There is no concept more evil, more corrupting than that of the continuum. Why at the very bottom of physics do we bury the most unprovable of assertions. All science is counting. No science counts for ever. Real numbers can never be more than conjecture.
But of course, you answer: it works so well! The conjecture has withstood ages, from Newton to Einstein to (fill in modern genius here.) Sure we can never prove the conjecture, but if it continues to serve us in building models of the world why should we get rid of it. Why, for that matter, should the dictates of science lead to dictates about physical reality?
The only crack we see in the idea of the continuum comes from quantum theory. Here, if the circumstances are right, we get discrete answers for different configurations. So, as many have suggested, when we try to construct a quantum theory of spacetime, perhaps there will be a discretization.
But even hear we come of short of ridding physics from the unprovable assertion. Even here we find, when we use the rules of quantum theory, that all probabilities are allowable. Again real numbers find themselves at the center of the theory.
Remarkably, there are ways in which one can get rid of both of these continuums (at least in a limited sense.) These are Roger Penrose’s spin networks. For sufficiently complicated spin networks, the networks posses two properties: they approximate directions and the approximate quantum probabilities. Combinatorial rules give rise to quantum probabilites and the full real span of probabilities is not postulated a priori. Combintaroial rules give rise to quantum probabilities which describe an object with with discrete degrees of freedom which approximate direction in three dimensional space. Funny but that they remain no more than a curiousity, or a way to find orthogonal sets of states in loop quantum gravity.
Real numbers. Bah. I’d rather believe in fairies. Us of the digital era, we are such pains in the rear.
O brave new quantum world!
1. Nothing about the physical world is provable.
2. What “corruption” follows from the continuum?
If we didn’t have the continuum, we’d have a vague
ultraviolet cutoff and that’d be just as bad.
1. Of course (caveat: why of course: I can imagine a universe in which this is not true. But I can’t currently imagine that this is our universe!) So why not admit this from the start and ban unprovable hypothesis. Note that the use of real numbers appears to be of a higher level of unprovability than say, than, say, that physical object X is in one of 1…10^6 states.
2. The “vague ultraviolet cutoff” you refer to comes from a picture of destroying the contiuum by making it a lattice. I suggest no such structure. Why can’t the contiuum approximation be broken in a more sophisticated manner?
Shouldn’t a discretized space time be phenomenolgically observable?
From my ancient experience with simulating [partial] differential equations, I remember that discretization, brings about exotic dispersion relations among other “errors”.
Will this happen to propagation of light (or gravitational waves, or anything for which physics offers a PDE) in a discrete space? The discrete “space” space, means a discrete “momentum” space, and the dispersions mentioned above will be only within the discrete limit in the momentum space and therefore unobservable, but maybe I am just confusing two things.
I disagree that the only crack (or pole?) in the continuum comes from quantum theory. Were Nature continuous, we’d expect an infinite amount of computation to be doable in a finite time — and we might want to take it as axiomatic that it isn’t. Also, we could pose physical-sounding questions, the answers to which depend on the truth or falsehood of the continuum hypothesis.
Of course, the above arguments assume we’re talking about continuity in measurable quantities. Continuity in amplitudes is so much more benign — which doesn’t prevent it from keeping me up at night anyway.