Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: what is mathematics? again
Replies: 4   Last Post: Dec 6, 2012 10:53 PM

 Messages: [ Previous | Next ]
 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
what is mathematics? again
Posted: Dec 2, 2012 7:18 AM

This is a continuation to post: What are sets? again

seen at:

Mathematics done before the 20_th century can be seen to be about
forms, i.e. universals exemplified by all sets bearing some isomorphic
relation between them. This way the individual properties of the
exemplifying objects would be abstracted away and what remains is the
pure form of them. Also this will make those universals free standing
in the sense that the field of the isomorphic relation dictating their
exemplification is the universe of all objects itself. Now any
relation that would come to act in the same way would be expected to
be dictating some form.

So informally speaking mathematics is a "discourse about form",

Moreover I think those forms are 'structural' universals in the sense
that if whatever exemplified by form A is part of an object
exemplified by form B then A is a part of B.

So mathematics under that assumption would be "Discourse about
structural forms".

Anyhow mathematics will be mentioned in this account simply as

Now lets take some examples of the above and see how it relates to
known pieces of mathematics.

Lets take the natural cardinal numbers.

Now a "bijection" can be regarded as a sort of 'isomorphic' relation
between sets in general (i.e. whether those are sets of ordered pairs
or not).

Now we can proceed in a Fregean manner and define a cardinal as:

x is a cardinal number <-> [Exist z. for all y. y exemplifies x <-> y
bijective to z]

It is clear that the bijective relation has the universe as its field.

so cardinal number is a form.

Now the natural ordinals like 1st, 2nd, 3rd, ... Those are a little
bit more complicated, each is exemplified by isomorphic relations. for
example 2nd is the universal exemplified by all relations isomorphic
to {(0,1),(1,2)}

So it is also a structure since the field of that relation is the
whole universe.

The Omega-th ordinal would be the universal exemplified by all
relations isomorphic to {(0,1),(1,2),(2,3)...}

Of course one can use classes to define ordered pair in a general
manner, like saying:

Exist F: [for all y. y atom of F <-> Exist a,b,c: y=(a,b,c)] &
for all y1,y2,a1,b1,c1,a2,b2,c2 (y1=(a1,b1,c1) & y2=(a2,b2,c2) -
>[c1=c2<->a1=a2 & b1=b2])

where (,,) follows some concrete fixed definition of ordered triples.

However the definition of c in relation to a,b is not related to the
definition of the (,,) itself.

The above sentence is to be abbreviated as: Exist F: F is ordered pair
defining function.

Now we can define "ordered pair" in general in the following manner:

p is an F_ordered pair <-> F is ordered pair defining function & Exist
a,b: (a,b,p) in F

p is an ordered pair <-> Exist F,a,b: F is ordered pair defining
function & (a,b,p) in F.

b is a projection of an ordered pair <-> Exist F,a,p: F is ordered
pair defining function & [(a,b,p) in F or (b,a,p) in F]

p is an ordered pair of a,b <-> Exist F: F is ordered pair defining
function & (a,b,p) in F.

R is an F_binary relation <-> for all y. y in R -> y is an F_ordered
pair.

R is a relation <-> Exist F. R is an F_binary relation

In this way we can define all relations isomorphic to some relation
regardless of the type of the implementation of the ordered pair those
relations are defined after.

Now higher degree isomorphic relations between sets of relations is
also possible, this is achieved by system isomorphism. where two
tuples (C,R1,R2,..) (C',R'1,R'2,..) of the same size are said to be
isomorphic iff a bijection from C to C' preserve all relations. This
introduces forms exemplified by systems which are usually denoted as
'structures' in structuralists views.
Those structures also fulfill the informal account here about forms
and so they are mathematical objects.

What is crucial for this definition if it is to be made rigorous is to
figure out what
constitutes an "isomorphic" relation that is a candidate for dictating
forms, and also how to spell out "field" of that relation being the
universe itself?

Here is a trial.

We'll call any relation R to be form defining iff there exist a class
X of all sets that have the relation R to each other and if the
transitive closure of X is V.

R is form defining <-> Exist X. X is a collection of atoms & (for all
sets y,z: y atom of X & z atom of X <-> y R z) & TC(X)=V.

For any form defining R define R* as:

R* = X <-> X is a collection of atoms & (for all sets y,z: y atom of X
& z atom of X <-> y R z) & TC(X)=V

form(\$) <-> Exist R. R is form defining & for all y. y exemplifies \$ <-
> y atom of R*

where "exemplifies" is a primitive binary relation.

The reason why those relations are form defining is because of them
been determined by factors that range over the whole universe of
discourse, then the individual characteristics of sets other than
those formal ones would be abstracted away, so only the form those
sets possess would be extracted, and as said earlier because the
process contain some feature that make it cross about all the universe
of discourse, then those forms are free standing, and those are the
ones mathematics is concerned with, otherwise they could be claimed to
be forms limited to some part of the set hierarchy, much as "cat" is a
form limited to some part of the animal kingdom. So being Free
standing discriminate them from ordinary forms peculiar to specific
part of the set hierarchy.

So mathematics is "discourse about form" with this it is meant any
theory that can be interpreted in the set hierarchy (which is as
mentioned above a logical background theory) having all its objects
being interpreted as forms. So for example PA is a piece of
mathematics since it can be interpreted in this theory with an
interpretation in which all its "objects" are interpreted as "forms"
defined after "bijection" relation in the Fregean manner. So it is a
case of discourse about form, thus mathematical!

So here there is a line of separation between what is foundational and
what is mathematical, the set\class hierarchy is foundational i.e. it
belong "essentially" to logic! it is a sort of extended logic,
although it definitely use some mathematics to empower it and actually
it needs a mathematician to work it out, yet this doesn't make out of
it mathematical, the piece of mathematics used in those
foundational theories is just an application of mathematics to another
field much as mathematics are used in physics. So what I'm saying here
is a theory like ZFC is not "Essentially" about mathematics, it is not
even a piece of mathematics, it is a form of LOGICAL theory.

So Set theory is a kind of LOGIC. However one can easily see that such
form of logic can only be handled by mathematicians really, but still
that doesn't make out of it a piece of mathematics as mentioned above.

Mathematics is the study of "form" as mentioned above, it is
"implemented" in the set\class hierarchy, it provides a discourse
about forms whether simple or structural. All known branches of
mathematics: Arithmetic, Real analysis, Geometry, Algebra, Number
theory, Group theory, Topology, Graph theory, etc... all can be seen
as discourse about form, since all its objects can be interpreted as
forms.

Anyhow it is reasonable for branches of mathematics to be developed
along some Foundation back-grounding in logic, and then the
mathematical forms be implemented on that background logic, this can
be seen clearly with topology which starts from set theory and then go
higher to deal with forms like continuity and connectedness. However
it can be seen to be essentially about the higher concepts it tries to
manipulate, the back-grounding in sets is just the logical part of it,
since what it tries to manipulate is a sort of "form", then topology
is essentially mathematical.

Also I wanted to raise the issue that "any" consistent theory is
speaking about a model that is "possible" to exist! So if we secure a
might possibly exist. And that's all what mathematics needs to bring
about. Whether those forms really exist or not? this is not the
discipline of mathematics. So consistency yields "possible" existence,
and that's all what mathematics should yield, i.e. forms that could
possibly exist.

How those forms are known to us? the answer is through their
exemplification as part of the discourse of consistent theories about
form. Whether they are platonic in the sense of being in no place no
time, etc.., that is not relevant, we come to know about them by their
exemplifications which are indeed not so abstract and can be grasped
by our intellect. How can such an abstract notion be exemplified by
such concrete objects, that's not the job of mathematics to explain.

So the above account give some loose informal account on what is
Mathematics.

Zuhair

Date Subject Author
12/2/12 Zaljohar@gmail.com
12/2/12 Roy Woo
12/2/12 Charlie-Boo
12/2/12 Charlie-Boo
12/6/12