I have tried to answer some of your objections below.
Many thanks! I have divided my response into two posts, one on the more formal problems and the other on the more conceptual, metaphysical, and similar problems.
Would not the cause then not be the formulae governing the dynamics, and the effects the actions of the two photons?
The mathematical models in their general form don’t describe actual effects, and the specific formulations used in experiments are
1) not complete until after the fact and
2) have no influence on the photons.
In a similar manner, Einstein’s equations that almost are General Relativity theory describe how matter curves spacetime and simultaneously how spacetime curvature “moves” mater, these are descriptions. Even if completely accurate, they cause nothing.
If there is no space, is it not the case that there is no space-time? Please give an example of your latter point which you have not explained.
Every “point” in space in pre-relativistic physics could be exactly determined by values given to 3 coordinates x, y, & z. This space was 3D Euclidian space. Spacetime is not 3-dimensional but 4-dimensional (neither the geometry nor the possibility of extra dimensions affects my argument and discussion of them is therefore omitted). Every point in any 4-dimensional space, regardless of geometric structure (Euclidean, Minkowskian, Riemannian, complex, etc.), requires four coordinates. Another way of saying this is that while you can have planes and lines in 4D space, they are defined by how they span “space” along 4-dimensions and can only be described thus. What we experience as “space” is 3D, but relativity theory (and its empirical support) tell us that no such “space” exists. Spacetime is “space”, but unlike most mathematical spaces it has the unique property of also being the actual “space” we inhabit (though it does not seem like we do to us). In this 4D “space”, there cannot exist any 3D space, every point is defined again by 4 coordinates, and thus to the extent by “space” in “if there is no space, is it not true that there is no space-time” the answer is “it is not true.” What we call spacetime is convenient nomenclature: it distinguishes the physical, Newtonian space of classical physics from the “space” of relativistic physics (in which this classical space doesn’t exist and neither does time).
I am familiar with the mathematics you refer to.
Yet you use notation and terms unique to such mathematics, specifically relations and the “mRn” notation used to indicate a relation, it arguments, and its arguments “order”. It seems strange to not be familiar with the mathematics you use. However, luckily this isn’t too much of a problem. It suffices to say a relation of the type you refer to has certain properties your use doesn’t adhere to. An example your definition of a relation in terms of two “rules”.
The use of set theory in general is more problematic as the whole of your mathematical presentation depends upon it and much of your explanation relies on it. Also, it is so very related to formal logic that the founding work on symbolic/formal logic by Frege, a monumental achievement, came crashing down because of a problem in the ways it allowed sets to be defined (a problem pointed out to Frege by Russell in a letter). As it is an example of how vital seemingly trivial nuances are in proofs & mathematics in general (as well as logic & set theory), I will briefly summarize the problem Russell noticed.
A set, as you know, can be described simplistically as a collection of elements or objects. However, not every collection can be a set, and Frege did not adequately address a particular constraint. Russell showed him that his system allowed one to define the following set: “the set that contains all and only those sets which do not contain themselves” (this is really just the classic Barber’s Paradox).
Let S be Russell’s set and assume that S doesn’t contain itself. By definition, S contains all sets that don’t contain themselves, so S must contain itself. Yet if it contains itself, it contains a set that contains itself, again violating the definition. Thus Frege’s system allowed for paradoxes that defeat the entire purpose of formal languages/systems.
Your use of sets depends heavily on concepts that you do not define even informally (such as “concepts”). So not only do you rely on arbitrary set-theoretic notions for your proof, you misuse these.
Given the set F of all fathers and the set C of all children c, fRc means "f is the father of c where c is the cousin of Peter"?
For the same reason that a + b = c doesn’t mean a + b =c – d. Given any relation R and at least one well-defined set on which R is defined, for any elements a & b the relation aRb can have one and only one meaning/value and that must be because there can exist 1 and only 1 interpretation of R in terms of its arguments (the symbols/letters on either side of R)
Symbols (and their variants) such as +, -, /, ^ , *, etc., are usually relations. Take the set N of natural numbers. For any elements of N (x,y), we can define a relation xRy s.t. the result is some natural number z equal to x + y. What we can’t do is define xRy to be “x + y – a”.
Another way to think of the problem is in terms of piecewise functions. I can define f(x)= x/y except when y= 0, in which case the function equals 0.
However, even if I didn’t use the right notation I would have to specify exactly what the function is equal to for all values of x & y including specifying the otherwise undefined case in which y = 0.
It should be m causes n or vice versa.
That may be what you meant. It isn’t what you wrote. You would have been better of simply stating something like this:
"there cannot be two uncaused causes, since, as stated, for every two things, there is a
relationship between the two things"
Even then, though, this is an assertion. You haven’t proven it, but you do rely on it for your proof.
can I not rephrase it as follows (implying two relations)?:
Given the set A of all ancestors a and the set D of all descendants d, aRd means "a is a female ancestor of d" or "a is a male ancestor of d".
No. You are now including in your set properties of its elements that the set does not “contain”, which is to say you are essentially using an entirely different set. Given the sets A & D and any pair (a,d) whatsoever, I have no way of knowing whether aRd describes a female or male descendent.