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Roy Woo
Posts:
1
Registered:
12/2/12


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


Mark it On Sunday, 2 December 2012 20:18:57 UTC+8, zuhair wrote: > 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



