Dear Alma Ionescu,
Your writing style is exquisite and you phrase the problem well: "It would be very nice to know once and for all if math is a human invention. ... If it is ... we could cross it out of our list of places to search for deeper meaning. If it isn't, and there hidden logical rules to the universe ... then we can see it as an intrinsic part of reality."
I suggest that logical structure [the basis of AND and NOT 'gates'] is inherent at all scales, from telomeres, and RNA-DNA-protein operators, to crows counting, to microcomputers; and that counters [the essential basis of QFT's 'Number operator'] are ubiquitous. From there Kronecker says we do all the rest. I doubt very seriously that all of the physical counters and counting [and adding / subtracting] operations of the non-human portion of the universe should be ignored or written off. With Kronecker, I see the embellishment of these bases into "mathematics" as human invention and discovery. It is the logical structure [both static and dynamic] of physical reality that enables the relations we classify as 'math'.
To draw a boundary between the physico-logico-mathematical structuring of physical reality and the abstracting and symbolic representation and manipulation of such seems not right. The only thing that man brings to math is the degree of consciousness that allows him to move beyond insect's and lower animal's direct use of math to formal representation and manipulation of symbols.
Although "when one makes a complex plane, fractals grow in it uninvited, without having been planned, placed there, or even wanted, like leaves on a light bulb." Is this really any different than motion pictures growing from the light bulb, without having been planned, placed there, or even wanted? Actually, it takes quite a bit of purposefully created superstructure to bring out motion pictures or fractals. And it is not clear to me that either is more 'implicit' than the other. Both flow from the scale independent logical structure of physical reality, and I doubt that either has any meaning in any sense if one entirely removes physical reality from the picture.
You note that "math lies dormant", but the "motion pictures" lay just as dormant. [This analogy is probably ruined by the existence of lanterns before light bulbs, but you get the picture.]
You remark that "math made so much sense that it was worthy of worshiping..." If math is inherent in the self-consistent logical structure of physical reality, this should not really change that sentiment.
I very much like your abstraction that "the square root of 2 emerged, and for all his efforts Pythagoras could not fix it." One can rearrange, but not alter the nature of physical reality.
You ask so many cogent questions and make so many observations, one cannot respond to them all, but a very important observation you make is:
"Mathematical physics is only as good as physical insight."
I apply this reasoning to Bell's theorem, and hope you will read my essay and provide feedback.
I very much enjoyed your many questions and astute observations.
Best regards,
Edwin Eugene Klingman