Date: Dec 2, 2012 7:18 AM Author: Zaljohar@gmail.com Subject: what is mathematics? again This is a continuation to post: What are sets? again

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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

"Discourse about form".

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

consistent discourse about form then, we are speaking about forms that

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