On Apr 5, 7:26 pm, Zuhair <zaljo...@gmail.com> wrote: > On Apr 5, 5:22 pm, fom <fomJ...@nyms.net> wrote: > > > > > > > > > > > On 4/5/2013 8:47 AM, Zuhair wrote: > > > > On Apr 5, 2:25 pm, Zuhair <zaljo...@gmail.com> wrote: > > >> I just want to argue that > > > >> "Mathematics is analytic processing fictional or real" > > > Zuhair, > > > In the link > > >http://plato.stanford.edu/entries/existence/#FreRusExiNotProInd > > > you can find the argument for the descriptivist theory of > > names used to address the question of negative existentials. > > > In the link > > >http://plato.stanford.edu/entries/logic-free/ > > > 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.