Dear Alexei
I thank your very kind words! I can see that "beauty" is something in which you live in... and it emerges in anything you do or think.. I will try to follow your example!
You know, mathematics is a logic language, strictly logic; however, to where it leads depends on the hypothesis and assumptions on which it is applied. Because it is logical, it leads to "understandable" models provided that the hypotheses and assumptions are "understandable"; however, if those hypotheses and assumptions are not understandable or incorrect, mathematics leads to models of reality that are not understandable.
If we accept that the universe is as simple as it can be, than hypotheses and assumptions must be understandable, no "magic" in them; and also the models mathematically obtained.
From this point of view, a model that is not understandable implies that the hypotheses/assumptions made are not correct.
Mathematics has also the possibility of fitting whatever set of data - it is just a matter of considering enough parameters. This is very important and is the first phase of discovery process because it allows having control over phenomena and organizing data. These mathematical models are usually "not-understandable", they present logical inconsistency and parameters that obviously cannot represent a physical entity. However, many consider that these models of data are correct models of reality; and so they consider that the universe is "not-understandable". That seems to be the case of Wigner, as expressed in the statement you cited: "mathematics is "the correct language", that is why it is "unreasonably effective" in physics".
Therefore, while some (the mainstream) consider that the unreasonably effectiveness of mathematics in physics is the proof that it is "the correct language", for others, like me, it is the proof that the hypotheses and assumptions made are wrong because mathematics can only lead to "reasonable" models.
Now, how do you know which approach is correct?
If you take just a quick look to my essay, you will probably suspect that I may know unexpected things about the universe. How is that possible? By looking for understandable models of the universe. That is the ultimate beauty of mathematics, the fact that it allows us to go finding the nature of the universe by looking for models that are understandable. And one does one step in the correct direction, one feels the amazing beauty of it.
All the best!
Alfredo Gouveia Oliveira