Why have mathematicians used differential equations to model nature instead of difference equations

Question

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such.

Yet, they could have been just as successful in modeling natural phenomena with difference equations instead of differential equations (Just choose a very small $\Delta x$ instead of $dx$). Furthermore, difference equations don't require complicated epsilon-delta definitions. They are simple enough for anybody who knows high school math to understand.

So why have mathematicians made things difficult by using complicated differential equations instead of simple difference equations? What is the advantage of using differential equations?

My question was inspired by this paper by Doron Zeilberger

""Real" Analysis is a Degenerate Case of Discrete Analysis" http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html

Answer 1

Although small discrete systems are easy to work with, continuum models are easier to deal with than large discrete systems. Whether or not nature is fundamentally discrete, the most useful models are often continuous because the discreteness can only occur in very small scales. Discreteness is useful to include in the model if it occurs in the situation we are interested in. I think this is to a large extent a question of scales of interest.

For example, if I have a mole of gas in a container, I could well model it as individual particles. But if I want a simpler model to work with and I am only interested in the behaviour at scales well above the atomic one, the usual "continuous" fluid mechanics is a good choice. This is because at such scales the gas is essentially scaling invariant (it obeys similar laws if you zoom in) and thus calculus becomes applicable (and very powerful). This is of course not true if I go all the way to the atomic scale, but I am not interested in that scale, so it does not matter if my model treats gas in the same way at those scales as well. Large scale continuous quantities like pressure and density give a good understanding (including the ability to make good predictions quickly) and that should not be neglected. (Of course, if I want something more coarse, I can go to a thermodynamic description. Either way, modelling includes a step where the number of particles is taken to infinity to simplify mathematics.)

The "scales of interest" phenomenon happens in both directions; we may neglect both too small and too large scales. For example, it might be a good idea to model a long rod by an infinitely long one (thus in a sense removing discreteness from the model). Then one can apply Fourier analysis or any other such tools that assume that the rod is infinitely long and mathematics becomes easier. This is maybe more common with respect to time than length: Fourier or Laplace transforms with respect to time are used for systems that have finite lifetime. If we are not interested in very large scales, we can assume our system to be infinitely large.

Discrete models are probably useful if nature has genuinely discrete structure (regarding the physical system in question) and we are interested in phenomena at the scale where discreteness is visible. But seen on a larger scale, a discrete model would contain something (particles or some other discrete structure) that we cannot measure and might not even be interested in. Something that cannot be measured and does not have a significant impact on the behaviour of the system should be left out of the model. This is related to the observation that continuum models often work well for large discrete systems.

Let me conclude with an observation that is easy to miss because we are so used to it: At human scales nature seems continuous.

Answer 2

First, a historical remark: it was not until relatively recently in the history of science that people were convinced that the atomic theory of matter is correct. I believe the tide was turned by a paper by Einstein in 1905 which explained Brownian motion (as actually observed by Robert Brown) using the assumption that water is made up of molecules. Before that many scientists held the belief that the universe really is continuous, and even those who didn't had trouble arguing with the predictive and explanatory success of continuous models.

Aside from that, the premise underlying this question ignores many deep and fundamental issues associated with passing back and forth between the continuous and the discrete. The sentence "just choose a very small $\Delta x$ instead of $dx$" sweeps under the rug some profoundly difficult mathematical problems. Some examples:

• Global dynamical properties of a system are often hard to see in discrete models. For instance numerical stability issues make it very hard to discretely analyze hyperbolic systems. There are also some behaviors that just don't show up in a naive discretization - for instance, it is not at all obvious why the second law of thermodynamics is consistent with the atomic theory of gases (wherein the equations are symmetric in time).
• While there are a number of standard ways to replace an ordinary differential equation with a difference equation, the corresponding technique for partial differential equations (the finite element method) is extremely challenging and is the basis for a lot of current research in numerical analysis.
• Approximate solutions are actually not simpler than exact solutions in many (most?) cases. Consider the isoperimetric problem: find the planar curve of a given length which encloses the largest area. This can be reduced to solving a system of ordinary differential equations (the Euler equations). If you do it analytically you get a circle; if you do it discretely you get a sequence of curves which give better and better approximations of a circle. How is the latter simpler? This is a serious issue in physics: continuous models often have lots of symmetry that you lose when you discretize them.

I'll also point out that one of the hardest problems in modern mathematical physics - finding a quantum theory of gravity - has so far resisted the "just choose a very small $\Delta x$ instead of $dx$" approach.

Answer 3

Physicists use lattice approximations all the time.

But lattice models will typically break part of the symmetry of the system, which is a disadvantage both from a theoretical point of view and from a practical point of view. For example, it is not possible to make a lattice model rotation invariant (whereas most laws of physics are rotation invariant...)

Answer 4

You might be interested in some of the answers to this conceptually similar question:

What is the high-concept explanation on why real numbers are useful in number theory?

My understanding is that people often prefer the continuous over the discrete for what boils down to essentially model-theoretical reasons. Explicitly, it is often the case that statement A about discrete objects can be `traded' for statement B about objects living in a differentiable (smooth? analytic? complex analytic?) world, so that the truth of statement B implies the truth of statement A. But B is easier to prove than A, because the mathematical world in which B resides has more structure, is perhaps better studied, and admits a larger number of perhaps more powerful proof and calculation techniques than the world in which A resides. There might be honest mathematical logic arguments to back this up, like a Speedup Theorem such as one of those mentioned in François Dorais's excellent answer.

Taking this a step further, by definition, people who create models are approximating reality. Of course they will prefer to model reality in ways which give them the maximum number of fast and effective techniques to perform computations and to explain phenomena. So unless people are concerned with a phenomenon which is "essentially discrete", and sometimes even if they are concerned by "essentially discrete" phenomena (e.g. in the context of queueing theory), whenever possible people will tend to prefer differentiable/smooth/analytic/complex analytic models to discrete models.

An additional point is that if you are a consumer rather than a manufacturer of a mathematical model, you can usually just "plug and chug" without getting bogged down in foundational details. Therefore I doubt the relevance to nature-modeling "consumers of mathematics" of D. Zeilberger's argument that "discrete is better because it has shorter and more conceptually satisfying definitions".

Answer 5

The short answer why calculus, and mathematical analysis, developed the way it did, is that Newton and Leibnitz did not have computers, and especially, did not have symbolic computation. This lead to lots of pseudo-problems (to do with the so-called infinitely small, and infinitely large), and made, for better or for worse, modern mathematics, both pure and applied. Of course, even today it is convenient to have differential equations, but these too can be considered as discrete combinatorial objects. My main point was philosophical, and mainly pedagogical. Discrete Calculus is much easier to understand, since it is so concrete, and with computers, can be made even more concrete. Even in the informal approach to calculus taught to scientist and engineers there is lots of unnecessary overly-abstract notions of limit, continuity etc. One can also define continuity for discrete functions (f(x+h)-f(x) is not "too large"), and differentiability ((f(x+h)-f(x))/h is "continuous") etc. Technically things get a bit complicated, and for explicit answers it is useful to use O(h) (with a NEW, discrete) meaning of O(h), and state that the degenerate case h=0 gives the traditional formulas, and since h is SO small, it could be taken as the "REAL" thing.

Answer 6

There are very weak definitions of continuity available for series. Of course, one can easily construct functions that are continuous but not differentiable. However, the majority of USEFUL constructs if they are defined continuously ARE differentiable.

In this case, why prefer to use only difference equations, as Y.C. said above in the comments? They are not easier to work with. The easiest equations to work with are complex differentiable, as Painleve once said. Furthermore, it is questionable whether the real world is discrete. We don't know. The equations in quantum mechanics are continuous or differences of continuous equations.

Also, modern topological foundations operate with continuous concepts primarily. If we do no require continuity, we cannot understand many restrictions on what is and is not possible.

A great illustration is the book : C. Truesdell, 1948, A Unified Theory of Special Functions, based upon the functional equation $\frac{\partial}{\partial z}F(z,a)=F(z,a+1)$, the Princeton University Press

It managed to eliminate much useless specialization by revealing a general complex continuous fact that could be used to construct a general method of expressing almost any special function in terms of another one arbitrarily chosen.

EDIT:

K.C. in comments above also says much of what I was going to add here. He should post it as an answer? For this is an important reason why $\mathbb{R}$ is used in physical models.

The substance of the argument can be found in Leibnitz's 1676 papers (which also introduced the diagonal argument and 1-to-1 correspondence as measures of length!)

Seeing how much of the discussion seems to admit that mathematics models nature only approximately view (NOA-V), I would like to give a (representative?) reference for the reasons supporting the opposite viewpoint (call it the Leibnitz-Euler-Dirac View = LED-V):

C. Truesdell, 1966, Method and Taste in Natural Philosophy, Six Lectures in Natural Philosophy, Springer.

This book Six Lectures is misleading in it's title and is almost pure mathematics, not philosophy. It an even better example of the power of continuity in models, a brilliant use of continuity to solve ergodicity questions for phase spaces with finite states.

Answer 7

From Arnold's “Old and recent stories”:

The textbook of Zel'dovich (physicist) defines derivative as `the difference quotient, where difference of the argument is sufficiently small.'

He did not want to consider any limits, because in his words, `there is no sense to consider the difference smaller than $10^{-10}$: the structure of the space and time in such closeness is not described by the mathematical continuum.'

-- We are, - he said - always interested in the ratio of the finite differences, and the derivatives of mathematicians - it's just approximate mathematical formulas to calculate the ratios of finite differences.

Answer 8

There are those who pursue this in one direction or another, and often they come from the same ultrafinitist leanings of Doron. Also, there are approaches, like Loop Quantum Gravity, that end up constructing a discrete space-time through more fundamental ontology. And there are related formalisms in more standard theories, like through the calculation of holographic information content from the Holographic Principle of quantum gravity, which also arrive at finite information content.

Examples of the more "brute" ways this has been pursued (read as: more from the philosophical bent but with little mathematical formalism) can be seen in papers such as:

To the finite information content of the physically existing reality

Computational capacity of the universe

Clearly, these do not provide much more than the argument of finite content, which is Zeilberger's jump point as well. But there has been definite work in the direction of discrete geometric reasoning as well. Kustaanheimo's work in the middle of last century is often taken as the starting ground here. The standard reference is:

Kustaanheimo, P., 1951, ‘A Note on a Finite Approximation of the Euclidean Plane Geometry’, Societas Scientiarum Fennica. Commentationes Physico-Mathematicae, 15 (19): 1-11.

This has led to a body of work by others who have continued in this vein. A nice summary is found at the:

Stanford Encyclopedia of Philosophy entry on Finitism in Geometry

Note that when you look at these approaches, they do not always (or even often) strictly turn limit dynamics to difference dynamics. There are some approaches where this is what occurs, and several approaches are possible that do not impose lattices or large-scale symmetry breaking, but they usually require ontologies that more radically break from the classical.

Finally, it should be pointed out that although these approaches are often pursued by those with ultrafinitist leanings, there are a large number of contributers who do not have such foundational positions.