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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
Sets as Memory traces.
Posted: Feb 6, 2013 6:14 AM
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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.

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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