I. In my experience, interesting and relevant mathematics is more often stimulated by interesting experimental results or questions than vice versa.
II. Interesting mathematical ideas can misdirect you into scientific dead ends - they can become answers in search of a problem.
III. Complicated or lengthy mathematical procedures are very easy to use as a cover-up for shoddy or dishonest science.He gives the following examples:
I. random matrices, general relativity, quantum Hall effect, localisation, Kondo effect
II. "particle democracy" = "self-consistent dispersion theory"
quantum critical points!
III. Attempts of an unnamed "honored professor" [presumably W. Goddard III's] to calculate superconducting high-Tc using computational quantum chemistry.
I agree with Anderson's concerns, but offer one minor disagreement about the title and context of his essay. It was meant to be a response to Eugene Wigner's famous essay "The unreasonable effectiveness of mathematics in the natural sciences". However, I think Wigner's paper was wrestling with profound philosophical questions not singing the praises on a highly mathematical approach to theoretical physics.
On the other hand, Phil said he embarked on the exercise "with a very negative attitude" and with his "usual determination to put the cat among the pigeons" due to his difficult relationship with his Princeton colleague Wigner. (This is described at several points in the book.)