[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.

[gjm11]

I am intrigued by your ideas and wish to subscribe to your newsletter. How does Stokes' theorem in a discrete setting just amount to associativity?

(I wondered about the sort of discrete differential geometry found e.g. here https://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf and here https://arxiv.org/pdf/math/0508341.pdf but in that setting the situation seems to be more "Stokes' theorem is true by definition".)

[enriquto]

I am intrigued by your ideas and wish to subscribe to your newsletter.

Lol, this is not the first time that I am mocked here :)

How does Stokes' theorem in a discrete setting just amount to associativity?

Manifolds are modeled by graphs, and calculus on manifolds becomes linear algebra using the matrices naturally associated to these graphs.

Consider a graph with n vertices and m edges.

The most important matrix is the oriented incidence matrix B, of size mxn, that has a single +1 and a single -1 on each row, indicating the vertices connected by the corresponding edge.

Scalar fields := functions defined over the vertices = vectors of R^n

Vector fields := functions defined over the edges = vectors of R^m

When you interpret matrices as linear operators:

    B     : R^n -> R^m is the gradient operator
    -B'   : R^m -> R^n is the divergence operator
    -B'.B : R^n -> R^n is the laplacian

A subset of the vertices is given by a binary vector M \in R^n. The integral of a scalar field f over M is M'.f

The outwards boundary of a subset M is -B.M. The flux of a vector field F through this boundary is (-B.M)'.F

Stokes theorem is thus the trivial identity (-B.M)'.F = M'.(-B'.F)

(I use a dot for matrix products because the star breaks the formatting. You can also define the divergence without the minus sign, but I like my laplacians to be negative-definite, it feels weird otherwise.)

[gjm11]

I didn't actually intend my comment as mockery. I'm very much for finding correspondences between discrete and continuous things. I just didn't see how to set things up so that Stokes turned into associativity.

I think this is in fact the same thing as the "true by definition" found in the second of my links above -- but I hadn't thought hard enough about how that cashes out concretely to see that you can express it as the associativity of a three-way product. Nice.

[enriquto]

stokes theorem is often written (in a continuous setting) as

    < dM, F > = < M, dF >

(the adjoint of the boundary operator is the exterior derivative). In the discrete case, these integrals are actually products of matrices. There is nothing too deep here.

Edit: regarding the "true by definition" issue, you can do one of two things. (1) Build the definition of the boundary and exterior derivative independently and then verify that Stokes theorem holds. Or (2) build the definition of only one of these two operators, and then define the other one as its adjoint. In that second case, Stokes theorem is true by definition. But it doesn't matter too much, these are just two different ways to write the same thing.

[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/

[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

[BeetleB]

>Hence interesting is solving discrete problems with "continuous" tools - like e.g. http://ac.cs.princeton.edu/home/

I'm often interested in the opposite: Solving continuous problems by going to the discrete domain.

I'm not a mathematician, but I did enjoy taking math courses and dabbling a little.

My personal highlight was when I was struggling for months to solve a continuous variable problem, but then one day I decided to "pixelate" it and converted it to a discrete problem. I solved the discrete problem, and got the answer to the continuous problem by taking the limit of the solution.

The problem was: If you choose n numbers at random (uniformly) in the interval (0,1), what is the expected value of the maximum?

I showed this problem to a number of people (including math professors) who struggled with it. Finally, one day, a colleague solved the problem in 5 minutes and 3 lines using continuous math. (It's not a clever solution either - surprising so many people missed it).

Still, I feel content with my discrete proof (which was about 2 pages). Since then I've often thought I should collect interesting continuous problems solved this way and put them on a web site, but never did. :-(

[jeremysalwen]

So is this the solution? The probability that all numbers are less than x is equal to x^n. So then you take the derivative of that to get the probability that the maximum us is exactly x. n x^n-1. Then calculate the expected value as integral from 0 to 1 of x n x^n-1 = n/n+1.

[BeetleB]

Too lazy to think deeply about it, but it sounds right. You start with the cumulative distribution function and get the pdf from it, which looks like what you're doing.

Really simple solution. Problem is simple enough that this could be a standard HW problem in a probability course. Yet so many people (including myself) did not see it. We kept doing multiple integrals (n integrals for n points) and tried using induction on it.

This was actually a subproblem of the real problem. The real problem was: Given n points chosen randomly on a circle, construct the n-sided polygon. What is the probability that the center of the circle is inside the polygon? Since I worked on it, the problem has shown up on the Internet in various places (usually for the special case of n=3 - triangles, but I think I've seen the general one posted here and there). I don't recall if anyone came up with the same solution I did for the general case - I think one site had it.

[aoki]

it is (a standard HW problem). for bonus points, there's a connection between order statistics of the uniform distribution (such as max) and the beta distribution, of which the solution above is an example.

[ChrisLomont]

>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.

[Chris2048]

"not known to be discrete" does not imply "It's not unreasonable that there are discrete and continuous aspects to the universe"; it is not known that anything is continuous either.

[ChrisLomont]

The theories that best model every known experiment are continuous. General relativity models large scale phenomena to incredible precision, and is continuous. Predictions made by it around 100 years ago are still being tested and found true, such as gravitational waves.

The Standard Model, which models everything else, is continuous. A specific part of it, QED, is the most accurate theory known, agreeing with experiment to better than 10 parts in a billion.

There is no known violation of these two theories, which explain all of the observable universe. There is an issue on how to glue them together, and one of the leading methods, string theory, is also continuous. I am unaware of any theory that can replace these that has no continuous parts, such as symmetries.

> it is not known that anything is continuous either

Every physics theory that we use to explain the universe is continuous, and not a single one is not, so I'm betting that there are continuous things in reality.

[Chris2048]

Other leading theories are non-continuous. LGQ example is an active area of research.

> Every physics theory that we use to explain the universe is continuous

Are you suggesting thete are no discrete models?

Any discrete model may be similar to leading continuous models, so the explanatory power of both isn't relevant unless it can only be produced by continuous models.

[ChrisLomont]

>Other leading theories are non-continuous.

Which theories discretize all variables? I'm well aware people make models all the time, but they usually fail to explain all we observe or are toy models, like 2D TQFT models, not designed to mirror reality, but to fiddle with to see if they extend.

LQG, for example, treats space as quantized at the Plank length, but this has already been disproven experimentally [1,2], forcing LQG to reassess. LQG still has continuous parameters, such as symmetry, and is based on the same continuous math as GR and TQFTs. It's simply trying to discretize (somewhat) spacetime, but it certainly has not yet succeeded at that.

Hogan's holometer experiment [3] also has shown that the scales LQG wanted to discretize spacetime at are incorrect.

So there is at least 2 different experiments invalidating a central piece of LQG. Of course LQG can simply retrench and claim scales are smaller than the Plank length.

There is no experimental evidence pointing to LGQ as reality. It has not even been shown to reproduce GR in the semi-classical limit, so it may end up simply being mathematical fantasy. LQG also has lots of other technical problems that may in the end throw it on the heap of failed theories, of which there have been many. It makes no prediction not already covered under GR.

So, yes, I realize there are people trying to discretize the universe, but so far none of these theories have reproduced what we observe in experiment, and the only theories that have reproduced all we observe in experiment are continuous.

>Any discrete model may be similar to leading continuous models, so the explanatory power of both isn't relevant unless it can only be produced by continuous models.

Which model using only discrete variables can make the same predictions of GR or SM? I cannot think of one, I cannot find one, so it may be that so far the only models that work are continuous.

Heck, even going simpler - what discrete model can reproduce only GR? LQG so far cannot, despite significant effort trying to make it do so. So that surely puts the burden of proof on showing discrete models as equivalent to continuous ones.

So I'm game for something to break current models, because that would be cool. However, nothing has, and no theories is really even close to replacing SM and/or GR.

[1] https://arxiv.org/abs/1102.2784

[2] http://hal.in2p3.fr/tel-01219651/document

[3] https://en.wikipedia.org/wiki/Holometer

[Chris2048]

> but it certainly has not yet succeeded at that

The claim here is actual "continuous" objects likely exist. How can you demonstrate this?

All continuous models can be reproduced in discrete models up to the resolution of "all current observations", How are discrete models inherently unable to match continuous ones?

[q1bz2p]

I have reviewed [1] and [2] but it isn't clear to me how they prove that space isn't quantized at the Planck length.

Could you explain how these experiments disprove that?

[crimsonalucard]

>Discrete math is important because the universe is discrete. Continuous math is an approximation that sometimes, but not always, is rather convenient.

This is not true. There are "things" in the universe that are discrete (for example: matter). But whether the universe itself is discrete is something we don't know.

What is the smallest discrete measurable length in the universe? We don't know.

[dbcurtis]

Would you concede that the Planck length is a good approximation of the smallest measurable length in the universe?

[JoeAltmaier]

The change in radius of a black hole when 1 bit of information is added is much smaller than 1 Planck length. The Planck length is just a convenient unit of measure in physics. It has no relevance to limits of space or time at all.

Here's an article on the subject:

https://www.quora.com/Is-there-anything-smaller-than-a-Planc...

[dbcurtis]

Interesting. I am now better informed.

[Sean1708]

Why would the Planck length be the smallest measurable length in the universe?

Edit: To be more clear: Is there a theoretical reason why the Planck length would approximate the smallest measurable length?

[mabbo]

> 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?

[enriquto]

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.

[wslh]

> Discrete math is important because the universe is discrete.

Is time discrete?

[lmm]

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.

[vog]

I'd rather ask: Is space discrete? We know there are discrete particles, but AFAIR they can potentially move to any position in the continuum, i.e. for each particle, x, y and z are in the domain of reals (or some huge real interval), not some discrete subset of it.

Or, am I mistaken?

[lmm]

It's the same question (since time and space are not fully distinct), and the answer's equally unknown.

[dcow]

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?

[jonsen]

Discontinuous, probably.

[randcraw]

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...

[daveslash]

The "Concrete Math" book title is a play on words - combining continuous and discrete. From the preface: "When DEK taught Concrete Mathematics for the first time.... [h]e announced that, contrary to the expectations of some of his colleagues, he was _not_ going to teach the Theory of Aggregates, not Stone's Embedding Theorem, not even the Stone-Cech compactification. (Several students from the civil engineering department got up and quietly left the room). [Edit]: Typo/Spelling fix.

[AstralStorm]

Neurons themselves are working on physics at not quite quantum scales, so probably a good model would be timewise continuous stochastic.

[woopwoop]

>Manifolds are just graphs with many vertices.

Okay, I'll bite. How? What is the definition of the tangent space? Dimension?

[contravariant]

Yeah you'll want hypergraphs (or even better simplicial complexes) in general but you can make sense of manifolds as discrete objects. In fact I'm fairly sure you can triangulate any (smooth?) manifold.

The tangent space can be defined in terms of derivations, as soon as you define what a smooth function on the 'discrete' manifold should look like (you may have to define the derivative at a face, rather than a vertex, or have the function take values on faces rather than vertices).

[woopwoop]

Sure. I think the notion of a manifold wouldn't be very interesting if it didn't have some kind of discrete analogue. But, for me at least, smooth manifolds are much easier to think about than simplicial complexes or PL manifolds.

[enriquto]

the tangent space is the set of edges

vector fields (sections of the tangent bundle) correspond to real-valued functions defined on edges

dimension is always 2 ;)

if you want higher dimension you have to consider higher-dimensional cliques beyond edges (that are 2-cliques): triangles, tetrahedra, and so on. But very often this is not necessary, even when discretizing 3d stuff. For example, for Poisson equation, and the associated classical pde, you only need the laplacian, which acts on functions, regardless of the dimension.

[contravariant]

That seems a weird choice of tangent space, surely a circle and a disk should have different tangent spaces?

[enriquto]

And they do. When you discretize a circle as a graph, all the edges are along the boundary. When you discretize the disk, you fill its whole interior with edges in all directions.

[woopwoop]

Is a triangle a circle or a disk?

[CogitoCogito]

> 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".

[VyseofArcadia]

The problem I've always had with real numbers is that the vast majority (i.e. uncountably many) of them are not computable, and only countably many of them are computable.

That means that almost every real number, all but a vanishingly small subset, cannot be represented by any means whatsoever. No formulas, no algorithms, nothing. On top of which, they're surprisingly complicated to construct. There's several different ways of doing it, and they're all complicated.

At some point you've got to ask yourself, "are they really even there?" (Of course, those who already have non-Platonic leanings will find that question amusing.) I start to think that maybe we'd be better served by dropping all the non-computable numbers and start doing almost everything in the field of computables.

[enriquto]

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).

[BeetleB]

Hello Doron Zeilberger. Did not know you read HN.

[dubya]

Apart from his interesting ideas about infinity, Zeilberger has things to say about math contests. See http://sites.math.rutgers.edu/~zeilberg/Opinion71.html I'd love to find a modern equivalent of the "gilyonot lematematika" he mentions.