Update to Ray 2.0:
The examples about the Italian school/Frege/Cauchy are all supposed to be examples of us formalizing intuitive processes (algebraic geometry, set theory, analysis) in the “wrong way”.
Reply to Ray 2.1:
I think you mostly understand my post but just to clarify, I don’t even get into ZFC. PA itself could be an incorrect formalization of arithmetic:
I am making the claim that addition and multiplication of numbers is a physical process that we know how to do (for instance by adding together jars of beans) and mathematics (number theory) is an abstraction of it, exactly analogous to how balls dropping to the ground is a physical process and Newton’s theory is an abstraction of it.
And in the same way that Newton failed to capture the limits of gravitational processes (for large masses /high speeds) , our formalism of addition might fail to capture the limits of the physical process of arithmetic (for large enough numbers).
I in fact make the stronger claim that all mathematics is an abstraction of physical processes/patterns in the same way.