Math Forum Discussions

Roy Woo

Posts: 1
Registered: 12/2/12

Re: what is mathematics? again
Posted: Dec 2, 2012 9:49 AM

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Mark it
On Sunday, 2 December 2012 20:18:57 UTC+8, zuhair wrote:
> This is a continuation to post: What are sets? again
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> seen at:
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> https://groups.google.com/group/sci.logic/browse_thread/thread/a78c4e246a5c8a58/4345de40b3794321?hl=en#4345de40b3794321
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> Mathematics done before the 20_th century can be seen to be about
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> forms, i.e. universals exemplified by all sets bearing some isomorphic
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> relation between them. This way the individual properties of the
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> exemplifying objects would be abstracted away and what remains is the
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> pure form of them. Also this will make those universals free standing
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> in the sense that the field of the isomorphic relation dictating their
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> exemplification is the universe of all objects itself. Now any
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> relation that would come to act in the same way would be expected to
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> be dictating some form.
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> So informally speaking mathematics is a "discourse about form",
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> Moreover I think those forms are 'structural' universals in the sense
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> that if whatever exemplified by form A is part of an object
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> exemplified by form B then A is a part of B.
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> So mathematics under that assumption would be "Discourse about
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> structural forms".
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> Anyhow mathematics will be mentioned in this account simply as
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> "Discourse about form".
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> Now lets take some examples of the above and see how it relates to
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> known pieces of mathematics.
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> Lets take the natural cardinal numbers.
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> Now a "bijection" can be regarded as a sort of 'isomorphic' relation
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> between sets in general (i.e. whether those are sets of ordered pairs
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> or not).
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> Now we can proceed in a Fregean manner and define a cardinal as:
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> x is a cardinal number <-> [Exist z. for all y. y exemplifies x <-> y
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> bijective to z]
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> It is clear that the bijective relation has the universe as its field.
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> so cardinal number is a form.
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> Now the natural ordinals like 1st, 2nd, 3rd, ... Those are a little
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> bit more complicated, each is exemplified by isomorphic relations. for
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> example 2nd is the universal exemplified by all relations isomorphic
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> to {(0,1),(1,2)}
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> So it is also a structure since the field of that relation is the
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> whole universe.
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> The Omega-th ordinal would be the universal exemplified by all
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> relations isomorphic to {(0,1),(1,2),(2,3)...}
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> Of course one can use classes to define ordered pair in a general
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> manner, like saying:
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> Exist F: [for all y. y atom of F <-> Exist a,b,c: y=(a,b,c)] &
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> for all y1,y2,a1,b1,c1,a2,b2,c2 (y1=(a1,b1,c1) & y2=(a2,b2,c2) -
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> >[c1=c2<->a1=a2 & b1=b2])
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> where (,,) follows some concrete fixed definition of ordered triples.
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> However the definition of c in relation to a,b is not related to the
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> definition of the (,,) itself.
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> The above sentence is to be abbreviated as: Exist F: F is ordered pair
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> defining function.
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> Now we can define "ordered pair" in general in the following manner:
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> p is an F_ordered pair <-> F is ordered pair defining function & Exist
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> a,b: (a,b,p) in F
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> p is an ordered pair <-> Exist F,a,b: F is ordered pair defining
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> function & (a,b,p) in F.
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> b is a projection of an ordered pair <-> Exist F,a,p: F is ordered
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> pair defining function & [(a,b,p) in F or (b,a,p) in F]
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> p is an ordered pair of a,b <-> Exist F: F is ordered pair defining
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> function & (a,b,p) in F.
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> R is an F_binary relation <-> for all y. y in R -> y is an F_ordered
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> pair.
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> R is a relation <-> Exist F. R is an F_binary relation
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> In this way we can define all relations isomorphic to some relation
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> regardless of the type of the implementation of the ordered pair those
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> relations are defined after.
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> Now higher degree isomorphic relations between sets of relations is
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> also possible, this is achieved by system isomorphism. where two
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> tuples (C,R1,R2,..) (C',R'1,R'2,..) of the same size are said to be
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> isomorphic iff a bijection from C to C' preserve all relations. This
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> introduces forms exemplified by systems which are usually denoted as
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> 'structures' in structuralists views.
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> Those structures also fulfill the informal account here about forms
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> and so they are mathematical objects.
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> What is crucial for this definition if it is to be made rigorous is to
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> figure out what
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> constitutes an "isomorphic" relation that is a candidate for dictating
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> forms, and also how to spell out "field" of that relation being the
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> universe itself?
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> Here is a trial.
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> We'll call any relation R to be form defining iff there exist a class
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> X of all sets that have the relation R to each other and if the
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> transitive closure of X is V.
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> R is form defining <-> Exist X. X is a collection of atoms & (for all
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> sets y,z: y atom of X & z atom of X <-> y R z) & TC(X)=V.
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> For any form defining R define R* as:
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> R* = X <-> X is a collection of atoms & (for all sets y,z: y atom of X
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> & z atom of X <-> y R z) & TC(X)=V
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> form($) <-> Exist R. R is form defining & for all y. y exemplifies $ <-
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> > y atom of R*
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> where "exemplifies" is a primitive binary relation.
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> The reason why those relations are form defining is because of them
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> been determined by factors that range over the whole universe of
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> discourse, then the individual characteristics of sets other than
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> those formal ones would be abstracted away, so only the form those
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> sets possess would be extracted, and as said earlier because the
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> process contain some feature that make it cross about all the universe
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> of discourse, then those forms are free standing, and those are the
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> ones mathematics is concerned with, otherwise they could be claimed to
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> be forms limited to some part of the set hierarchy, much as "cat" is a
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> form limited to some part of the animal kingdom. So being Free
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> standing discriminate them from ordinary forms peculiar to specific
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> part of the set hierarchy.
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> So mathematics is "discourse about form" with this it is meant any
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> theory that can be interpreted in the set hierarchy (which is as
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> mentioned above a logical background theory) having all its objects
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> being interpreted as forms. So for example PA is a piece of
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> mathematics since it can be interpreted in this theory with an
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> interpretation in which all its "objects" are interpreted as "forms"
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> defined after "bijection" relation in the Fregean manner. So it is a
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> case of discourse about form, thus mathematical!
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> So here there is a line of separation between what is foundational and
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> what is mathematical, the set\class hierarchy is foundational i.e. it
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> belong "essentially" to logic! it is a sort of extended logic,
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> although it definitely use some mathematics to empower it and actually
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> it needs a mathematician to work it out, yet this doesn't make out of
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> it mathematical, the piece of mathematics used in those
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> foundational theories is just an application of mathematics to another
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> field much as mathematics are used in physics. So what I'm saying here
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> is a theory like ZFC is not "Essentially" about mathematics, it is not
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> even a piece of mathematics, it is a form of LOGICAL theory.
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> So Set theory is a kind of LOGIC. However one can easily see that such
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> form of logic can only be handled by mathematicians really, but still
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> that doesn't make out of it a piece of mathematics as mentioned above.
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> Mathematics is the study of "form" as mentioned above, it is
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> "implemented" in the set\class hierarchy, it provides a discourse
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> about forms whether simple or structural. All known branches of
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> mathematics: Arithmetic, Real analysis, Geometry, Algebra, Number
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> theory, Group theory, Topology, Graph theory, etc... all can be seen
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> as discourse about form, since all its objects can be interpreted as
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> forms.
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> Anyhow it is reasonable for branches of mathematics to be developed
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> along some Foundation back-grounding in logic, and then the
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> mathematical forms be implemented on that background logic, this can
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> be seen clearly with topology which starts from set theory and then go
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> higher to deal with forms like continuity and connectedness. However
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> it can be seen to be essentially about the higher concepts it tries to
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> manipulate, the back-grounding in sets is just the logical part of it,
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> since what it tries to manipulate is a sort of "form", then topology
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> is essentially mathematical.
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> Also I wanted to raise the issue that "any" consistent theory is
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> speaking about a model that is "possible" to exist! So if we secure a
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> consistent discourse about form then, we are speaking about forms that
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> might possibly exist. And that's all what mathematics needs to bring
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> about. Whether those forms really exist or not? this is not the
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> discipline of mathematics. So consistency yields "possible" existence,
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> and that's all what mathematics should yield, i.e. forms that could
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> possibly exist.
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> How those forms are known to us? the answer is through their
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> exemplification as part of the discourse of consistent theories about
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> form. Whether they are platonic in the sense of being in no place no
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> time, etc.., that is not relevant, we come to know about them by their
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> exemplifications which are indeed not so abstract and can be grasped
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> by our intellect. How can such an abstract notion be exemplified by
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> such concrete objects, that's not the job of mathematics to explain.
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> So the above account give some loose informal account on what is
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> Mathematics.
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> Zuhair