```Date: Feb 6, 2013 1:17 PM
Author: Zaljohar@gmail.com
Subject: Re: Sets as Memory traces.

On Feb 6, 2:14 pm, Zuhair <zaljo...@gmail.com> wrote:> Suppose that we have three bricks, A,B,C, now one can understand the> Whole of those bricks to be an object that have every part of it> overlapping with brick A or B or C, lets denote that whole by W. Of> course clearly W is not a brick, W is the totality of all the three> above mentioned bricks. However here I want to capture the idea of> 'membership' of that whole, more specifically what do we mean when we> say that brick A is a 'member' of W. We know that A is a part of W,> but being a part of W is not enough by itself to qualify A as being a> member of W, one can observe that brick A itself can have many proper> parts of it and those would be parts of W of course (since part-hood> is transitive) and yet non of those is a member of W. So for a part of> W to be a member of W there must be some property that it must> satisfy. I'll add another primitive binary relation in addition to> part-hood, and that binary relation I'll call as "contact". So we'll> be working within a kind of Mereotopology. However axioms to> characterize contact relation would be different from those of> Mereotopology. Here only disjoint (non overlapping) objects are> allowed to be in contact. When we say A is in contact with B then we> mean that for some x,y where x is a part of A and y is a part of B> there do not exist a gap between x and y, of course as said above> provided that A and B are disjoint objects.>> Now we come to stipulate the sufficient condition for membership of a> whole, this is:>> x is a member of y iff x part of y And (there do not exist a part of y> that is in contact with x) And every proper part of x is in contact> with some part of x.>> This supplies us with the impression that x is a 'solid' entity and> yet x is separate (not in contact) from some other parts of y should> those exist.>> However this interpretation of membership using this kind of> mereotopology has its shortcomings, the greatest is that it is> limiting in the sense that only one level of membership is possible,> that is between solid blocks and collections of them, any collection> of several blocks would not be able to be a member of any object since> it does have separate proper parts. So this would only be enough to> interpret flat sets.>> If we desire having a hierarchy of sets being members of other sets> and if we want also to keep the above background of thinking of> matters in terms of parts and wholes and contact etc.., then we need a> more complex approach, one of those would be to invoke TIME in the> picture!>> This without doubt would complicate the whole picture but yet it does> supply us with some interpretation of sets and their membership.>> Now instead of having a binary relation C to represent contact, we> upgrade that to a triple relation symbol C(x,y,t) to signify x in> contact with y at moment t.>> This would revolutionize how matters are tackled here. So for example> if at moment t1 we have a whole apple P being a solid block i.e. it is> not in contact with any object and all its proper parts in contact> with some proper part of it, then P would be said to be a block at t1.> However this doesn't mean that apple P would also remain in this block> status, possibly at moment t2 the same apple P had been cut into two> separate halfs, so at moment t2 P is a whole of two separate blocks P1> and P2 and no longer being as a solid block, so at moment t2 P cannot> be an element of any object, while at moment t1 P could have been.>> This development would introduce us to the concept of MEMORY traces!> and of course the introduction of a new binary primitive 'memorized> in' or 'carved in'.>> A memory trace is supposed to be a record of objects in block status.> So for example suppose that an rock was immersed in some mud at moment> t1 and thus left its print on that piece of mud, then after a while> that rock was broken into two smaller parts, and one part also fell> down on another part of the mud and made another print on that piece> of mud, so suppose that mud remained like that for years, now this> piece of mud have the prints of the whole rock at solid status and> also of a part of that rock at solid status, this piece of mud would> be considered as a 'memory trace'.>> Now we would coin another interpretation of 'set' as a 'memory trace'.> Membership would be of objects in solid status carved in the memory> trace.>> so x carved in y  or x memorized in y is taken to mean that x is a> solid block at some moment t and memorized as such in y.>> So we have the axiom.>> x M y -> Exist t. x is solid at t.>Even more appropriate is to stipulate 'carved in' as a three placerelation symbol, so Cv(x,y,t) would mean x is carved in y at moment t.Now we can 'define' a binary relation M standing for 'memorized in'as:x M y <-> Exist t. x is solid at t & Cv(x,y,t)here of course what is meant by x M y is: x memorized as a solidobject in y. And of course we can interpret set membership by therelation M defined above, and of course sets would be memory traces.Zuhair> So membership can be interpreted as this memorizing relation and sets> can be interpreted as 'memory traces'>> It is natural to assume identity of memory traces after what is> memorized in them.>> It needs to be stressed that memory traces are NOT the wholes of what> is memorized in them! since the whole of an apple and a half of it is> the apple itself and it is not different from the whole of three> thirds of it, but the memory traces of those are different!>> However if an object do not change its solid status over time, i.e. if> we have the following property:>> for all t. x is solid at t>> then wholes (i.e. totalities) of such objects can be taken to be> memory traces of them since time is not having any differential effect> on those kinds of objects.>> So all in all, sets here can be interpreted as memory traces and set> membership as memorizing objects in solid block status.>> Zuhair
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