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


Re: Mathematics and the Roots of Postmodern Thought
Posted:
Mar 28, 2013 1:15 AM


On 3/27/2013 1:09 PM, fom wrote: > On 3/27/2013 12:51 PM, david petry wrote: >> On Tuesday, March 26, 2013 9:33:57 PM UTC7, fom wrote: >>> On 3/26/2013 9:06 PM, david petry wrote: >> >> >>>> Goedel did write a short (about 1 page) article giving his analysis >>>> of the Liar paradox. >> >> >>> I do not see it in what I own. That does not mean it is >>> not there. It is just not in any titles or titles that >>> seem related relative to your description. >> >> It's very possible that the "article" I'm talking about was not a >> standalone article but rather a section or chapter of a larger article. >> > > Not everything gets into "collected works" of authors. > > But, he gave many small lectures and many short articles. > The titles in the table of contents are uninformative. > So, I will just have to keep my eyes open. > > I will check the larger papers now that you make that point. > >
He does mention the Liar paradox in "On Formally Undecidable Propositions of 'Principia Mathematica' and Related Systems". I do not think this mention justifies your criticism on pragmatic grounds, however. You may decide.
He writes:
"The analogy of this argument with the Richard antimony leaps to the eye. It is closely related to the 'Liar' too;^* for the undecidable proposition [R(q);q] states that q belongs to K, that is, by (1), that [R(q);q] is not provable. We therefore have before us a proposition that says about itself that it is not provable [in PM]^**. The method of proof just explained can clearly be applied to any formal system that, first, when interpreted as representing a system of notions and propositions, has at its disposal sufficient means of expression to define the notions occurring in the argument above (in particular, the notion 'provable formula') and in which, second, every provable formula is true in the interpretation considered."
The two footnotes:
"* Any epistemological antinomy could be used for a similar proof of the existence of undecidable propositions"
"**Contrary to appearances, such a proposition involves no faulty circularity, for initially it [only] asserts that a certain welldefined formula (namely, the one obtained from the qth formula in the lexicographic order by a certain substitution) is unprovable. Only subsequently, (and so to speak by chance) does it turn out that this formula is precisely the one by which the proposition itself was expressed"
My impression is that you read something else with more detail than the introduction of his original paper. I will check a little further.
Since your argument is pragmatic, and since you may appreciate a little support from some ignored subject matter, I will share with you some passages that speak to the reasonableness of your criticism (even if it falls on deaf ears).
This is from an "elite philosopher" Francois Recanati who quotes Kaplan, Stalnaker, and Hintikka.
"In what sense is it possible to separate the relation between words and the world from the use of the words? There is no doubt that the relations between words and the world hold only in virtue of the use which is made of the words in the relevant speech community: meaning supervenes on use.^* That is something the logical empiricists fully admitted. Still, a distinction must be made between two things: the conventional relations between words and what the mean, and the pragmatic basis for those relations. Though they are rooted in, and emerge from, the use of words in actual speech situations, the conventional relations between words and what they mean can be studied in abstraction from use. Such an abstract study constitutes semantics. The study of the pragmatic basis of semantics is a different study, one which belongs to pragmatics or (as Kaplan put it) metasemantics:
'The fact that a word or phrase has a certain meaning clearly belongs to semantics. On the other hand, a claim made about the basis for ascribing a certain meaning to a word or phrase does not belong to semantics... Perhaps, because it relates to how the language is used, it should be categorized as part of ... pragmatics..., or perhaps, because it is a fact about semantics, as a part of Metasemantics.' (Kaplan, 1989)
In the same Carnapian spirit Stalnaker distinguishes between descriptive semantics and foundational semantics:
'"Descriptive semantics" ... says what the semantics for the language is, without saying what it is about the practice of using that language that explains why that semantics is the right one. A descriptivesemantic theory assigns semantic values to the expressions of the language, and explains how semantic values of the complex expressions are a function of the semantic values of their parts... "Foundational semantics" [says] what the facts are that give expressions their semantic values, or more generally, ... what makes it the case that the language spoken by a particular individual or community has a particular descriptive semantics.' (Stalnaker, 1997)
The uses of linguistic forms on which their semantic depends, and which therefore constitute the pragmatic basis for their semantics, are their *past* uses: what an expression means at time t in a given community depends upon the history of its uses before t in the community. But, of course, pragmatics is not merely concerned with past uses."
The footnote reads:
"The relations of reference which are studied in semantics are neither directly observable nor independent of what men do and decide. These relations are in some sense themselves established an 'upheld' through human behavior and human institutions... In order to understand fully the basis of semantics, we are thus led to inquire into the uses of our symbols which bring out the ways in which the representative function of our language comes about" (Hintikka, 1968)
For what this is worth, there is nothing "crackpot" with respect to what motivated certain specific features of your argument. And, while I am myself little more than a "crank" poster on these newsgroups, I hope you see that I respect your criticism along these lines.



