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Topic: Mathematics and the Roots of Postmodern Thought
Replies: 1   Last Post: Mar 28, 2013 1:15 AM

 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 UTC-7, fom wrote:
>>> On 3/26/2013 9:06 PM, david petry wrote:
>>
>>

>>>> Goedel did write a short (about 1 page) article giving his analysis

>>
>>

>>> 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
>> stand-alone 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.
> 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 well-defined
formula (namely, the one obtained from the q-th
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
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 descriptive-semantic 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 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.