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Why Discrete Math Is Important

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Re: Why Discrete Math Is Important

#11

Post by JonathonW »

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…

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.

Re: Why Discrete Math Is Important

#12

Post by 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.

Re: Why Discrete Math Is Important

#13

Post by saagarjha »

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.

Re: Why Discrete Math Is Important

#15

Post by craigching »

JonathonW wrote: post #11
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…

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 t…

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!

Re: Why Discrete Math Is Important

#16

Post by simonbyrne »

kccqzy wrote: post #9
comstock wrote: post #2

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...

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.

Re: Why Discrete Math Is Important

#17

Post by luckyt »

JonathonW wrote: post #11
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…

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 t…

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.

Re: Why Discrete Math Is Important

#18

Post by gizmo686 »

comstock wrote: post #2

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...

Discrete calculus is a thing [0], although it is not so much a reformutaltion of infinitesimal calculus as it is its own field that borrows heavily from calculus.

[0] https://en.wikipedia.org/wiki/Finite_difference

Re: Why Discrete Math Is Important

#19

Post by psandersen »

luckyt wrote: post #17
JonathonW wrote: post #11

Earlier quoted context omitted.

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 t…

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.

Re: Why Discrete Math Is Important

#20

Post by glockenspielen »

craigching wrote: post #15
JonathonW wrote: post #11

Earlier quoted context omitted.

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 t…

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
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