Date: Apr 4, 2013 4:44 PM Author: fom Subject: Re: Matheology § 224 On 4/4/2013 2:47 PM, Virgil wrote:

> In article

> <6bfe8608-1196-4d6e-8efa-934f67140e1f@a14g2000vbm.googlegroups.com>,

> WM <mueckenh@rz.fh-augsburg.de> 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.

>

Correct.

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).