Continuity

Full Title or Meme

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

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.