Date: Jan 2, 2013 7:49 AM Author: Zaljohar@gmail.com Subject: Re: Mathematics as discourse about form: On Jan 2, 7:37 am, Charlie-Boo <shymath...@gmail.com> wrote:

> On Jan 1, 6:55 am, Zuhair <zaljo...@gmail.com> wrote:

>

> > Add a new primitive to the language of ZFC,

> > this primitive is the binary relation "exemplifies" denoted by

> > the infix dyadic symbol ~

>

> Zuhair, you are trying to define one nebulous term (mathematics) with

> another nebulous term ("form"). In fact, "form" is so nebulous that

> it doesn't even have a definition itself, so now you have to define

> what IT is. That's not how definitions work.

>

> You define something using familiar, well understood, accepted terms

> and concepts. Then we immediately gain the wealth of knowledge of

> firsthand experience using these terms and concepts. And personal

> experience can even give meaning to something inherently

> unfathomable. What does it mean to think? That is fairly nebulous.

> But since we all do it most of the time, we can accept references to

> "thinking".

>

> That is why I keep pointing out that you have not established the

> essential properties of your system - that it defines exactly math, no

> more, no less: there is no intuitive meaning to your terms because you

> are coining them as you go along. You even use odd syntax where $

> appears in an expression next to other punctuation characters.

>

> Both Hilbert and Einstein tell us any theory can be explained in

> simple terms if we understand it enough. That is because

> understanding means to equate something with our past experiences

> which we know so well and are so comfortable with. Not introducing

> another undefined term. Undefined term Math = = Undefined term Form =

> = ???

>

> C-B

>

>

>

>

>

>

>

> > Define:

>

> > R is a form defining relation iff

> > R is an equivalence relation &

> > For all x. ~x=0 -> For all s. Exist y. y R x & s in TC(y)

>

> > TC(y) refers to the 'transitive closure of y"

> > defined in the usual manner as the minimal

> > transitive superset of y.

>

> > To the axioms of ZFC add the following axiom scheme:

>

> > Forms: if R is a binary relation symbol, then

>

> > [R is a form defining relation ->

> > For all x. Exist! $: For all y (y ~ $ <-> y R x)]

>

> > is an axiom.

>

> > Each $ here is said to be a form defined after R and x.

> > /

>

> > Now the idea is that For any theory T if T can be

> > interpreted in ZFC+forms in a manner such that

> > each object in T is interpreted as some form in

> > ZFC+forms and all primitive relations of T are

> > interpreted by defined relations in ZFC+forms;

> > then T is a mathematical Theory.

>

> > Example: Peano Arithmetic "PA" is interpretable in

> > in ZFC+forms by an interpretation where all objects

> > of PA are interpreted as forms each defined after

> > equivalence relation bijection and some set in

> > the following manner:

>

> > Define (natural number): # is a natural number iff

> > (Exist x. x is finite & for all y. y ~ # <-> y bijective to x)

>

> > It is just straightforwards to interpret all primitive predicates of

> > PA by 'defined' relations in ZFC+forms.

>

> > So PA is a mathematical Theory.

>

> > This is what I paraphrase as "PA supplying a consistent

> > discourse about form". What I mean by "consistent discourse"

> > is being interpretable in a consistent extension of logic,

> > here ZFC+forms is the consistent extension of logic and

> > since PA is interpretable in it then it provides a consistent

> > discourse of its objects and since its objects are interpretable

> > as forms in ZFC+forms then PA is said to provide a consistent

> > discourse about form. Any theory that is so interpretable

> > in any consistent theory that can define forms is accordingly

> > said to be a theory that provides a consistent discourse

> > about form and thus it is "MATHEMATICAL".

>

> > So roughly speaking this account views mathematics as being

> > about "Logic of Form".

>

> > Para-consistent (inconsistency tolerant) discourse might be

> > interesting if proves to be indispensable by the use

> > of consistent discourse, and thus theories supplying

> > para-consistent discourse about form would be

> > designated also as MATHEMATICAL. Actually I tend to think

> > that any discourse about form other than the trivial discourse

> > of proving everything if proves indispensable by use of other kinds

> > of discourse, then it would be mathematical

>

> > So Mathematics can be characterized as:

>

> > Non trivial discourse about form.

>

> > The difference between mathematics and science is that the later is

> > about establishing the TURE discourse about the objects it negotiates,

> > while mathematics is about supplying any NON TRIVIAL discourse about

> > the objects it negotiates (i.e. forms) whether that discourse is what

> > occurs in the real world (i.e. True) or whether it doesn't (fantasic).

> > Mathematics supplies a non trivial discourse of form, so it supplies a

> > language about form, and as said above more appropriately put as

> > supplying: Logic of form, whether that logic is consistency based or

> > paraconsistent. The truth of that, i.e. reality matching of the

> > discourse about those forms is something else, this really belongs to

> > a kind of physics rather than to mathematics. The job of mathematics

> > is to supply the necessary language that enables us to speak about

> > forms, not to validate its truth.

>

> > Anyhow that was my own personal opinion about what constitutes

> > mathematics.

>

> > Zuhair