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Topic: Then answer to Frege's two objections to formalism.
Replies: 17   Last Post: Apr 9, 2013 7:56 AM

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Posts: 2,665
Registered: 6/29/07
Re: Then answer to Frege's two objections to formalism.
Posted: Apr 5, 2013 2:38 PM
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On Apr 5, 7:26 pm, Zuhair <> wrote:
> On Apr 5, 5:22 pm, fom <> wrote:

> > On 4/5/2013 8:47 AM, Zuhair wrote:
> > > On Apr 5, 2:25 pm, Zuhair <> wrote:
> > >> I just want to argue that
> > >> "Mathematics is analytic processing fictional or real"
> > Zuhair,
> > In the link
> >
> > you can find the argument for the descriptivist theory of
> > names used to address the question of negative existentials.

> > In the link
> >
> > you will find the statement that classical logic presupposes
> > equivalence between denotation and existence.  This comes
> > from the influence of Russellian description theory
> > and the analysis of negative existential statements.

> > This presuppostion is embraced in the axioms for first-order
> > logic given by

> > Ax(P(x)) -> P(t)
> > P(t) -> Ex(P(x))
> > The link on free logic will explain how it differs from
> > classical first-order logic in relation to names and
> > existential import.

> No problem at all, all of those are just variants of assumptive
> fictional reality (as far as first order logic is concerned, or free
> logic), that has nothing to do with mathematics per se, all
> mathematics has to do with is the CONSEQUENCES of those assumptions
> according to fictional assumed rules of logical inference, so it is of
> the kind of If-The-Ism. There is no commitment to any of those
> assumptions per se, there is only an assumptive record that has
> consequences and it is those consequences in relation to those
> assumptions that are the Analytics concerned with those methods that
> mathematics is concerned with. All what is in the links you've
> referred to is just changing the rules of the game we are to follow
> all made in a fictional world. All of those are not matters that
> mathematics is ought to be concerned with, they are indeed
> philosophical matters, but as said not mathematical.
> Zuhair

In other world you are free to choose whatever Real or fictional game
you desire to start with give them the rules you want and assume that
they are consistent, then what matters is what theorems you'll get. So
mathematics would be concerned with statements of the sort: If G then
t where G is the fictional game you desire to define (primitives,
axioms, rules of inference, formation rules, etc..) and t is the
theorem derived in the Game. IF G is inconsistent then the above
sentence is trivially True, if G is consistent as assumed then if t is
not a theorem then the whole statement is False (absolutely so), and
if t is a theorem then the statement is true(absolutely so).

So mathematics is the collection of all analytic truths, all of which
have the degree of absoluteness, that no other discipline can have,
sciences belong to empirical truths which are generally not as
absolute as analytic truths, also philosophy lies in intuitive
gestures like necessity, Generalization, etc... that are also not
absolute as analytic truths are. So mathematics is the highest in rank
of absoluteness of truths of its claims.

Matters which may arise in mathematical discourse like is ZFC
consistent? and the alike are NOT mathematical issues per se, they are
philosophical in nature since although the question is about an
analytic issue, yet it is proved by Godel that those issues are not
decidable by any analytic means, so the choice of axioms would mainly
depend on philosophical insight or some Proof system built upon such
insights, the choice of the initial axioms of those is not the job of
mathematics, although the consequences of a proof system relative to
its premises would be mathematical. Those philosophical issues are
what may be called as Meta-mathematics. Of course for mathematical
statement to be non trivial we ought to have a good meta-mathematical
apparatus, but this is another issue.


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