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Why Discrete Math Is Important

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Re: Why Discrete Math Is Important

#41

Post by ginnungagap »

andars wrote: post #8
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...

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.

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.

Re: Why Discrete Math Is Important

#42

Post by geezerjay »

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.

> Obviously infinite objects are irrelevant to modeling the real world.

I disagree. The concept of infinite objects are essentially object sets with an unknown limit. This represents a general case from which we can draw important concepts.

Re: Why Discrete Math Is Important

#43

Post by hsrada »

atsushin wrote: post #26

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. :(

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.

Re: Why Discrete Math Is Important

#44

Post by mastry »

hsrada wrote: post #43
atsushin wrote: post #26

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. :(

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.

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

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

Re: Why Discrete Math Is Important

#45

Post by dcow »

gizmo686 wrote: post #5
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 doesn't. In my experience, "Discrete Math" is not offered as a math class, but rather as a computer class. In effect, it is the "math for computer science majors" class.

That was not my experience despite it being a requirement for the CS major.

Re: Why Discrete Math Is Important

#46

Post by EtDybNuvCu »

ginnungagap wrote: post #41
andars wrote: post #8

Earlier quoted context omitted.

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.

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, order…

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

Re: Why Discrete Math Is Important

#47

Post by dcow »

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 (…

I too have a problem with continuous math. However I have to wonder if we didn't have our senses, would our imaginations be discrete or continuous?

Re: Why Discrete Math Is Important

#48

Post by jonsen »

dcow wrote: post #47
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 (…

I too have a problem with continuous math. However I have to wonder if we didn't have our senses, would our imaginations be discrete or continuous?

Discontinuous, probably.

Re: Why Discrete Math Is Important

#49

Post by gosubpl »

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 (…

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/

Re: Why Discrete Math Is Important

#50

Post by coldtea »

fjsolwmv wrote: post #23
psandersen wrote: post #19

Earlier quoted context omitted.

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.

Computers don't just do floating point. And when they do it, at the lower levels, they don't do it as floating point.
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