Can physics get rid of the continuum?

Question

Almost every physical equation I can think of (even though I don't actually feel comfortable beyond the scope of classical mechanics and macroscopic thermodynamics, as that's enough for dealing with everyday's engineering problems) is expressed assuming continuous domains at least for one variable to range over; that is, the real and complex number sets are ubiquitously used to model physical parameters of almost any conceivable system.

Nevertheless, even if from this point of view the continuum seems to be a core, essential part of physical theories, it's a well-known property that almost all of its members (i.e. except for a set of zero Lebesgue measure) are uncomputable. That means that every set of real numbers that can be coded to compute with -as a description of a set of boundary conditions, for example-, is nothing but a zero-measure element of the continuum.

It seems to me that allowing that multitude of uncomputable points, which cannot even be refered to or specified in any meaningful way, makes up an uncomfortable intellectual situation.

I wonder if this continuum-dependent approach to physics can be replaced by the strict use of completely countable formalisms, in a language which assumes and talks of no more than discrete structures. What I'm asking is if the fact that we can only deal with discrete quantities, may be embedded in physical theories themselves from their conception and nothing more being allowed to sneak in -explicitly excluding uncomputable stuff; or if, on the other hand, there are some fundamental reasons to keep holding on to continuous structures in physics.

Answer 1

I tend to think there is no way or even point in getting rid of the continuum. Even if an unbelievable breakthrough in physics happened to fulfill that desire, I bet old continuum theories would be more preferable "for dealing with everyday's engineering problems".

However, that's the point of my answer (which is more like an extended comment), continuum entities exhibit or can be modeled via more discrete structures. The trick is to employ more algebra.

For example when you (or your CAS) are calculating derivatives symbolically you are just applying some simple algebraic rules. You aren't calculating limits, messing with infinitesimals, instead you are finding the algebraic derivative.

Moreover, finding antiderivatives in this setting also has nothing to do with calculus. Mathematica or any other CAS uses sophisticated algorithm originated in algebra coupled with lots of heuristics to find indefinite integrals.

Or for example when multiplying or adding polynomials you don't treat them as functions $\mathbb R \to \mathbb R$.

$$\pi x^2 + 1$$

is not a very continuous object — it's just a polynomial in $\mathbb Z[x, \pi]$, it can be encoded as $[((0,0),1),((2,1),1)]$.

Differential geometry (the part calculus on manifolds) is just like calculus, but it is often presented using lots of algebra.

Of course, underlying physical interpretation is continuous, but treating "continuum" objects doesn't always require dealing with their continuity. Ultimately, you wrote the equations without having to count $\mathbb R$.

UPD As for the computability, discrete doesn't mean computable. See for example Hilbert's tenth problem — there is no algorithm to find integer roots of a polynomial with integer coefficients. Of course, if you consider such problem as discrete enough. So computability problem may arise almost everywhere and its relation to physics is indeed very interesting.

Answer 2

This is an interesting question. I don't know if it's answerable in a very strict sense, and this is also why you might see alot of comments. This is basically another one, which got too long.

Firstly, username Yrogirg makes some good points in that many mathematical theories which are modeled using the set of reals (and I think I even used these terms in the technical sense here) entail many facettes which are really agebraic in nature. You can easily implement derivations by representing the quantities of interest as distinct symbols and translating the associated abstract rules of computation to manipulation of these symbols.

You formulate the physical problem and then its sulution by taking a look at the structure of the theory and you deduce the way to get from the object "boundary/starting conditions" $\mathcal{B}$ to the object "expectation value of observable" $\langle A \rangle$ (and stating the boundary for the system is also really only providing input which has been taken from another observable). Everything in between $\mathcal{B}\rightarrow\langle A \rangle$ is computational in nature in the sense above, so in view of your question one probably only has to bother about the real-ness of the objects on the left and the right. Measuring is fundamentally about comparing two things and as the rationals $q=\frac{n}{m}$ lie dense in the reals and so no human can tell the difference, I see no reason to be worried.

The same point is made if I say that if you define a curve to be perfectly smooth in the mathematical sense and then draw a picture of it by successively approximating it using finite lines. The experiment distinguishablity of the real and the approximated curve, for each level of sophistication, can be overcome by enough time and data storage.

The point you raise regarding the null-measure of the reals is not a problem as, with respect to the observables, I don't see why you'd need to manually integrate over points. Again, the numbers express or store information about a comparison. You might for example compare areas (how often does the first fit into the second?) or shades of gray (what is darker/makes the measuring device react stronger?), and if you "compare points" you really express distances, which means you'll compare lengths. Here you get into the whole unit business.

As a remark, although I think it's not very welcomed on this site, there actually are some papers (in subfields, in which some peolpe with names are involed too) which contain statements like

"What is needed is a formalism that is (i) free of prima facie prejudices about the nature of the values of physical quantities—in particular, there should be no fundamental use of the real or complex numbers..."

To sum it up, I think your first sentence "Almost every physical equation I can think of is expressed assuming continuous domains at least for one variable to range over" is exactly what it is. A method of representation. The method of getting at the results is implied by the mathematical relations associated with the symbols.

I personally don't take anything in physical theory particularly literally, but it's a useful and accessible language in any case, so that doesn't really influence how you do physics.

Answer 3

I am not an expert, so you should take my answer with grain of salt. I think the main thing to realize is that precisely because celestial mechanics/hydrodynamics can be formulated as continuum theories, they are "easy" to discretize and solve numerically. With the advent of computers this has become even more obvious than it has been before. It is in fact so easy to solve Newtons equations numerically that it is done in one of the first of Feynman's lectures for example. Differentiability is a promise that

$$x(t+\delta t) = x(t) + v(t)\delta t$$

will be a good discretization, if you take a small enough $\delta t$. This view of analysis is of course not emphasized until you take lectures on numerical analysis. From this viewpoint the real numbers are just a polite fiction, part of "Cantor's paradise" that mathematicians won't leave as Hilbert said.

To summarize: I think it is misguided to attempt a reformulation of continuum physics, the continuum theories lead to consistent discretized versions, that can be proven to approximate them to arbitrary precision.

Answer 4

As the incredible difficulty to finding solutions to the three dimensional Ising model proves beyond doubt, discrete problems are fundamentally just as hard as continuum problems, in many cases probably even more so. More interestingly, physically relevant solutions of the higher dimensional Ising model correspond to modes of certain continuous equations, and the actual "beef" of the theory is in finding these intricate correspondences between systems that look very different on the surface, and yet, share very fundamental properties.