Basudeba,
I have proposed that if maths and physics assume that processes are identically repeatable, or that two identical objects may exist, then they are not precisely describing nature. That does not of course makes mathematics itself 'wrong'!
My falsifiable evidence is this; Ask a top player to firmly break a frame of snooker balls by hitting at velocity u. I propose that any attempt to recreate the final complex pattern of balls in the same way will fail. Experiment 2; ALL grains of sand created by the same process in a desert will be different in some way. I go further; Experiment 3; Smash two coconuts together and film with a high speed camera. You may repeat for 1,000 years!!, but even if (4) we say we'll smash two 'identical' coconuts together (only metaphysically possible) we won't achieve the precise same outcome!
Now what I point out is that at present we DO NOT HAVE any 'category' or law beyond the 'Law of the Excluded Middle' so cannot rationalise the deviations. Can you perhaps suggest one?
You may be thinking, well the mathematical description is precise, it's trivial if nature varies slightly so why worry? I suggest that thinking hides the truth. It is NATURE that's primarily important here, not Maths! Wigner missed the point. There are fundamental non-trivial truth's we've missed by thinking as we do. You say "all paradoxes are wrong". I agree. It's our poor thinking at fault.
Maths is an abstract (metaphysical) 'approximation' of nature, where A=A is essential, and I agree all you say of it's domain. But when we are considering the (physical) entities nature alone, and REAL interactions, then we need a different descriptor, which can only be the 'squiggle' approximation of =, ;A~A, which is then equivalent to a Bayesian inverse distribution as a quantum PA distribution. Effectively this shows we have underparameterized the complexities of nature with our mathematics. In the EPR case, the statistical method used cannot then access and quantify the additional degrees of freedom in nature. (The proposed approach has exposed solutions to many resolutions of paradoxes and anomalies in astrophysics).
Do you now better understand? I look forward to reading your essay and chapter on Bells Inequalities.
PS. To tune yourself in to the well developed philosophies on this on FQXi you'd do well to read the winning essay last year and, for instance, the McEachern and Sycamore essays.
Best wishes
Peter