Dear Stefan,
Let's see if I can convince you of point 3 for math and 1 and 4 for the universe.
Point 3 is as follows: in math, all theorems are already there just waiting for mathematicians to reveal them. It is only an accident of history that Pythagoras theorem is named like that. This mathematical fact was there the moment you specify the axioms. Frozen means that in one axiomatic system mathematical statement do not have a transitory existence: they do not switch between true and false and mathematical objects do not pop in and out of existence. In this sense, math does not evolve; it is only our understanding of it that evolves. The history of math is only the history of mathematicians. Godel's theorem was still true in ancient Greece, but the people living in that time did not discover it.
Point 1 of reality: You can hear on the radio that x people died in a car crash, never that x people got hurt by the theorem that there are an infinite numbers of primes. We can touch, see, feel the world around us; it is made out of concrete, not abstract things. This is precisely the opposite of mathematical structures. A complex number for example it is not a pair of a real and an imaginary numbers. This is just one possible representation of it. You can represent complex numbers by 2X2 matrixes and this is even a better representation because it has a natural explanation of sqrt(-1) (which was the main roadblock in the acceptance of complex numbers as a valid mathematical structure). Complex numbers and mathematical structures in general are just abstract objects independent of their concrete representation. In the real world a concrete instance is what matters: a physical chair, not the idea of a chair. In the real world we try to make ideas and dreams reality, a mathematician distills the essence by considering only formal abstract structure.
Point 4 of reality: What we observe in the real world has the same truth value for all observers. The Roman Empire did fell, and its impact was different from different people's points of view: the common citizen of Rome, the soldier in the invading army, etc. but they all can agree on this fact. To shorten the explanation, please see my earlier reply to Georgina on Aug. 4, 2009 @ 02:24 GMT (the car in the garage "counter example" for how the world might look if events were contextual and appearing different for different observers.)
Now back to your questions. You say:
Your argumentation goes like this:
1. There exists a platonic realm of fixed relationships, unanimated.
>Yes
2. There exists a universe which is dynamic and in some parts animated.
>Yes, but I disagree with some parts. (It is in all parts)
3. Hence, there must be a difference between both, because the term animated and the term unanimated cannot be the same at the same time and working out these differences proves the necessity of both, the animated and the unanimated realms.
>This is not what I am saying.
4. Because of the fact that it is indeed possible to divide reality in animated and unanimated parts, the necessity of the points 1. and 2. are proven and therefore also point 3.
>This is not what I am saying.
You also say:
1. The existence of a platonic realm of fixed realationships is a possibility, but not a necessity that can be proven with certainty.
I disagree with this. See my point 3 above. The value of pi as 3.141... was 3.141... regardless of our universe, regardless on when we discover its value, and regardless of other mathematical structures like topological and logical structures. This value exists outside space and time. A universe, a star, a child can be born, not the value of pi or the theorem that the sum of the angles in triangle in flat geometry is 180 degrees.
What I am doing is ultimately old fashion physics and not a subjective interpretation. I only present the philosophical implications here, but I am not a philosopher and I am only doing it to explain my new paradigm.
Why is this physics and why it is not subjective? Because all the 3 principles are validated by all experiments performed to this date. In my heuristic rule I am asking for the differences between math and reality. This means that whatever differences are found (the 3 principles so far) they have to pass all past, present, and future experimental tests. This is why this is ultimately objective physics and not subjective philosophy. In computers there is the GIGO principle: garbage in, garbage out. If the 3 principles are invalidated by experiments, they are the garbage in and the conclusions will only be the garbage out. Next, the 3 principles generate mathematical consequences and they have already proven a large chunk of the necessity of our universe. By a leap of faith now, I am speculating that ALL core characteristics of our universe will be obtained in this way. If true, this will mean that our universe cannot be except in the way it is: it has uniqueness and this is big. If true then we have mathematically succeeded in proving that there is no God. God's hands were tied at the moment of creation and he had no freedom into creating this universe, the blue print for the universe was already there just like the value of pi: there was no other way possible except to have 3 spatial dimensions, one time dimension, quantum mechanics, electromagnetic, weak, strong, and gravitational forces, etc, etc.