Kevin Karn wrote: > R. Srinivasan wrote: >[...] > > > > 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. > > Yes, I see what you're saying. But why not reject both set theory, and > NW's theory about m? Why is it important to concern ourselves with > infinity, or with m (which, for all practical purposes, is an analog of > infinity)? Why not just ignore them both? > > (In terms of the theory of computation, I would state the position this > way: There are no infinite tapes. Therefore, it is sufficient to study > finite automata. This doesn't mean we have to establish a certain m, > which is the maximum size of any possible FA. It just means we don't > concern ourselves with impossible things, like infinite tapes. You can > reject infinity without specifying m. You just ignore them both. What > this looks like to the casual observer is a scientist simply studying > real machines, and avoiding metaphysical tangents.) > > What is the scientific reason why we have to pay attention to concepts > like infinity and m, which are so utterly removed from our experience? > Surely we don't have to take a stand on these issues for practical > reasons. So why is it necessary to take a stand?
Turing machines that halt do not require infinite tapes -- so these are not infinite objects. What about TM's that do not halt, but for which a proof of not-halting is available? E.g. the TM goes into a simple and obvious loop. I feel these are not problematic either -- they might be treated as finite objects if we let them run for a certain large number of steps, and then pause and look for (and find) a proof of not-halting -- either via human input or via an automated proof-checker -- and then return an error condition and halt. So we could treat these as finite objects, provided the proof of not-halting exists. The really problematic case is with Turing's proof that there must exist TM's that do not halt, but for which a proof of not-halting is not available. These are unavoidably infinite objects and are problematic. The logic NAFL that I have proposed does not permit these and so Turing's proof of their existence must use some infinitary reasoning not acceptable in NAFL. It is obvious that even the concept of "all" Turing Machines is problematic in NAFL if Turing's result is true, for NAFL does not permit one to quantify over infinitely many infinite objects (given that Turing's result implies that there must exist infinitely many TM's that do not halt, with no proof of not-halting available). So you can see that Turing's proof cannot go through in NAFL.
Regarding why we need infinities, like real numbers -- there are many applications of real numbers like the Newton-Leibniz calculus, which have been undeniably useful. The problem with classical real analysis is that it leads to paradoxes (like Zeno's paradoxes, the Banach-Tarski paradox). So can one do real analysis without these paradoxes, using only finitary methods? That is what I claim is possible in my cited arXiv paper, but it would be a limited version of real analysis in the sense that large portions of classical real analysis would be excluded.