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Topic: Sets as Memory traces.
Replies: 10   Last Post: Feb 11, 2013 12:48 AM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: Sets as Memory traces.
Posted: Feb 6, 2013 1:17 PM
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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 place
relation 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 solid
object in y. And of course we can interpret set membership by the
relation 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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