The following is a kind of topology but in Mereological terms, however it is a dynamic form of Mereotopology. It is basically the study of "part-hood" and change of "contact" relations, the first is a binary relation that is present
in our daily language when we say for example body parts, part of a course, parts of a house, etc... Here what is intended is the simplest kind of part-hood, so part-hood relation is symbolized by "P" and x P y is taken to mean x is a part of y. Part-hood doesn't change over time.
Axioms about part-hood stem from the naive contemplation of part-hood, they are:
Axiom 1) A part of a part of an object is a part of that object
Formally this is: x P y ^ y P z -> x P z
Axiom 2) If X is not a part of Y, then there exists a part k of X such that k is disjoint of Y (i.e., k and Y have no common part).
Axiom 3) If X is part of Y and Y is a part of X, then X is identical to Y.
Axiom 4) For any property Q that holds of at least one object, there exists an object X that is a sum of all objects satisfying Q, i.e. every object satisfying Q is a part of X, and every object Y that have every object satisfying Q as a part of would have X as a part of it.
This is called General Extensional Mereology.
Now we come to axiomatize timed-contact relation, here contact means "superficial" contact or simply "touch" and it can vary with time as we see in everyday life; naive thought about contact results in the following axioms:
Axiom 5) If X touch Y at moment t then Y touch X at that moment.
Axiom 6) If X touch Y at any moment, then X and Y are disjoint.
Axiom 7) If x is a proper part of X and x touch Y at moment t and X and Y are disjoint, then X touch Y at moment t.
Axiom 8) If X touch Y at moment t and X have proper parts, then there must exist a proper part x of X such that x touch Y at moment t.
To formalize this we present touch as a three place relation symbol denoted by C(x,y,t) to mean x is in contact with y at moment t and present the above axioms in that notation.
Now we come to interesting definitions:
x stick to y iff for all t: C(x,y,t)
i.e. x stick to y if at all moments x is in contact with y.
Now we come to imagine the possibility of existence of a kind of very solid objects, concrete so to say, those that have proper parts that stick to each other in a non changing manner, i.e. an object in which every two proper parts in contact with each other would remain so over all moments of time, and the object itself is not the sum of two disjoint non contacting (i.e. separate) proper parts.
Formally speaking:
x is concrete iff (Not exist z,y,t such that: z,y are disjoint and z not in contact with y at t and x is the sum of z and y) and (for all r,s: r,s are proper parts of x ^ r touch s at some moment t-> r stick to s)
Of course it is a theorem that "a non-split part of a concrete object is concrete".
A non-split object is an object that is not the sum of two non contacting disjoint proper parts.
So proper parts of concrete objects are either sticking to each other or
are departing apart (i.e. not in contact with each other all the time), i.e.
there is no change of contact relations between proper parts of a concrete
object.
Now we define "concrete individual" as:
x is concrete individual iff x is concrete ^ not exist y (y stick to x)
Now concrete individuals themselves can be 'dependent' or 'free'
A dependent concrete individual is one for which there always exists some object that is in contact with at some moment of time, i.e. there is no moment of time where it is free from contact. Of course it is clear that its contacts must vary over time, otherwise it won't be an individual in the first place.
A free concrete individual is one for which there is a moment t such that there do not exist an object that is in contact with it.
From hereafter we'll use the word 'individual' to mean concrete individual.
Various kinds of dependency relations can be defined over dependent or even over free individuals.
Even Free individuals can have some styles of dependency relations. For example a free individual can be dependent on contact with some individual in order to mediate contacts with individuals. So for example a Free object a would only have contact with an object c at moment t if it has contact with object b at moment t, in this example the free individual a would be said to have a free dependent contact relation after b, since without contact with b it cannot be in contact with any individual at any moment of time. So although a is free yet it had some dependency in contacting. Of course if a is freely depending on b (as above example) it doesn't always entail that b is freely depending on a. So here a 'direction' has been established were there is a form of contact relation between objects that is not necessarily symmetrical, and in this kind of relations one can explain many things, or more appropriately speaking we can have a room to construct many style of dependency relations, most notable of them is the "SET" dependency relations, and that kind of free dependency relation can to some extent explain set theoretic structure although it doesn't motivate its detail.
So a set dependent relation is the following:
a is set dependent on X iff a is a free individual & X is a sum of free individuals &
For all t,b (C(a,b,t) -> for all x (x is free individual part of X -> C(a,x,t)))
In other words the free individual a can only have contact with any individual b at moment t if it it has contact with ALL free individual parts of X at moment t.
Here the free object a would be called: 'set-dependent liaison' of X.
It is prudent to stipulate non existence of more than one set-dependent liaison for each sum X of free individuals.
Now one can proceed and define set-membership as a relationship between an individual part of X and the set-dependent free liaison of X, and here according to this definition it would be a relation between free individuals.
Or one can also proceed in Lewis style and define set-membership as a relationship between X itself and a sum Y of set-dependent liaisons
having the set-dependent liaison of X as a part of it. Here this definition would render set membership as a relation between sums of free individuals.
Of course naive thought about set dependent relations would stem from contemplating a hierarchal structure of sets of sets of sets... etc and those
doesn't follow from the naive background of the kind of mereotopology explained here, but nevertheless it would provide a nice envisioning into it,
it can almost make us SEE sets.
I personally like the application of dynamic Mereotopology that is concerned with 'continuity' and 'discreteness' especially those of edge-node graphs and building a hierarchy of them by breaking continuous graphs and then recombining them and linking them, I think this discipline is more fruitful than the above 'dependency' over time relations of individuals, although both are very interesting.
Could mathematics be traced to investigation of part-hood and change of contact?
May be!
Zuhair Al-Johar
Jan 9, 2016