The great Felix Klein urged us quite a number of years ago to join hands with our students and share the pleasures of “delightful and instructive walks through forests, fields, and gardens … without digging up the most profitable plants to replant them in prepared soil according to the principles of rational agronomy.” 
And he furthermore added: “Apart from the majority of enthusiastic students there are always a few students who are not entirely satisfied, who criticise and question. These are the ones dearest to my heart. For I see in them what I consider to be the true goal of all teaching: independent thought.” 
I feel that the intellectual mathematics philosophy is conceived in the spirit of these passages.
 Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Teil I. Springer-Verlag, Berlin, 1928. Quoted from the English translation: Development of Mathematics in the 19th Century. Math. Sci. Press, Brookline, MA, 1979, page 152.
 Felix Klein, Über Aufgabe und Methode des mathematischen Unterrichts an den Universitäten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 7, 126–138. Quotation from page 133, my translation.