Difference between revisions of "Continuity"

Revision as of 16:41, 20 October 2024

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This is a discussion of the need for continuous functions to model physical reality.

Context

Problems

• Particles in physics are discreet, but almost every physical equation 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 that allowing that multitude of uncomputable points, which cannot even be referred to or specified in any meaningful way, makes up an uncomfortable intellectual situation.

Can this continuum-dependent approach to physics 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.

Think of it as a case for Occam's Razor: why would we assert that such an infinite amount of points "are out there", if we can't possibly refer to them? We have no intuition about, and no possible way to reach the full continuum. Even if I'm taking too much a philosophical stance with this, it makes me think of "here be dragons" warnings at the corners of Middle Age's maps.

Solutions

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" B

to the object "expectation value of observable" ⟨A⟩
(and stating the boundary for the system is also really only providing input which has been taken from another observable). Everything in between B→⟨A⟩
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=nm
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.

one of the first of Feynman's lectures for example. Differentiability is a promise that

(t+δt)=x(t)+v(t)δt

This is not a requirement that physical reality is continuous, only that we can predict where the particle is headed, RIGHT NOW!

In the real world, adding a little bit of thermal or quantum jitter removes this property of point particle differential equations, as does smoothing out things to continuous fields with smoothing dynamics.

References