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.
\nBut 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?
\nThe 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.
\nBut 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.
\nRemarkably, 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.
\nReal numbers. Bah. I’d rather believe in fairies. Us of the digital era, we are such pains in the rear.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 … <\/p>\n