There are, of course, many exceptions to this "rule". From the preface to my favorite calculus book:
My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.
- Richard Courant, Differential and Integral Calculus [1]
Courant, who trained — under Hilbert, no less — as a pure mathematician, never tired of pointing out the importance of applications to pure mathematics and vice versa. In addition to the calculus books, see, e.g.,
jasomill|6 years ago
My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.
- Richard Courant, Differential and Integral Calculus [1]
Courant, who trained — under Hilbert, no less — as a pure mathematician, never tired of pointing out the importance of applications to pure mathematics and vice versa. In addition to the calculus books, see, e.g.,
https://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943...
[1] https://archive.org/details/DifferentialIntegralCalculusVolI...