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Why Discrete Math Is Important

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Re: Why Discrete Math Is Important

#31

Post by yters »

fjsolwmv wrote: post #24
glockenspielen wrote: post #20

Earlier quoted context omitted.

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.

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

Re: Why Discrete Math Is Important

#32

Post by 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.

Re: Why Discrete Math Is Important

#33

Post by bordercases »

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…

It gains a bit more unity when you try to interpret it through certain lenses.

In particular, one problem with discrete mathematics as it's taught, is that it doesn't separate the methods of counting from the set of objects that you need to count.

There are a couple of ways around this. One good way is to look at all combinatoric identities as referring to the number of ways you can connect some set to some other set. Sometimes they're called "choices", "mappings", functions, whatever. You can talk about the function and sets separate from the numbers, and the numbers drop out of properties of the set. Doing this removes a layer of interpretation and guesswork even if it ups the abstraction a bit.

Additionally, discrete math just looked at as the math of algorithms also gets you far. Sedgewick's Analysis of Algorithms book is actually a discrete math book in disguise, since it gives a system of notation that can describe basically any combinatorical object separate from the counting method -- and then maps it to the counting method.

https://www.amazon.com/Introduction-Analysis-Algorithms-2nd-...

Re: Why Discrete Math Is Important

#34

Post by [deleted] »

yters wrote: post #31
fjsolwmv wrote: post #24

Earlier quoted context omitted.

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.

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

[deleted]

Re: Why Discrete Math Is Important

#35

Post by 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.

Re: Why Discrete Math Is Important

#36

Post by kccqzy »

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…

Oh I’m interpreting GP’s mention of standard math sequence as something everyone in math/CS/engineering needs to study, not what math majors specifically need to study.

Re: Why Discrete Math Is Important

#37

Post by jonsen »

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.

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?

Re: Why Discrete Math Is Important

#38

Post by 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.

Re: Why Discrete Math Is Important

#39

Post by tw1010 »

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

I would be careful taking that stuff too seriously. It's ok as entertainment though.

Re: Why Discrete Math Is Important

#40

Post by AstralStorm »

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.

Even then, only small subset of discrete maths often called numerical methods is needed. (Discretizaton, discrete linear and nonlinear algebra, stability.) Going at a problem from fully discrete math point of view is often suicidal (results in unworkable algorithms) as integrals or difference equations are much more useful in practice than say discrete combinatorics. (Mostly used in cryptography.)

Graphs are sometimes useful in a narrow set of CS problems, as are similar structures. However, these are often not taught at discrete maths courses.

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