"Patricia Shanahan" <email@example.com> wrote in message news:MN9ug.firstname.lastname@example.org... > Daryl McCullough wrote: >> R. Srinivasan says... >> >>> Consider my objection made in the previoius post and cited below, >>> namely, that it is *in principle* impossible for any human being to >>> know what m is (leave alone construct it), based on the criteria laid >>> out by Norman Wildberger. For NW suggests that the number of particles >>> in the universe should determine m, the maximum natural number. But >>> what are particles, what is the universe? We human beings simply >>> *postulate* stuff like "particles" and the "universe" through our >>> theories. There is *in principle* no other way for us to access these >>> entities -- even the experiments we make only access these particles >>> indirectly, through our theories. Therefore if at all it is possible >>> for us to know what m is, it is only through our theories; but in order >>> to have a theory T about particles and the universe, T must already >>> have access to natural numbers as postulates. So if at all one can >>> arrive at a construction for m, it is possible only by circular >>> reasoning and one has to resort to Platonism or realism (particles >>> "really" exist, the universe is "really" what our theories say it is, >>> etc.) to overcome the circularity. In any case m can *in principle* be >>> specified only non-constructively by us humans, as noted below. So NW's >>> thesis is subject to the same sort of objections that he makes about >>> set theory. >> >> I think I agree. Some people want to restrict mathematics >> to what is *physically* possible. e.g.: Since it is physically >> impossible to have an infinite number of objects, or to perform >> an infinite number of actions, then completed infinities don't >> exist. Since it is physically impossible to compute the value of >> a nonrecursive function, then only recursive functions exist. >> Since it is physically impossible to construct a non-measurable >> set of reals, all sets are measurable. >> >> However, what is physically possible and what is not is determined >> by the laws of physics, and we don't *know* those laws, a priori. >> If mathematics is to apply to the real world, we should try to make >> our mathematics general enough to cover all possibilities. Allowing >> only finitistic mathematics is placing an a priori restriction on >> what the possible laws of the world might be. How do *know* that >> that's the way the world is? > > I agree, but would go a step further. Even if we know something cannot > be a law of the world, in its full messy detail, it may be useful to > study as an abstraction that cannot really exist. > > I don't believe infinite memory computers are physically possible, but > infinite memory models of computation are a valuable simplification that > lets us ignore the possibility of running out of memory. > > The set of all sets of the natural numbers may be an impractical object > to deal with directly, but we want to be able to reason about many of > its elements. Assuming the whole set exists, and examining rules for > manipulating sets of natural numbers on that basis, is simpler than > trying to come up with a definition of its "practical" subset. > > Patricia
Prime numbers don't appear in Nature (or rather, I know of only one recently found example). This doesn't mean the study of number theory is invalid because it has no "physical" models.