For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?
What is Mathematics?, p. xix
It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together.
Richard Courant, "Mathematics in the Modern World", Scientific American, Vol 211, (Sep 1964), p. 42
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
Empirical evidence can never establish mathematical existence--nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful.
Richard Courant in: The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba, Springer-Verlag, 1996, page 148