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 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