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Topic: Matheology § 223: AC and AMS
Replies: 102   Last Post: Apr 18, 2013 12:26 AM

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 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology § 223: AC and AMS
Posted: Mar 18, 2013 5:57 PM

On 3/18/2013 2:05 PM, WM wrote:
> On 18 Mrz., 17:36, fom <fomJ...@nyms.net> wrote:
>

>> Science is based on principles.
>
> One may believe in principle that it is possible in principle to find
> in principle a second prime triple. But in mathematics we prove that
> these principles are violated.

>>
>> One may choose *in principle*
>>
>> One may name *in principle*
>>
>> Although related, they are not the
>> same.

>
> They are exactly the same for immaterial objects.
> Or what do you understand by "choosing a number" - in principle?

Now, just so we are clear. You have asked a question.
on materials in the literature that support my own ideas.

In the future, I would appreciate reciprocation in kind.

The syntax of definability in mathematics, except in
so far as it has been obfuscated with careless uses
of other syntactic forms in model theory, corresponds
with the syntax of definite description. One may find
a specific discussion of the definitional syntactic form
I have in mind here in "Some Methodological Investigations
on the Definability of Concepts" by Tarski.

There is a qualified version of naming in description
theory called a "descriptively-defined name" whose
debate often revolves around the planet Neptune. As a
name, 'Neptune' had entered the vocabulary of astronomers
before it had been materially witnessed as a material
object. This example is interesting to description
theorists precisely because it conflicts with various
semantic theories based on Russellian acquaintance.

One of the problems with descriptions -- and definitions
in general -- is that there can be a multiplicity of them.

In "Language Acts" Searle attempts to address this problem
with a cluster theory of descriptions.

In "Naming and Necessity" Kripke attempts to describe
a scenario skeptical of Searle's theory. Although
never formally introduced, Kripke's scenario is now
called the causal theory of naming.

In "Pragmatism and Reference" David Boersema begins his
book with an analysis of these positions. At the end of
his analysis, he says the following in (dialectical)
support of Searle's position:

"There is another reason to suggest that although thesis (5)
[from Kripke interpreting Searle's arguments],

(5)
The statement 'If X exists, then X has most of the Phi's"
is known a priori by the speaker

may indeed be a thesis of Searle's view, he would be
glad to accept it as such (and in fact this could be
seen by Kripke as not a point against Searle's view).
It is this: if one believes that names have essences
(or, rather that objects named have essences), and if
one believes that at least one of the descriptions
associated with this name 'picks out' this essence, then one
might be more inclined to accept the claim that the
disjunctive set is analytically true of the object, and, as
a corollary, that the sufficiently reflective speaker knows
this analytic truth a priori.

"Thesis (6): [from Kripke interpreting Searle's arguments],

(6)
The statement 'If X exists, then X has most
of the Phi's" expresses a necessary truth
(in the idiolect of the speaker)

As noted earlier, Kripke holds that this necessity thesis,
like the aprioriticity thesis (5) is false. Again,
if we make the amendment from 'most of the Phi's' to
'some of the Phi's,' such a statement is true of
Searle's view, but it is not so obvious that this is
so objectionable or unintuitive. If Kripke is saying
that for the cluster account it must be the case that
X exists, then some description must be believed to
be true of X (and is true of X), then indeed Searle
is committed to the thesis (assuming Searle is
committed to names having meanings). But, again, as
with thesis (5), this seems to be a commitment to a fact
about language and the use of names, not a commitment
to facts about any objects. Furthermore, if one believes
that at least one description of the disjunctive set of
descriptions associated with a name picks out this
essence, then this thesis may not only be acceptable,
but desirable."

Now, Aristotle talks about "essence" in his writing
and the remarks above are referring to his use. But, just
as Frege tried to ground his systems with the "truth of actuality"
Aristotle tried to ground "essence" with "substance".

These things are unnecessary in mathematics if one places
the coherence of mathematical facts prior to the actuality
of mathematical facts.

All that matters is that for any name used by mathematicians,
it is related to a name that can be understood as 'essential'
in the sense of these requirements on "naming in principle"

If one treats mathematics "in principle" as being organized
according to "what is prior and what is posterior" relative
to the use of formal sentences in a deduction, then every
object to which a mathematician would refer has a "first"
description. Thus, there is a principled way to understand
how names in mathematics could have essences.

A foundational theory such as set theory provides the
context. As a foundation, it is presupposed when mathematics
is done informally.

The argument here culminates in an admissibility criterion.
In order to be a faithful model, a model of set theory must
be shown to have a *canonical well ordering*.

Then, all of the names used by mathematicians fall under the
theory of "descriptively-defined names" with the issue of
multiplicity addressed by the faithfulness criterion.

In the post,

news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdnZ2d@giganews.com

you will find a discussion of these matters
with regard to set theory addressed in through
examination of finite situations.

you will find that I tried to start a more
general discussion of how this assumption
relates to the axiom of choice through the
implicit dependence of the satisfaction maps
on extra-logical names.

why would a mathematician care about such
things?

In every book on set theory, one finds

"ZF has no universal class"

"the set universe V"

This is simply nonsense. My formal theory
is designed to address that by extending
any and every model by exactly one point.
It does so by focusing on descriptions and
names.

It is a theory that will probably go to the
grave with me.

Date Subject Author
3/14/13 Alan Smaill
3/14/13 mueckenh@rz.fh-augsburg.de
3/14/13 Virgil
3/14/13 fom
3/14/13 mueckenh@rz.fh-augsburg.de
3/14/13 fom
3/14/13 mueckenh@rz.fh-augsburg.de
3/14/13 fom
3/15/13 mueckenh@rz.fh-augsburg.de
3/15/13 fom
3/15/13 mueckenh@rz.fh-augsburg.de
3/15/13 Virgil
3/15/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 Virgil
3/17/13 fom
3/17/13 Virgil
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 Virgil
3/17/13 fom
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3/17/13 Virgil
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3/18/13 fom
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3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/18/13 Virgil
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
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4/17/13 Virgil
3/18/13 Virgil
3/18/13 Virgil
3/18/13 fom
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
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3/18/13 Virgil
3/19/13 fom
3/18/13 Virgil
3/18/13 fom
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
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3/18/13 Virgil
3/18/13 fom
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
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3/19/13 Virgil
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3/18/13 fom
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4/17/13 Virgil
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3/14/13 Virgil
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