In Einstein's words, 'elements of reality' are 'free creations of the mind'. On the other hand, Einstein believed into a deeper-lying theoretical framework when he spoke of objective reality. Accordingly, we may expect paradoxes due to incompatibilities between different points of view.
There are several cases where a mathematical model is correct and complete but not appropriate to its application.
Let me start with the joke of more people leaving a room and than it contains.
My second example are so called ideal filter. They are non-causal. A non-causal system may exhibit an output before the input.
As long as the theory of signals and systems is logically explained to students step by step, they do not tend to complain much. However, B. Girod and R. Rabenstein admitted serious problems in their textbook 'Einfuehrung in die Systemtheory': Students have to persist. I see a main obstacle for understanding some consequences from the lacking insight that already our usual notion of time implies double redundancy.
While R. Feynman is famous for explaining very clever, he is perhaps even more known for having uttered:Quantum mechanics is impossible to understand. Shouldn't we suspect the reason for that in not yet correctly resolved questions?
In so far, I would like to agree with the article.
Being just an old engineer, no physicist, I naively try and translate the essence of =exp(ipx) roughly into
. I wonder because Fourier or cosine transform turns a discrete function of t into a continuous function of f and vice versa. What about spin, I realized the first maximum not at -1/2 or +1/2 but at +1/2 or +3/2.
Once again: I am not a physicist, and I am not a mathematician, too.
Nonetheless, engineers like me understand that a pound sugar never exactly equals a second pound sugar. Numbers are something ideal. It is easy to choose the number 2. However, it is impossible to give 2.000... with actually infinite accuracy. I see a main problem already in the notions of one and zero.
Mathematicians prefer to consider these problems and Buridan's donkey outside mathematics. Well, this might even be correct if mathematics understands itself like a gamble with arbitrarily set rules. Application in particular in physics requires to reveal and honestly delete inconsistencies at its roots. I refer to Cantor's obviously and admitted by Fraenkel untenable definition of a set.
One must not take two mutually excluding points of view at a time. The same applies with Schroedinger's cat.
Orthogonal variables mutually exclude and complement each other as also do discrete and continuous ones. As long as we look at a function of t, its envelope is imaginary.
What about S. Goldstein, I found most frequently quoted Physica Today 51-3, 42 (1998); 51_4, 38 (1998).