An "isomorphism" between continuous and discrete mathematics

Question

First of all, I should inform everyone that I am not a mathematician and my question might sound not at all rigorous or maybe even absurd to many of you. But I have been thinking about this problem for some time and I decided to ask it here.

I am an engineer and use math, mostly calculus and differential equations, to model physical systems.

The problem is this: What if nature is "discrete" and the underlying continuity assumption of calculus renders it (or will render it in the future) impossible to model physical systems accurately?

Not getting into what a physical system is, I think it would be possible to solve this problem purely mathematically if one can prove (or disprove) that a discrete mathematical system has inherent properties that a continuous mathematical system cannot map or vice versa. So, is that possible? Does this question even have a rigorous meaning?

PS: I asked my mathematical friend this question and although he was very uncomfortable with my "non-rigorous" sequence of arguments, he did give a very interesting comment. He said that the essence of continuous mathematics, the only thing that sets it apart from discrete mathematics, is the "completeness axiom" which states that a bounded set must have a supremum. [For expample, the set $ (X: X^2 < 2)$ does not have a supremum if we stick to rational (discrete) numbers.] Is that true?

It was this same friend that mentioned the word "isomorphism" to me which I interpreted to mean something along the lines of "functionally similar". Just as complex numbers are similar to 2-d vectors on a plane. So I took the audacity to put that word up on the question.

Please be gentle in your criticism!

Answer 1

The objective of this answer is to show that, contrary to the common paradigm in Physics, discrete models are more informative than continuous models. The Discrete is "nearer to the truth", so to speak, than the Continuous. Where it is supposed, of course, that the Continuous and the Discrete are just two different means of looking at the same physical phenomenon. We shall motivate our (perhaps somewhat exaggerated) point of view at hand of two examples.

Longitudinal Waves

MSE reference:

• How to construct longitudinal from transversal waves and vice versa?

The main picture from Question & Answer is copied here for convenience:

Where it is clearly suggested that the black dots are the Discrete / longitudinal wave, and the sinusoidal curve is the Continuous / transverse wave.
Now the following fact is proved in the above Q&A reference. Let $\;\lambda\;$ be the wavelength of the longitudinal wave and $\;A\;$ its amplitude. Then, independent of any discretization we have: $$ A < \frac{\lambda}{2\pi} $$ Most astonishing is that I have never encountered this restriction anywhere in literature. But apart from this, it is clear that the restriction only comes from the Discrete underpinning the continuous. It cannot be derived from the continuous model alone (i.e transverse wave) that an amplitude may be limited by the wavelength.

Upwind Schemes

MSE reference:

• Upwind differencing scheme in Finite Volume Method (FVM)

The following is from the famous book by S.V. Patankar: Numerical Heat Transfer and Fluid Flow; Hemisphere Publishing Company U.S.A. 1980 (page 84).

The scheme is sometimes said to be based on the "tank-and-tube" model (Gosman, Pun, Runchal, Spalding, and Wolfshtein, 1969). As shown in Fig. 5.2 [below], the control volumes can be thought to be stirred tanks that are connected in series by short tubes. The flow through the tubes represents convection, while the conduction through the tank walls represents diffusion. Since the tanks are stirred, each contains a uniform temperature fluid. Then, it is appropriate to suppose that the fluid flowing in each connecting tube has the temperature that prevails in the tank on the upstream side. Normally, the fluid in the tube would not know anything about the tank toward which it is heading, but would carry the full legacy of the tank from which it has come. This is the essence of the upwind scheme.

Especially take notice of the phrase: the fluid in the tube would not know anything about the tank toward which it is heading, but would carry the full legacy of the tank from which it has come. With other words, the upwind scheme is equipped with "knowledge" about past and future. In the continuous analogue, on the contrary, the so-called arrow of time is completely absent. Thus, when taking the limit for tank and tube sizes to zero, directional information is lost forever. Therefore in the upwind case, the Continuous, instead of the better and the more exact, rather seems to be sort of sloppyfication of the Discrete.
It is demonstrated in the same MSE reference that upwind schemes are no way less accurate than, for example, central differences. It's even possible to reproduce "exact" (read: analytical) solutions with them. So the argument that upwind schemes are only $O(h)$ instead of $O(h^2)$ is just ge-$O(h)$, as we would say it in good old Dutch :-)

Answer 2

Interesting post.

What if nature is "discrete" and the underlying continuity assumption of calculus renders it (or will render it in the future) impossible to model physical systems accurately?

I think it's natural to regard nature as a "continuous system", but I'm not a physicist myself, so I'm not sure about deep concerns about this matter. What I would like to point out is that even though we use continuous models, we can perform computations using discrete approximations as precise as we need them, so I really doubt that using continuous models can blur our understanding of physical systems.

Addressing your question about proving fundamental differences between discrete and continuous systems, I can give you at least some intuition for a very specific "systems". Consider for example the set of integer numbers and the set of real numbers. You have a good intuition about what an isomorphism is: two "structures" are isomorphic if they are "in escence" the same, under some clear definitions and where the "escence" depends on the context. If we go back to our example with $\Bbb R$ and $\Bbb Z$, an "isomorphism" between them should preserve some properties that these two sets do not share! for example, an "isomorphism" should preserve cardinality, but it is the case that (despite the fact that both sets have an infinite number of elements) these sets have different cardinality.

The property that your friend mentioned is another example, but I'm not sure I'd regard the rational numbers as a "discrete" structure since, in the usual definition of a discrete set, we require that the "elements have enough space between them", which is not the case with $\Bbb Q$.

Answer 3

One may worry about many things, for instance, some people are worried about the world around us being "real" (rather than some giant computer simulation). On the practical side of the issue, I would say that one should worry about things (dealing with relation of our math and the world around us) that are more concrete, like "are bridges that are build using our math models stable?", "do our our weather (or climate) models work?", etc. Once you think this way, then you realize that neither one of these things are affected by the "discrete/continuous" dichotomy. For instance, weather predictions are based on combinations of (a) differential equations (which one normally puts in the "continuous" box), (b) linear algebra (where does it belong, is a separate matter, see below) and (c) computers (since 1940s they decidedly belong to "discrete", but, say, before WWW2, they were primarily analog, hence "continuous"). Accuracy (or rather lack of thereof) of our weather/climate predictions primarily depends on the following: (a) our PDEs are admittedly insufficient and (b) unstable, which implies that linear algebra approximations are admittedly insufficient to accurately approximate solutions of the differential equations.

Now, to the dichotomy (continuous/discrete) itself: Once you think about it for about 10 minutes, you realize that it is simply bogus. Consider linear algebra (something that I think you are familiar with). Does it belong to "discrete" or "continuous"? If you think in terms of finite difference methods for solving PDEs, then your first reaction would be "of course discrete". But, in reality, it is not. Linear algebra that one uses for approximate solutions of PDEs depends critically on real numbers that those, belong to "continuous". So, the correct answer would be "none of the above".

Next, most people (I think, you are included) when they say "continuous" really mean something like "real-analytic functions on some domains in $R^n$" or just domains in $R^n$ with real-analytic boundary (as domains for such functions. (OK, maybe piecewise real-analytic.) After all, this is what shows up in most PDEs motivated by mathematical physics (at least, say, pre-20th century and that's all what a non-nuclear engineer would need). However, the rational numbers (already mentioned in Daniel's answer) do not belong to this box and neither they belong to "discrete" box. To make things worse, there are natural continuous concepts such as Brownian motions and fractals, which again belong to "none of the above" box. Neither classical PDEs that you are familiar with (dealing with infinitely differentiable coefficients of PDEs and solutions) nor discrete (say, finite) math, provides a good tool for dealing with such things. Furthermore, there is a serious circumstantial evidence that our world (to the extent we understand it) is fractal on both microscopic (quantum mechanical) and macroscopic (clusters of galaxies) scale.

As for an "isomorphism" between "discrete" and "continuous", just forget it. In some very special cases one can assign some meaning to it (like "piecewise-linear approximations to solutions of some PDEs converge to actual solutions"), but not in general. (And, just to repeat myself, much of the math belongs to "none of the above".)

In conclusion, mathematics develops (in part, but only in part) in response to complexities of the outside world. So far, our experience tells us that different types of problems are best handled by different math tools ("continuous", "discrete", or "none of the above"). So far, if you want to build stable bridges, you use a combination of smooth PDEs and linear algebra. To study, say, stock market, you use stochastic PDEs ("none of the above"). And so it goes.

Answer 4

The following question is copied from an old thread in sci.math

Suppose you have a giant vase and a bunch of ping pong balls with an 
integer written on each one, e.g. just like the lottery, so the balls 
are numbered 1, 2, 3, ... and so on. At one minute to noon you put 
balls 1 to 10 in the vase and take out number 1. At half a minute to 
noon you put balls 11 - 20 in the vase and take out number 2. At one 
quarter minute to noon you put balls 21 - 30 in the vase and take out 
number 3. Continue in this fashion. Obviously this is physically 
impossible, but you get the idea. Now the question is this: At noon, 
how many ping pong balls are in the vase?

The answer given by most mathematicians sounds as follows:

Every ball placed into the vase (at a well-defined time before noon) is taken out from the vase (at a later, well-defined time before noon). Therefore at noon the vase is empty, there are no balls in the vase at noon.

However, we can say that the number of balls $\;B_k\;$ at step $\;k=1,2,3,4,\cdots\;$ is: $$ B_k=B(t_k) = 9 +\, ^2\log(-1/t_k) \quad \mbox{where} \quad t_k = -1/2^{k-1} $$ And if we properly interpolate these values, then we have a continuous function that does not have the following limit: $$ \lim_{t\to 0} B(t) = \lim_{t\to 0} \left[ 9 +\, ^2\log(-1/t) \right] = 0 $$ So (according to standard mathematics) the discrete is not isomorphic to the continuous at noon.

Answer 5

My main math specialization is topological algebra and I share your doubts. I feel that continuous and discrete objects are not similar but essentially different and need different math intuitions to deal with. Concerning nature. An engineer should be satisfied that our theories and models are practically successful with a good precision, but this is not a philosophical answer. :-) On a fundamental level the problems of our theories about nature which you are speaking about, was firstly formulated, as far as I know, in Zeno aporiae, showing contradictions in naive motion understanding. It took about two thousands years to eliminate these contradictions by inventing analysis (by Newton and Leibniz). But the understanding of motion in modern physics differs from classical. I was told that already about a century ago was established a fact that in the microcosm the only motion manner is a wave motion, but not a particle motion. On the other hand there are attempts to construct discrete nature theories, for instance, the theory of loop quantum gravity. In modern physics we have paradoxes in quantum mechanics such as wave function collapse, Schrödinger's cat state and quantum entanglement. I think they can be solved provided the quantum mechanics is incomplete, because the nature is far from our intuition, for instance, if it is non-local or fractal.