fom
Posts:
1,968
Registered:
12/4/12


Re: LOGIC & MATHEMATICS
Posted:
May 27, 2013 10:50 AM


On 5/27/2013 4:51 AM, Zuhair wrote: > On May 27, 7:51 am, fom <fomJ...@nyms.net> wrote: >> On 5/26/2013 11:17 PM, zuhair wrote: >> >> >> >> >> >> >> >> >> >>> On May 26, 4:49 pm, Nam Nguyen <namducngu...@shaw.ca> wrote: >>>> On 26/05/2013 3:52 AM, Zuhair wrote: >> >>>>> On May 26, 11:03 am, Nam Nguyen <namducngu...@shaw.ca> wrote: >>>>>> On 26/05/2013 12:52 AM, Zuhair wrote: >> >>>>>>> Frege wanted to reduce mathematics to Logic by extending predicates by >>>>>>> objects in a general manner (i.e. every predicate has an object >>>>>>> extending it). >> >>>>>> [...] >> >>>>>>> Now the above process will recursively form typed formulas, and typed >>>>>>> predicates. >> >>>>>> Note your "process" and "recursively". >> >>>>>>> As if we are playing MUSIC with formulas. >> >>>>>>> Now we stipulate the extensional formation rule: >> >>>>>>> If Pi is a typed predicate symbol then ePi is a term. >> >>>>>>> The idea behind extensions is to code formulas into objects and thus >>>>>>> reduce the predicate hierarchy into an almost dichotomous one, that of >>>>>>> objects and predicates holding of objects, thus enabling Rule 6. >> >>>>>>> What makes matters enjoying is that the above is a purely logically >>>>>>> motivated theory, I don't see any clear mathematical concepts involved >>>>>>> here, we are simply forming formulas in a stepwise manner and even the >>>>>>> extensional motivation is to ease handling of those formulas. >>>>>>> A purely logical talk. >> >>>>>> Not so. "Recursive process" is a nonlogical concept. >> >>>>>> Certainly far from being "a purely logical talk". >> >>>>> Recursion is applied in first order logic formation of formulas, >> >>>> Such application isn't purely logical. Finiteness might be a purely >>>> logical concept but recursion isn't: it requires a _nonlogical_ >>>> concept (that of the natural numbers). >> >>>>> and all agrees that first order logic is about logic, >> >>>> That doesn't mean much and is an obscured way to differentiate between >>>> what is of "purely logical" to what isn't. >> >>> Yes I do agree that this way is not a principled way of demarcating >>> logic. I generally tend to think that logic is necessary for analytic >>> reasoning, i.e. a group of rules that make possible to have an >>> analytic reasoning. Analytic reasoning refers to inferences made with >>> the least possible respect to content of statements in which they are >>> carried, thereby rendering them empirically free. However this is too >>> deep. Here what I was speaking about do not fall into that kind of >>> demarcation, so it is vague as you said. I start with something that >>> is fairly acceptable as being "LOGIC", I accept first order logic >>> (including recursive machinery forming it) as logic, >> >> Compositionality is an intrinsic part of the modern >> conception of logic. Originally, the idea had been >> to recognize this in the analysis of natural language, >> and, this is contemporary in the sense that linguistic >> analysis is directed at identifying a presumed underlying >> logical form. >> >> This is actually part of the basis for the FregeHilbert >> distinction. Frege viewed logical form as corresponding >> to underlying thoughts that could be taken as "the same" >> across differing colloquial language use. Thus, the >> Fregean notion differed from formalist semantical theories >> that ground the modern recursive definition of formal >> languages. >> >> Frege's identity puzzles are deprecated in the modern >> paradigm that treats the axiom >> >> Ax(x=x) >> >> as an ontological statement of selfidentity. But, >> it is simultaneously a semantical statement of >> consistent interpretation of syntactically identical >> inscriptions. >> >> The relevance of this is that the denotations of >> a logical system must be understood as a wellorder. >> If one uses the symbol, >> >> 'a' >> >> to act as a denoting symbol it can no longer be >> available as a denoting symbol when the next >> designation of such is required. So, the >> courseofvalues interpretation of a domain >> involves the presupposition of a sequential >> listing of denotations. > > Despite the detail of that, the question of whether or not predicates > have extensions in the Fregean manner is itself a purely logical > issue. I don't see it a mathematical issue, nor do I see it as a > mathematical interpretation of logic or anything like that.
The intent had been that you had no reason to concede the "mathematics of succession" since the nature of logic in its analytical role arrive at the same structural form. There had been no intention to dispute your views.
> So whether > one stipulate a theory in its favor or against it, then that > stipulation is a logical answer. And given that identity is itself a > logical concept,
I disagree with that particular view, but I am not disputing your use of it as such.
> then there is no harm at all in considering those > extensions as part of logic as far as they are stipulated in a > consistent manner, one can view them as an IMAGE of that consistent > fragment of second order logic with identity on the OBJECT world, so > it is copying the content of predication in that consistent fragment > of logic, so it do have a purely logical content, there is no way to > call that mathematical or any other discipline that logic can be > applied to. Extensional logic is logic. Now that mathematics is found > to be interpretable in it, then this yield that mathematics is a part > of extensional logic. Do you see any clear mathematical motivation > with the system I've presented? >
Since my views are nonstandard, my answer to your question is "yes". But, since I am aware of Frege's work and respect it profoundly, my qualified answer is "no".
I find it intriguing that Frege's number classes are representable in Quine's New Foundations. It may be that a "shadow mathematics" will arise because of the renewed interest in Frege and the recent serious consideration of Quine. Only time will tell as these matters are sorted out further.
Just so you understand some of my "nonstandard" commentary, recall that Frege's "number of a concept" is taken by analogy from "direction of a line". Along similar lines, Cantor took the identity of completed infinities in analogy with the singlepoint compactification of the plane and his notion of "equipollence" between powers of an infinite set generalizes perspectivity in projective geometry.
I understand certain basic relations geometrically,
news://news.giganews.com:119/Jr2dnbdYvtfPdlrNnZ2dnUVZ_tdnZ2d@giganews.com
and I understand compositionality through an intensional axiomatized equational theory,
news://news.giganews.com:119/IqudndogJ8VB1zNnZ2dnUVZ_qydnZ2d@giganews.com
Truth table semantics begins with geometric forms instantiating all possibilities for logical equivalence,
news://news.giganews.com:119/zsCdnW9U7v4BOlzNnZ2dnUVZ_hmdnZ2d@giganews.com
and ends with a canonical ordering that fixes a representation for a complete connective,
news://news.giganews.com:119/AuqdnYcXm8eaLVzNnZ2dnUVZ_hdnZ2d@giganews.com
These particular constructions actually become quite complex. Certain group theoretic constructions reflecting the choice of representation provide a syntactic labeling of the free orthomodular lattice on two generators.
My understanding of propositions is based on a free DeMorgan algebra,
news://news.giganews.com:119/Jr2dnbNYvtf9cFrNnZ2dnUVZ_tdnZ2d@giganews.com
and, in fact, I view DeMorgan algebra as foundational rather than Boolean algebra,
news://news.giganews.com:119/Jr2dnbBYvtedcFrNnZ2dnUVZ_tdnZ2d@giganews.com
The ortholattice O_6 which appears in these constructions is fundamental to models of logic as discerned by Pavicic and Megill in
http://arxiv.org/pdf/quantph/9906101v3.pdf
Although I have nothing to show for my efforts, I have taken the issue of demarcation very seriously. After all, what exactly is meant by "foundational" if one is starting somewhere in the middle?
So, please do not feel that any of my comments have been intended to argue with your investigation. As noted before, I am aware of the Fregean point of view and I respect it. You are doing interesting things with your analyses, even if I am not quite fluent enough in the details to always make some helpful comment or criticism.

