The example provided in the previous post about why mathematics is a set of circular arguments is wrong. Although funny, Bob showed me convincingly that it is not proper mathematics. He showed it this way:
Suppose a=b then
2a + 3b = 3a + 2b <==>
2(a -b) = 3(a -b) <==>
2(a - b) - 3(a - b) = 0 <==>
(2 - 3)(a - b) = 0 <==>
(2 - 3) = 0 or (a - b) =0
and then there is no problem anymore.
This same day I found a new example and I think this one is much better. For this example I rely on an explanation how it is possible that there are different values of the density of the vacuum in the universe. The exact link can be found here. The density of the vacuum in the universe must have some value. It is measured to be close to zero. Using quantum field theory different answers come up. The moral of the story is that the answer is dependent on 'the assumptions and the reasonings that went into the answer'. Mathematics can not decide on its own if a statement is correct or not. Using proper mathematics one can get answers that are even not near the measurements found in reality. These examples show that it can go anywhere. Quantum field theory has made a lot of very precise statements about reality. It is using proper mathematics. Still can it come up with totally different values, which are different from the measurements found. That is only possible if the statements within mathematics are truely independent from reality. Otherwise the math is just wrong like my previous example and that is very hard to imagine.