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


On Jan 2, 7:37 am, CharlieBoo <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 = > = ??? > > CB > > > > > > > > > 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". > > > Paraconsistent (inconsistency tolerant) discourse might be > > interesting if proves to be indispensable by the use > > of consistent discourse, and thus theories supplying > > paraconsistent 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



