia.org/wiki/Tarski–Grothendieck_set_theory)

~~~
ginnungagap
Sure, but ZFC by itself also deals with every interesting set, I was just
being nitpicky of my own assertion.

I'm not familiar with TG, what's the relation between it and ZFC+some large
cardinal axiom?

------
compsciphd
1) I loved my undergraduate discrete math class.

2) who can't love a class that teaches you how to understand the math behind
poker :)

------
hnzix
Symbolic logic / truth tables is the single most useful subject I have ever
taken wrt programming. It provides an intuitive understanding of conditionals
so they can be expressed simply and clearly.

------
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.

~~~
2sk21
Fully agree - the whole AoPS range of books and courses are just amazing.
Highly recommended for all kids.

------
atsushin
I wish I had paid more attention to or had a better instructor for my discrete
mathematics course, I find many of the topics covered in it extremely
fascinating now, years later. :(

~~~
hsrada
Surely, online resources for these courses must exist? I guess it's just about
finding the will and time to put in double the effort because of a lack of
instructor/conducive environment.

~~~
mastry
The Coursera/UC series on discrete mathematics [1] looks like a good
introduction.

[1] [https://www.coursera.org/specializations/discrete-
mathematic...](https://www.coursera.org/specializations/discrete-mathematics)

------
nv-vn
Highly agree. As a current high school student, I've gone out of my way to
study discrete math. Though I found calculus interesting, it's not
particularly applicable to any part of CS except for a few concepts. OTOH, DM
is incredibly useful for practically everything, which is what led me to seek
it out.

~~~
aoki
if your interests tend toward machine learning, you may run into calculus
again. in particular, if you keep going past the "train a neural network"
phase (for which very basic calculus is fine) to the optimization problems
that are under the hood of most ML algorithms, you wind up in the land of
functional analysis.

------
killjoywashere
> Discrete math shows up on most middle and high school math contests.

That seems a terribly weak reason for anything to be important.

~~~
baldfat
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?

~~~
dubya
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.

~~~
baldfat
I understand that BUT school administration is looking at Math contest through
standardized testing outcomes.

------
sidcool
My peev has been, how do I improve my problem solving skills, not necessarily
Mathematics wise. But in general.

~~~
fjsolwmv
Read _How to Solve It_ by Polya

~~~
sidcool
Bought it just now.

------
vidanay
I literally just stopped working on my discreet math homework tonight before
loading HN and seeing this article.

