I start with a little funny problem. Suppose I am dressed up and I want to change my dresses. What do I do? I first take off the outermost outfits and lastly I take off my innermost under garments. Why do I have to maintain that order? Why can't I do the reverse? For example I could take off my under garments first and then lastly I could take off my outfits. No that is impossible, unless and until I am allowed to tear my dresses or my dresses are super-stretchable. We can put the problem in a different way. Suppose I want to keep my coat in the cupboard. What do I do? I first open the cupboard and then I put my coat inside the cupboard. Can I do it the other way round? Suppose I put my coat first and then I open the cupboard. No that sounds really stupid! Yeas, that really sounds stupid but let me add that the problem can be formulated conversely. Suppose I want to take out my coat from the cupboard and close the cupboard. How do I do it? I take out the coat first and then I close the cupboard. It is impossible that I first close the cupboard and then I take out my coat. By all these examples I want to suggest the generic nature of the problem. Our problem is then to explain why we have to maintain certain order in doing all this kind of tasks.
Intuitively or unconsciously we seem to know the explanation of this problem. Vaguely, the explanation is that if we change the order of the tasks then the relevant objects have to go through some spatial overlapping with each other. Since such overlapping is not physically possible we cannot reverse the order of the relevant operations. Let us try to formulate this explanation.
Our physical world is populated by countable or discrete physical objects. These objects occupy space. Or we can say that these objects occupy certain definite regions of the space. The rule is then: never can the objects share their respective occupied regions, be it partial sharing or complete sharing. If x is such a discrete object then by r(x,t) I shall mean that the region occupied by x at time t. Furthermore I shall use the infix ∩ between two occupied regions meaning that the two regions overlap with each other. The rule can be then formulated as follows
(NLB) x ≠ y ⇒ ¬∃t. r(x,t) ∩ r(y,t)
Let us call it the non-intersecting law of bodies (NLB) and our explanation boils down to saying that our world is abide by this law. So the reason why I have to open the cupboard before taking out my coat is that if we try to do it in different order then NLB would be violated. It is evident now that it is only a physical necessity that we cannot do these sort of tasks in the reverse order.
This explanation may sound quite trivial. Nevertheless, it invites two immediate problems. First we may find some physical or semi-physical objects like clouds, waves, and shadows that seem to violate NLB. We may meet this objection by restricting the range of the variables. So we can say that the variables in the formulation of NLB range over certain physical objects, which we call bodies for maintaining certain compactness, density and so on. This will not affect the status of NLB as a physical law. We can fairly keep on claiming that all the middle size familiar physical objects are abide by NLB. The next problem seems to be a bit serious and I am not sure whether I can spell it out clearly. Vaguely the problem is this. Fine, one may say, we cannot do certain things in reverse order because of NLB. But, she may add further, Why do we have to violate NLB by reversing the order? In other words NLB is maintained in following some orders and it is violated following some different orders. But why? I suspect that by asking this why-question one wants to find an explanation in terms of some kind of structure of the world and I guess such structure will be more mathematical or algorithmic rather than metaphysical one. An answer to this why-question is beyond my ken now and so it will be beyond the scope of the present note. I shall, however, focus some other issues initiated by NLB and they will be pertaining more to metaphysics and ontology than to physics or mathematics.
I said that corresponding to a body x there is a region r(x,t). This raises two questions. What is the ontological significance of the region r(x,t)? Shall we call x as 3 dimensional or 4 dimensional object? The two questions seem to be connected and answering the former may shed some light on the latter question. I would, however, concentrate on the first question and leave the second question as an incidental matter. According to NLB for two distinct bodies x and y the corresponding regions r(x,t) and r(y,t) cannot intersect or overlap. If they ever overlap then the overlapped region, which I shall indicate as r(x,t) ∩f r(y,t), will be a region of mixture. What NLB says now is that there cannot be mixture. But mixture of what?