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.