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Topic: Mathematics is
Replies: 27   Last Post: May 25, 2012 4:36 PM

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 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: Mathematics is
Posted: May 25, 2012 4:19 AM

On May 25, 4:40 pm, Zuhair <zaljo...@gmail.com> wrote:
> On May 24, 10:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>

> > On May 22, 8:17 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>
> > > "Zuhair" <zaljo...@gmail.com> wrote in message
>
> > > > The investigation of behaviors that are formalizable in consistent
> > > > formal systems extending logic.

>
> > > I'd think that only captures the interests of formal logic.
>
> > > > Logic is the study of Tautology.
>
> > > As I have it, logic is the study of self-contradiction, itself informally
> > > defined after the notion of mutual incompatibility of predicates.

>
> > If we examine a Table of Tautologies..
>
> >www.tinyurl.com/SetTheory2
>
> > So we can make Induction a Tautology in the form A^B->C
>
> > TAUTOLOGIES
> > -----------
> > A        B             C            TYPE
> > a        a->c          c            Modus Ponens
> > d->e     e->f          d->f         Transitivity
> > !(!d)    TRUE          d            Double Negation
> > P(0)     P(n)->P(S(n)) P(n)         Induction Formula

>
> No induction formula is not a tautology.
>
>
>

> > LOGIC might be defined as the FORMAL *USE* of TAUTOLOGIES.
>
> When making an informal definition I like some general words, I
> defined logic as:
>
> The study of Tautology.
>
> This includes having a system that can capture any tautology, and also
> includes tautologistic implication from a set of non tautologies which
> is what you call
> the formal use of tautologies. So the word "study" informally covers
> both of those aspects.
>

Right, but the goal of logic is to be mechanistic, so taking logic as
a discipline for granted it's a practical field, a utility, much like
calling the Institute Of Sport where athletes go to 'study'.

>
>

> > and MATHEMATICS the set of FORMALIZ*ABLE* Studies.
>
> > for instance, it would impossible to have LOGIC (or mathematics)
> > without MODUS PONENS.

>
> > FACT1 & FACT1 IMPLIES FACT2
> > ergo FACT2

>
> > So LOGIC appears to use a PROVEN-THEORY, an EXTENSION of TAUTOLOGIES
>
> In the strict formal sense we are using the word logic here, we can
> say it is confined to those systems that merely capture all
> tautologies. Those systems can have nine or 5 axioms and together with
> Modus Ponens they can capture all tautologies. Also this will cover
> tautologistic implication of non tautologies. And for various reasons
> I tend to also include "equality" among Logic. So if you are working
> with first order languages then logic will be confined to tautologies
> and tautologistic implication and also have the axioms of identity
> theory added to it. This identity theory is an extension of
> tautologyies in your sense it is both consistent and complete and I
> accept it as a part of logic although identity axioms by themselves
> are not tautologies. So in reality identity is mathematical but I'll
> joint it to logic sense it is too simple actually.
>
> Now all systems obtained by adding sentences that are not tautologies
> as axioms (aka non logical axioms) to the axioms that can capture all
> tautologies (aka logical axioms) and axioms of identity theory, are to
> be considered as mathematical systems because simply what is
> formalizable in those is not logics in the strict sense it involves
> something external to logic, this thing is mathematics!
>
> So I'm looking at a spectrum here at the bottom of which are the
> logical theories then mathematical and those will branch to
> specialized mathematics that is of use to other disciplines of human
> knowledge (applied mathematics) or to fantasy mathematics (non applied
> end mathematics).
>
> You can view this as a TREE. The roots are logic the Stem and the
> branches are Mathematics and the fruits are applied mathematics and
> the leafs are  non applied too particular mathematics (like the
> solutions to some fantasy questions)

Sheer poetry Zuhair! misc.writing added for a reason!

>
> This Tree has much more leafs than fruits, therefore I tend to say
> that mathematics is a kind of Rationalization, in simple words
> Logicism.
>
> Zuhair

Herc
--
TM-SIZE MAX-1s OUTPUT
-------------------------
BB(2) 6
BB(3) 38
BB(4) 3,932,964
BB(5) 1.7 x 10^352
BB(6) 1.9 x 10^4933
...
BB(199) COOPERS NUMBER
BB(200) UNIVERSAL TURING MACHINE SIZE
includes PorkyPig Jnr's Number = CN+1