To my mind this statement needs to be proved (though of course not in the sense my mathematician friends might want). Since neither she nor I are mathematicians, we are perfectly placed to comment on mathematics and mathematicians! The importance of mathematics is not in doubt. As a physicist, the "unreasonable effectiveness of mathematics" is obvious to all practitioners of the subject. However using mathematics as a tool is very different from doing mathematics itself which is what mathematicians do. And therein lies the rub.
Mathematics, to put it somewhat simplistically, is purely Platonic -- that is, complete understanding is to be achieved, and in fact, is achieved through pure thought. Pure mathematicians needs no knowledge of the physical world to prove their theorems. In fact, many of them (like G. H. Hardy) would be horrified at the thought that their work had any connection, dependence or relevance to the real world. Unlike physics, which is based on observation and experiment (leading to a theory) and eventual falsification (leading to a new theory, in Popper's famous description) a correct mathematical theorem is for eternity. No new observation of the physical world, no new insights into nature's working can have the slightest effect on its correctness. The only thing that can topple a theorem is a realisation that there was a flaw in its proof. 'Laws' of physics, on the other hand, are continuously falling by the wayside, replaced by new ones, based on newer and better experimental observations of the real world. (Even String Theory, the most esoteric and mathematical of all physical theories needs to pay obeisance to some fundamental symmetries of nature).
In the absence therefore, of a scientific principle or method, pure mathematics cannot be classed as a science. It is a pure art form, existing by and of itself with no supporting scaffolding from nature. It therefore requires no scientific bent of mind from its practitioners (in the sense that it does not require that its practitioners demand rational explanations of all natural phenomenon). A stone thrown into the air at an angle could well follow a parabolic path because God so decreed it, rather than the laws of physics. It would make not an iota of difference to any theorem past, present or future. (In recent years, this supremacy of mathematics has been partly dented by its somewhat intricate dependence on other branches of science -- Jones Polynomials and Chern-Simons theory, or the proof of the four colour map theorem which required the use of computers to eliminate a few remaining counter examples).
There is therefore no evidence, in my humble opinion, that the study of mathematics either promotes or even requires a scientific temper. This is also the reason why mathematicians as a community are far more religious (Ramanujan being the classic but by no means the only example) than their counterparts in the physical sciences -- not because mathematics promotes a belief in the supernatural, but because it does not require you to relinquish your belief in it. This has also been my personal experience, though I obviously do not have statistics to prove this claim.
Disclaimer: I do not claim that physicists are not religious -- many are, surprisingly. However it is a matter of statistics. As a fraction, fewer physicists in my opinion are overtly religious or believe in non-rational explanations of natural phenomena, compared to their mathematics colleagues.
Note added: Morris Kline discusses some of these issues and many more and twentieth century mathematics in his book -- Mathematics -- The Loss of Certainty. In particular he discusses what he calls 'The Authority of Nature" in the last chapter and in and around page 333. I thank Rahul Siddharthan for acquainting me with this book.