Russell: > As an instance of the sort of problems you face, the number 0.1 can > be represented as a finite string in base 10, but cannot be > represented as a finite binary string (floating point number). Is 0.1 > a valid number then? Unless you completely do the Kronecker thing, or > course....

You ask if 0.1 is a valid number, but I ask, "For what?" Surely it is a valid concept, as is "altruism" is in ethics, or "29.92 inches of mercury" is in meteorology. But these are higher-level abstractions that describe objects found in the mind - not in nature. Let me just say, I do not believe that the number 0.1 is necessary to compute the motion path of an electron, or the beating of your heart. Likewise, we will never have to calculate the effect of one-tenth of a photon on the Earth because photons come in whole numbers. Now, if you're going to argue that photons might come in fractional values - in the form of "probabilities", then I must necessarily ask... What's a probability? How does the universe perform probabilistic calculations? On its infinite analog slide rule? By flipping a coin? What's a coin? How does it fly through the air? What is air? What's a slide rule? What's a fraction? It's a vicious cycle. We need to get down to some answers. We need to be able to compute it. Cellular automata do this just fine. Do you object to that? Joel