
Re: Then answer to Frege's two objections to formalism.
Posted:
Apr 6, 2013 6:22 AM


On Apr 6, 1:15 pm, Zuhair <zaljo...@gmail.com> wrote: > On Apr 6, 8:34 am, Zuhair <zaljo...@gmail.com> wrote: > > > > > > > > > > > On Apr 5, 2:25 pm, Zuhair <zaljo...@gmail.com> wrote: > > > > So: > > > > Mathematics is about analytics fictional or real, most interestingly > > > those in proximity with reality. > > > > Zuhair > > > It is interesting to investigate the say metaphysical basis for this > > virtualreal proximity. The psychological basis are clear, as any > > fiction it wont go viral unless it meets some demands that many people > > share. The pervasiveness of applicability of mathematics can be seen > > to be due to mathematicians limiting their logically driven virtual > > reality to mimic pervasive relations and properties throughout the > > physical universe, as I said for example an investigation of Part > > whole relation and Connectedness would be of course expected to have > > pervasive applications wide across, since every scientific discipline > > would encounter such relations. Similarly the study of a relation like > > membership and collections would definitely have pervasive > > applications, because those relations are already pervasive. Truly the > > mathematical study of those relations puts them under harsh strict > > rule guidance in a fictional world starting from premises that do > > possess high proximity to what is thought of them to be in the real > > world, the so called "naturalness" of those axioms, such conditions > > can be argued that the real world need not imitate or follow so > > harshly as regards those relations, but still it is the case that > > studying those pervasive relations cannot really be contemplated > > otherwise, and analytics of those relations under those harsh strict > > virtual grounds had been proved over time to be useful in > > understanding them in the real world! which mean that there is some > > connection between the logically driven fictional world and the real > > world they are approximating. However one to understand that this > > connection might not be identical the fictional world only PROXIMATE > > the reals world, and of course some fictional processing might lead us > > astray from the real world happenings but the error is small to be > > significant from the practical stand point. What are the metaphysics > > of that connection deserves to be studied. > > > So interesting mathematics can be understood as: Pervasive analytics > > real or fictional. > > > If we redefine virtual to be real or proximal fictional, then > > mathematics in the interesting sense burns down to > > > Pervasive virtual analytics. > > > So at the end here I've presented the answers to Freges' objections to > > formalism which are the main ones I understand. > > > Zuhair > > I'll draw two objections to this method. > > (1) It is silent on some questions of mathematical interest deeming > them as metamathematical. > (2) There are subjects of mathematical interest that are not > approximative. > > The first objection is concerned with questions like is ZFC > consistent? or is ZFC true? etc... > More appropriately put with the context of this method, questions > about justification of the primary rules of the game. Clearly this > method is not concerned with any prior specifications on choice of the > rules of the game, it opens the door wide for ALL such games, and > considers all Analytic output of those games to be mathematical > despite the metaphysical nature of the game or the justification drawn > for its premises, what matters to this method is just the fellowship > of the results from the premises whatever those premises are. But to > many mathematicians the nature of the game is itself important and > justifications around it does matter. The statement that for example: > Con(ZFC) >Con(Z) although absolutely true analytically yet it would > turn to be a trivial result if ZFC was not consistent, and ignoring > justification for the antecedent in those statements might lead us to > be involved with trivialities. > > The answer to this question lies in understanding that not every > question raised within the discourse of some discipline qualifies it > as part of that discipline. For example I need a microscope to > diagnose some infections, now a lens is a part of that microscope but > the detailed manufacture of that lens and questions raised about it is > not part of the clinical medical practice. The details of manufacture > of a drug is not part of internal medicine, yet the internist is > involved primarily in treatment of diseases using drugs. It is the > same matter here mathematics uses rule driven fictional realms to > harvest analytic facts raised within them, the fictional realm itself > and the justification for selecting its rules is not part of > mathematics. What the mathematician demands of course practically > speaking is for the metamathematician to tell him that the so and so > system have strong justification, so that he uses them without being > just wasting time with trivialities much like what the physician > demands from the pharmacist or the drug manufacturing company, that > there are evidence that the drug works. > There are still some technical answers to objection 1, that are:
(1) some weak theories can indeed prove their own consistency and can be traced back to logic itself. so those are not the target of objection 1.
(2) for those strong theories mathematics might play a rule in solving the problem of their justification by proving them to be consequences of theories that have a better justification, we can continue that until we reach into base theories that can have a philosophical justification similar to those lying behind analytic results themselves, so mathematics did cash in on intermediate basis here. But still the ultimate justification would be a non mathematical one since it is not analytic it would be a priori synthetic one lying on philosophical concepts, and it will not be part of mathematics per se.
> The other objections is that some theories have been of interest to > some mathematicians like ZFC for example, and this being a logically > driven fictional game is too far from being approximative of reality. > It is generally thought that approximating reality would require no > more strength than third order arithmetic which is way weaker than the > vastly strong ZFC, but still ZFC received mathematical interest. > > The answer is that ZFC although speaks about a structure that is > vastly huge in comparison to the real world, however still all of its > primitives and premises are to some extent clear and naturally > mimicking albeit leading to strong ideal worlds way beyond the natural > ones. Still the motivation behind such strong theories is that they > can shorten proofs about the fractions of them that are naturally > mimicking, so although some theories in third order arithmetic can be > proved taking say very long proofs by just using third order > arithmetic apparatus, yet using the stronger theory might prove them > in much shorter way! and in that manner they are practical and useful, > so they are worth studying. So those strong systems are hoped to > provide easier and shorter answers to the proximal virtual parts of > them, thus they can be understood as approximating also or more > appropriately put being useful to facilitate such approximations. > However still that objection lies against "interesting mathematics" > and not against the definition given here for "mathematics" in general > which is not needed to be mingled into such matters. > > Actually both objections are to be raised against "interesting > mathematics" rather than to the general definition of mathematics > given here, since the later will cover all those cases whether > interesting or not. Of course interesting mathematics have some meta > mathematical component in directing study of mathematical issues, but > the mathematics harvested at the end are just those that are devoid of > those assumptions, they are just the *fellowship* of results from > those assumptions which has nothing to do with the justification given > to those assumptions per se. > > Zuhair

