```Date: Dec 8, 2012 4:41 AM
Author: Zaljohar@gmail.com
Subject: Mathematics in brief

In philosophy "form" is a universal, however here the term "form" isused to designate a universal that is exemplified by all objectsbearing some kind of isomorphic relation between them provided thatthe collection of all exemplifying objects do have all objectsincluded in its transitive closure. I paraphrase that as: a universalthat involves the whole universe. By contrast some forms as used inphilosophy involves only a particular sector of Ontology, like forexample "cat" which can only be exemplified by animals and so it is arestrictive kind of "form". Here any speech about forms will bemeaning the non privet kinds of forms, i.e those that involves thewhole universe after some isomorphic relation as mentioned above.Mathematics is "discourse about form" with this it is meant any theorythat can be interpreted in the set hierarchy having all its objectsbeing interpreted as forms in the set hierarchy. So for example PA isa piece of mathematics since it can be interpreted in the sethierarchy with an interpretation in which all its "objects" areinterpreted as "forms" defined after "bijection" relation in theFregean manner. So it is a case of discourse about form, thusmathematical!So here there is a line of separation between what is foundational andwhat is mathematical, the set\class hierarchy is foundational i.e. itbelong "essentially" to logic! it is a sort of extended logic,although it definitely use some mathematics to empower it and actuallyit needs a mathematician to work it out, yet this doesn't make out ofit mathematical, the piece of mathematics used in those foundationaltheories is just an application of mathematics to another field muchas mathematics are used in physics. So what I'm saying here is that atheory like ZFC is not "Essentially" about mathematics, it is not evena piece of mathematics, it is a LOGICAL theory.So Set theory is a kind of LOGIC. However one can easily see that suchform of logic can only be handled by mathematicians really, but stillthat doesn't make out of it a piece of mathematics as mentionedabove.Mathematics is the study of "form" as mentioned above, it is"implemented" in the set\class hierarchy which provides a discourseabout forms whether simple or structural. All known branches ofmathematics: Arithmetic, Analysis, Geometry, Algebra, Number theory,Group theory, Topology, Graph theory, etc... all can be seen asdiscourse about form, since all its objects can be interpreted in theset hierarchy as forms.Anyhow it is reasonable for branches of mathematics to be developedalong some Foundation back-grounding in logic, and then themathematical forms be implemented on that background logic, this canbe seen clearly with topology which starts from set theory and then gohigher to deal with forms like continuity and connectedness. Howeverit can be seen to be essentially about the higher concepts it tries tomanipulate, the back-grounding in sets is just the logical part of it,since what it tries to manipulate is a sort of "form", then topologyis essentially mathematical.Also I wanted to raise the issue that "any" consistent theory isspeaking about a model that is "possible" to exist! So if we secure aconsistent discourse about form then, we are speaking about forms thatmight possibly exist. And that's all what mathematics needs to bringabout. Whether those forms really exist or not? this is not thediscipline of mathematics. So consistency yields "possible" existence,and that's all what mathematics should yield, i.e. forms that couldpossibly exist.How those forms are known to us? the answer is through theirexemplification as part of the discourse of consistent theories aboutform. Whether they are platonic in the sense of being in no place notime, etc.., that is not relevant, we come to know about them by theirexemplifications which are indeed not so abstract and can be graspedby our intellect. How can such an abstract notion be exemplified bysuch concrete objects, that's not the job of mathematics to explain.Zuhair
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