fom
Posts:
1,969
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. I am answering it in a reasonably academic style based 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 "descriptivelydefined 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 "descriptivelydefined 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.
In the thread,
news://news.giganews.com:119/ldadnVZoyPBMv9jMnZ2dnUVZ_sSdnZ2d@giganews.com
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 extralogical names.
So, to answer a question you will not ask, 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.

