Why Discrete Math Is Important • Hacker News Board

Re: Why Discrete Math Is Important

#51

Post by enriquto » Tue, Jan 23, 2018, 12:18 PM UTC

gosubpl wrote: post #49

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/

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.

Re: Why Discrete Math Is Important

#52

Post by wslh » Tue, Jan 23, 2018, 12:19 PM UTC

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 (…

> Discrete math is important because the universe is discrete.

Is time discrete?

Re: Why Discrete Math Is Important

#53

Post by CogitoCogito » Tue, Jan 23, 2018, 12:44 PM UTC

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 (…

> 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.

What "flaw" are you referring to? Also you're certainly free to use other definitions for real numbers, if you feel they better capture "reality".

Re: Why Discrete Math Is Important

#54

Post by enriquto » Tue, Jan 23, 2018, 1:10 PM UTC

CogitoCogito wrote: post #53

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 (…

> 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. What "flaw" are you referring to? Also you're certainly free to use other definitions for real numbers, if you feel they better capture "reality".

All definitions of real numbers are equivalent, as far as I know. And as much as I love the definitions of R as a crowning achievement of human civilization, they lead to some infuriating paradoxes, especially in measure theory (e.g., freiling's axiom of symmetry).

Re: Why Discrete Math Is Important

#55

Post by lmm » Tue, Jan 23, 2018, 1:20 PM UTC

wslh wrote: post #52

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 (…

> Discrete math is important because the universe is discrete. Is time discrete?

Unknown. There's at least one approach to quantum gravity based on the idea that it is (causal sets), but we haven't proven it one way or another.

Re: Why Discrete Math Is Important

#56

Post by ChrisLomont » Tue, Jan 23, 2018, 1:39 PM UTC

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 (…

>because the universe is discrete

Many aspects of the universe are not known to be discrete, such as space, time, energy, and much, much more. It's not unreasonable that there are discrete and continuous aspects to the universe.

Many things that pop science treats as discrete, such as an electron, are most accurately described as interactions of a continuous fields via topological quantum field theories. Treating the electron as a discrete thing only works for some experiments. Treating it as a field works for all experiments.

Re: Why Discrete Math Is Important

#57

Post by ginnungagap » Tue, Jan 23, 2018, 1:43 PM UTC

EtDybNuvCu wrote: post #46

ginnungagap wrote: post #41

Earlier quoted context omitted.

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

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?

Re: Why Discrete Math Is Important

#58

Post by randcraw » Tue, Jan 23, 2018, 1:55 PM UTC

jonsen wrote: post #48

dcow wrote: post #47

Earlier quoted context omitted.

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.

Or concrete, as suggested by Knuth, Graham, and Patashnik in their book, "Concrete Mathematics: A Foundation for Computer Science".

https://www.amazon.com/Concrete-Mathematics-Foundation-Compu...

Re: Why Discrete Math Is Important

#59

Post by mabbo » Tue, Jan 23, 2018, 1:57 PM UTC

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 (…

> But this is just a flaw in our current definition of real numbers.

The distance between discrete infinite sets (countable) and continuous infinite sets (uncountable) is a pretty large gap. Are you saying that there is no gap? There are pretty well studied proofs showing that uncountable sets are much bigger than countable.

What's your take on this? Got a counter proof to Cantor?

Re: Why Discrete Math Is Important

#60

Post by enriquto » Tue, Jan 23, 2018, 2:17 PM UTC

mabbo wrote: post #59

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 (…

> But this is just a flaw in our current definition of real numbers. The distance between discrete infinite sets (countable) and continuous infinite sets (uncountable) is a pretty large gap. Are you saying that there is no gap? There are pretty well studied proofs showing that uncountable sets are much bigger than countable. What's your take on this? Got a counter proof to Cantor?

What's your take on this? Got a counter proof to Cantor?

As much as I am a fan of all things about cardinals, measure and category (Oxtoby's booklet on this stuff has been on my nightstand for several months, to a great pleasure), I still fail to recognize its technological and physical implications.

My take on this is modelled after the famous article "The dawning of the age of stochasticity" by David Mumford, where he argues that the current definition of real numbers leads to infuriating contradictions, especially in the light of probability theory. This article left me with the impression that in a near future, we will see a clever definition of "real-valued random variable" that does not depend on the definition of real numbers and avoids all kind of disgusting paradoxes.