In article <m4ydnY_JbpdHENrMnZ2dnUVZ_vGdnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> 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 "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. > > 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 extra-logical 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.
One must be cautious when asking for a definition from fom.
WM has several times claimed that the standard bijection from the set of all binary sequences to the set of all paths of a Complete Infinite Binary Tree is a linear mapping from the set of all binary sequences regarded as a linear space over |R to the set of all paths of a CIBT.
While the mapping is easily shown to be bijective, it fails to be a linear mapping as WM describes it:
> The isomorphism is from |R,+,* to |R,+,*. Only in one case the > elements of |R are written as binary sequences and the other time as > paths of the Binary Tree. Virgil is simply too stupid to understand > that.everal flaws in WM's claim that the identity map on induces a linear map on 2^|N.
WM's flaws in making that claim work include, but are not necessarily limited to:
(1) not all members of |R will have any such binary expansions, only those between 0 and 1, so that not all sums of vectors will "add up" to be vectors within his alleged linear space, and
(2) some reals (the positive binary rationals strictly between 0 and 1) will have two distinct and unequal-as-vectors representations, requiring that some real numbers not be equal to themselves as a vectors, and
(3) WM's method does not provide for the negatives of any of the vectors that he can form, so his "space" does not qualify as a linear space in that way, either.
On the basis of the above problems, and possibly others as well that I have not yet even thought of, I challenge WM's claim to have represented the set |R as the set of all binary sequences, much less to have imbued that set of all binary sequences with the structure of a real vector space or the showed the identity mapping to be a linear mapping from one vector space to another.
Note that while WM's model doe not achieve what he claims for it, there is another model, which a reasonably competent mathematician should be able to find, which does make the mapping into an isomorphism of linear spaces.
It is a shame that someone so obviously of limited ability at mathematics as WM should feel himself so driven to try and correct his betters. --