Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: Sets as Memory traces.
Posted:
Feb 6, 2013 1:17 PM


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 parthood > 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 > parthood, 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



