Dear Tim Maudlin,
You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.
Yes I agree with you and thats why I have propounded Mathematical Structure Hypothesis to explain their origin in the same line.
Question is - Who decides the symphonic structure of that language? For any mathematical structure to be compatible to explain the physical structure, we need to match their intrinsic "laws of invariance" otherwise their applications would be wrong.
This is why in context of Skolem's paradox: "A particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong."
You have classified the mathematical structures into two categories based on Wigner's essay
1) One which are naturally suited to physical world e.g. Integers and what does their suitability imply about the physical world?
2) Others which are not e.g. advanced concepts e.g. complex numbers should have use in physics.
I have explained on the basis of Mathematical Structure Hypothesis that whether it falls in any category, its basically physical characteristics behind the development mathematical language which describes the physical characteristics of the physical world i.e. whether Integer or Complex numbers.
Wigner talks about Complex numbers as advanced concept but what decides the structure of complex numbers and why they are so effective in Quantum Mechanics.
Eugene Merzbacher in his book on QM has explained by deriving that for certain physical characteristic to be satisfied( for quantum waves,any displacement in the space & time dimension should not alter the physical characteristics of waves) and to satisfy these criteria, the mathematical parameters turns out to be "i"(complex number).
Here is the reason the structure of mathematical language has been matched/molded to suit the physical characteristic of quantum waves(physical world).Infact, its not the mathematics describing physics here rather their corresponding law of invariance. So, what is the law of invariance behind complex number. Its answer lies in the definition of why negative multiplied by negative turns out to be positive? Why not positive multiplied by positive also become negative? Here is hidden laws of physics behind the definition of mathematical operators structure and vice versa.
This is because mathematical structures abstractness and physical reality both are creations of the same thing Vibration, which my Mathematical Structure Hypothesis has propounded.
Anyway, your essay is indeed great.
Thanks & Regards,
Pankaj Mani