Your discussion provides an interesting example of one of Burnham's laws, "Who says A, must say B." One of the most useful applications of this "law" is that it allows us to look at the practical implications of statements (A) which, in and of themselves, are not necessarily objectionable; and to see if the implications (B) are reasonable.
For example, one of the common arguments against ID theories is that they're "not scientific," apparently because it's supposedly impossible to tell the difference between naturalistic and "designed" effects. And yet, by that standard, the entire biotech industry is "not scientific," which is of course preposterous. This doesn't "prove" ID, of course, but it does neatly dispose of that popular argument against it.
You point out very nicely that there is an apparent necessity of some arguments to divorce mathematics from physical reality. I suspect the underlying motivation for this is largely emotional, rather than logical: for there to be a connection between mathematics and physical reality, is to suggest that there is a "reality" separate and above what we can observe. This is at least a necessary condition for admitting the existence of God. And that's a troublesome proposition for those who would argue that God (if there even is a God) plays no active role in the universe.
In essence, the debate is over whether mathematics is a process of invention, or one of discovery. The "dismissal of math" (the 'A' of Burnham's law) basically defines math as a series of "invented tools" that are developed to explain observed phenomena. And so by that standard, the 'B' must be that mathematics has no inherent physical meaning -- and thus, mathematical predictions of as-yet unobserved physical events are not meaningful. And it's stated baldly in the statement you quoted above: "Mathematical models and calculations are only useful if they accurately and correctly represent the data."
And yet this clearly fails to account for the fact that predictive mathematical theories are often developed well in advance of the observational evidence that confirms them (Einstein's gravitational theories being an obvious example). And think of the billions spent on particle accelerators which are often justified on the basis of searching for particles predicted by mathematical theories....
And, as you also point out, there are many examples of "interesting mathematical results" that have no immediate application, and yet just happen to drop nicely into place at some later point, in some complex physial theory.
The (B) statement that "mathematical models and calculations are only useful if they accurately and correctly represent the data," is clearly at odds with the practice of science. "The Data" are very often collected in response to the predictions, rather than the other way around. That particular (B) ends up being preposterous!
Indeed, why would we put so much time and money into CERN and Fermilab if not to investigate the mathematical theories? And how can we not hold biotech theories to same scrutiny as the ID hypothesis? Excellent points, all.
What an absolutely outstanding, insightful essay-post, r9etb! It's a keeper!
Thank you so very much for posting it!