>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?