On 4/4/2013 2:47 PM, Virgil wrote: > In article > <email@example.com>, > WM <firstname.lastname@example.org> wrote: > >> Piffle. Induction is valid for all elements of the inductive set. > > Let WM try to give a formal statement of what he means by he inductive > principle, then show that that principle supports his own position. > > He cannot do so! > > Either WM's statement of the inductive principle will be corrupt or > his application of it will be. > > Cantor's diagonal argument says that any listing of infinite binary > sequences must be incomplete because on can for any listing construct a > non-member of that list. > > Neither WM, nor anyone else, has manages a valid counter-argument. > > Note that claiming that no infinite binary sequences can exist supports > the argument, as does a claim that any such listing is necessarily > finite. > > Thus WM's claim of non-existence of actual infiniteness SUPPORTS the > Cantor argument. >
And, one need not be ontologically committed to transfinite numbers to admit the mathematical interest motivated by the question
"If an uncountable infinity exists, what are its properties?"
It is merely investigated through an arithmetic for transfinite numbers.
I have just started looking at a book "Pragmatism and Reference" by Boersema. It discusses the descriptivist account of names and the causal account of names. On his analysis, both accounts require an "essence" in principle. But, on his analysis, both accounts fail because of "realism".
Description theory is intimately involved with the foundations of mathematics, and, because of the considerations arising from description theory, realism is implicit to the systems of logic that had been used to formalize mathematics. Realism is also implicit to the attempts at predicative foundations such as that arising from Russell's theory of types.
Because the vast majority of mathematicians could simply care less about these things, the real winner in the foundational debates is Poincare and his conventionalism. For the most part, logicism and formalism have reduced mathematics to a meaningless game of symbol manipulation. Although the three philosophies of logicism, formalism, and intuitionism used to be well-delineated, they have borrowed techniques in such a way that it would be difficult to disentangle modern practice into "pure" forms. So, what is left is conventionalism.
And, of course, when mathematicians are performing their tasks, some notion of objecthood is being granted to the reference of their statements. This might be considered a "working platonism" to distinguish it from the ontological commitments of platonism as a philosophical perspective.
Similarly, there are branches of mathematics such as finite combinatorics that would be effectively untouched by these questions. So there is little or no issue with the overlap of techniques in these areas.
The "holdouts" in realist debates seem centered on Cantor and transfinite arithmetic. But, the fact is that the problem of truth and reference is so difficult, it cannot even be agreed upon in the philosophical community for typical uses of language. In relation to description theory, the abstract objects of mathematics correspond to "descriptively-defined names" whose only "real" relation to truth is the description by which it is introduced into the language.
When logicism is fully applied, these descriptions fall under formal statements and definability through descriptions becomes definability with respect to a model. It is at this point that Quine's (somewhat circular) argument for using descriptions to eliminate names and then to re-introduce them comes into play.
Anti-realist non-existence claims are perfectly compatible with respect to all mathematics because you cannot go out and have a beer with ZERO or get a six-pack of ONES at the corner liquor store.
For what this is worth, the notion of "mathematical existence" in the sense of non-contradiction can be found in Hilbert's writing. But, it can also be found in Leibniz who takes any possible existence as a logical existence. And, it is in Kant that one finds the remark that logic (in the form of contradictions) provides a negative criterion of truth. One of the motivations for description theory had been the investigation of objects given through self-contradicting descriptions. The example of a "round square" is fairly common. It is a small step from eliminating self-contradictory descriptions to eliminating non-existing objects. Frege took it. Russell took it. Lesniewski tried and reversed himself later (nominalism).