R. Srinivasan wrote: > Patricia Shanahan wrote: > > Stephen Montgomery-Smith wrote: > > > guenther vonKnakspot wrote: > > >> Stephen Montgomery-Smith wrote: > > > > > > > > >>> Now, one of the respondents said that Norm had rejected the axiom of > > >>> infinity, and so how is he supposed to do number theory. But nowhere in > > >>> his article has he rejected the axiom of infinity. He has rejected the > > >>> whole notion that axioms are the way to go. I would paraphrase what he > > >>> said slightly differently - axioms (might) describe the natural numbers, > > >>> but they don't define them. There is clearly a problem that there are > > >>> numbers between 1 and googolplex that we can never write down or > > >>> meaningfully describe. What makes us think that they are really there? > > >>> The evidence is at best empirical. > > >> > > >> Think about your above paragraph for a minute. Do you really think that > > >> the Number 2 is more real than some natural number which has never been > > >> mentioned or even thought of? Do you really think that "2" is the > > >> Number 2 ? "2" is the representation in one particular language of the > > >> name in one particular language of the Number 2. The Number 2 is > > >> neither its name in any language nor the representation of any such > > >> name in any particular language. The Number 2 is something which does > > >> not exist physically and can not be experienced by the senses. It is an > > >> abstract concept as are all other numbers. > > > > > > I think your opening "think about it" sums it up best. The answer is > > > not obvious to me. I do personally hold some Platonic belief that each > > > of the natural numbers, even all of those between 1 and googolplex, have > > > an abstract existence. But I regard my viewpoint as more a statement of > > > faith (perhaps something that is quote self evident unquote) than > > > anything else. I think that questioning my viewpoint is a reasonable > > > thing to do and not at all crankish. > > > > Isn't there also a simplicity argument here? The set of all natural > > numbers, with no gaps and no end, is in many ways an easier thing to > > reason about than the set of natural numbers that someone, somewhere, > > has used or will use. > > > It is clear that if the set of (feasible) natural numbers that can be > "meaningfully described" (or constructed) exists, one must use some > kind of fuzzy reasoning to deal with such a set X. For X cannot have a > sharply defined maximum -- if it did have a maximum m, there is no > *conceptual* reason why m+1 should not be feasible.
There is a big difference between doing operations on numbers, and operations on letters.
For example, factoring letters is very easy. Suppose m is a million digit number which has two prime factors. I can factor it in one second, voila: m=pq.
Of course, this is completely useless, and is not factoring at all. It's just *pretending* to factor. Take the method to the code-breakers at the NSA, and they'll look at you like a fool. They aren't interested in phony-factoring with letters. They're interested in real-factoring, with numbers.
The same holds for this m+1 argument. Sure, you can always add one to any letter, just like you can factor any letter into other letters. You can't always add one to a number. To see that, try to express the m+1 argument again, without using letters. Express m and m+1 directly as a specific numbers. Then the problem is evident. This is exactly analogous to the fool who says: I can factor any number m into m=pq. He suddenly has a problem if you force him to use specific numbers instead of letters.
Letters are a deceiptful technique for pretending to do things you actually can't do.
> If one argues that > that all the resources available in the universe were used to construct > m, and on physical grounds m+1 cannot exist, it follows that no human > being can provide such a construction for m in his/her lifetime at > least. It then follows that no human being can provide a construction > for m-1, m-2, ... So one is again forced into the kind of > non-constructive reasoning that Wildberger is objecting to in the first > place. Such non-constructive reasoning with natural numbers will > probably be similar to non-standard analysis (m would be like a > non-standard integer) will definitely not be permitted in my proposed > logic NAFL. > > I really doubt if ultra-finitism is the way to go when considering the > philosophical objections to infinite sets. I believe that the correct > resolution lies in NAFL, with N taken as a proper class rather than as > a set of natural numbers. The objections to inifnitary reaoning are > correctly captured by the NAFL ban on infinite sets and quantification > over infinite (proper) classes. See my arXiv paper <math.LO/0506475>, > which shows how nevertheless real analysis can be formulated in NAFL, > and suggests a new paradigm for computability theory as well. I suggest > that academicians -- i.e., mathematicians, theoretical computer > scientists, theoretical physicists, philosophers, logicians -- address > the issues raised in my work. A first step is to accept that the logic > NAFL has been propsed and examine it critically, objectively and > professionally. Wonder when this is going to happen and who is going to > bell the cat. Pretending that NAFL doesn't exist is not going to make > it go away. > > Regards, RS