Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



what is mathematics? again
Posted:
Dec 2, 2012 7:18 AM


This is a continuation to post: What are sets? again
seen at:
https://groups.google.com/group/sci.logic/browse_thread/thread/a78c4e246a5c8a58/4345de40b3794321?hl=en#4345de40b3794321
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 Omegath 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 backgrounding 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 backgrounding 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



