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Re: Mathematics as discourse about form:
Posted:
Jan 2, 2013 7:49 AM
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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
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