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Mathematics as discourse about form:
Posted:
Jan 1, 2013 6:55 AM
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Add a new primitive to the language of ZFC,
this primitive is the binary relation "exemplifies" denoted by
the infix dyadic symbol ~
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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