Richard Courant

"For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?"

"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."

"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."

"Mathematical techniques to achieve numerical solutions for began to appear about the turn of the century. The first definitive work was carried out by Richardson, who in a paper delivered to the in London in 1910 introduced a finite-difference technique for numerical solution of . Called a "relaxation technique," that approach is still used today to obtain numerical solutions for so-called {[w|elliptic partial differential equation}}s (the equations that govern inviscid subsonic flows are such equations). However, modern numerical analysis is usually considered to have begun in 1928, when Courant, Friedrichs, and Lewy published a definitive paper on the numerical solution of so-called s (the equations that govern inviscid compressible flow are such equations)."