I'm assuming maximum good faith here but this only seems interesting/contradictory because you haven't defined the system rigorously. All the seeming contradictions are coming from the fact that you are applying Euclidian geometric constructions to set theory via this example involving covering things up with actual pieces of paper.
The parallel postulate doesn't deal with pieces of paper. It deals with lines in Euclidian space. These spaces have infinite dimension, and the lines have infinite length, so the very first deduction you are attempting to draw (about "half the logic" being represented) is literally meaningless. If you take an infinite Euclidian space and partition it into two halves with a line, they are each infinite in size. A Venn diagram is not a Euclidian concept, it is a diagramatic representation of set membership. So you can't say anything about logic purely from the idea you can partition a space in half.
Your next paragraph about the logic of camouflage is also meaningless for the exact same reason.
The section about an object on one side or the other of the line is simply a category error. If we have an object with non-zero area and a line in a Euclidian space then the object can be on either side of the line or the line can intersect with the object. Nothing about this changes the position of the object or the line and your "paper" construction simply adds some confusion to a very simple geometric system. If you're actually dealing not with a Euclidian space but with set membership then it's best to abandon this paper/line construction and just talk about set membership. If you have an object and two sets, the object can be in either set, both or neither. This is not a contradiction and the object has no "position" in this system. It's simply a question of definition. If you want to learn how set theory works I recommend you read Naive Set Theory by Paul Halmos.
The final paragraph is meaningless because all the things before are meaningless. If we're talking about Euclidian spaces these have nothing to say about logic (modified or otherwise) and if you are talking about sets then the parallel postulate is irrelevant because it is a geometric concept. I understand the Dedekind construction etc and really don't follow what you're saying about the parallel postulate being a statement about Cauchy completeness. From the fact that the coordinates of a point fully represent that point, the existance of lines in a Cartesian space (not the parallel postulate) is a statement about Cauchy completeness. That said, nothing you've put here affects the construction of numbers in any way.
If you want any of this to be actual mathematics you need to abandon your "paper" construction and define things properly.
You are repeatedly begging the question so what follows is falsehood. The premise is to reconstruct the intuition behind the parallel postulate. You can look at Euclid's Elements to get a better understanding of constructivism.
I am most certainly not begging the question. I have addressed each of your points directly. I have a pretty good idea of Euclid's elements and the parallel postulate. Constructivism in the philosophy of maths is the idea that it is necessary to find a specific example of something to prove it exists. That's an idea that significantly postdates Euclid.
Comments
I'm assuming maximum good faith here but this only seems interesting/contradictory because you haven't defined the system rigorously. All the seeming contradictions are coming from the fact that you are applying Euclidian geometric constructions to set theory via this example involving covering things up with actual pieces of paper.
The parallel postulate doesn't deal with pieces of paper. It deals with lines in Euclidian space. These spaces have infinite dimension, and the lines have infinite length, so the very first deduction you are attempting to draw (about "half the logic" being represented) is literally meaningless. If you take an infinite Euclidian space and partition it into two halves with a line, they are each infinite in size. A Venn diagram is not a Euclidian concept, it is a diagramatic representation of set membership. So you can't say anything about logic purely from the idea you can partition a space in half.
Your next paragraph about the logic of camouflage is also meaningless for the exact same reason.
The section about an object on one side or the other of the line is simply a category error. If we have an object with non-zero area and a line in a Euclidian space then the object can be on either side of the line or the line can intersect with the object. Nothing about this changes the position of the object or the line and your "paper" construction simply adds some confusion to a very simple geometric system. If you're actually dealing not with a Euclidian space but with set membership then it's best to abandon this paper/line construction and just talk about set membership. If you have an object and two sets, the object can be in either set, both or neither. This is not a contradiction and the object has no "position" in this system. It's simply a question of definition. If you want to learn how set theory works I recommend you read Naive Set Theory by Paul Halmos.
The final paragraph is meaningless because all the things before are meaningless. If we're talking about Euclidian spaces these have nothing to say about logic (modified or otherwise) and if you are talking about sets then the parallel postulate is irrelevant because it is a geometric concept. I understand the Dedekind construction etc and really don't follow what you're saying about the parallel postulate being a statement about Cauchy completeness. From the fact that the coordinates of a point fully represent that point, the existance of lines in a Cartesian space (not the parallel postulate) is a statement about Cauchy completeness. That said, nothing you've put here affects the construction of numbers in any way.
If you want any of this to be actual mathematics you need to abandon your "paper" construction and define things properly.
You are repeatedly begging the question so what follows is falsehood. The premise is to reconstruct the intuition behind the parallel postulate. You can look at Euclid's Elements to get a better understanding of constructivism.
I am most certainly not begging the question. I have addressed each of your points directly. I have a pretty good idea of Euclid's elements and the parallel postulate. Constructivism in the philosophy of maths is the idea that it is necessary to find a specific example of something to prove it exists. That's an idea that significantly postdates Euclid.
This is simply factually incorrect.