Dear Dr. Laurence,
thank you for your provoking post.
For me "reality" is definitely not a mathematical structure. You should specify better what you mean by reality: if reality is what you see (the shadows on the Plato's cave) or what you believe is out of there (the bodies projecting the shadows). In my essay I clearly stated that the main point in the scientific method is to clearly distinguish between theory and experiment. Such a distinction, far from being trivial as it may appear at first sight, is in my opinion the main reason for the stubborn attachment of many scientists to the view of space-time as "a stage where particles move"--a theoretical landscape that has been proved to be foundationally inconsistent, and it is the origin of the apparent paradox of the GR-QT conflict. We are taking about "theory" here, namely what we believe is out of the cave, the mechanisms by which we explain what we see. But what we believe is not what is "actually" out there (this is a nonsense: who is the Referee to assess reality of what we believe?) We shouldn't forget that we are the ones that build-up the "ontologies" as convenient tools for reasoning.
Now, at this point, you should agree that what we are talking about is "theory". And theory is written in the language of mathematics. Therefore, it is not "reality" that is an abstract mathematical construction.
Thank you again for the opportunity given to me by your post
Mauro