What is Mathematics

Profound study of nature is the most fertile source of mathematical discoveries.
(J. Fourier, Analytic theory of heat, 1830)

It is true that Mr. Fourier had the opinion that the principal purpose of mathematics was the benefit of the society and the explanation of phenomena of nature; but a philosopher like he should know that the sole purpose of science is the honor of the human mind, and under this title, a question about numbers is as valuable as a question about the system of the world.
(C. G. Jacobi, Letter to Le Gendre, 1830)

In practice this is of course not at all important, because it is negligible for the largest triangle on earth that can be measured; however the dignity of science requires that we understand clearly the nature of this inequality...
(Gauss, in a letter to a friend on a correction of .001" in the measurement of triangles on the earth surface).

Degeneration of mathematics begins with the ideas of Riemann, Dedekind and Cantor which progressively repressed the reliable genius of Euler, Lagrange and Gauss.
(C. L. Siegel, Letter to H. Weyl, 1959)

Carl Ludwig Siegel wrote once to Weil: It is completely clear to me which conditions caused the gradual decadence of mathematics, from its high level some 100 years ago, down to the present hopeless nadir... Through the influence of textbooks like those of Hasse, Schreier and van der Waerden, the new generation was seriously harmed, and the work of Bourbaki finally dealt the fatal blow. What would Siegel write today?
(F. Catanese, Monodromy and normal forms, http://arxiv.org/pdf/1507.00711v1.pdf)

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA).
(V. Arnold, in: Mathematics: Frontiers and Perspectives, AMS 2000)

You cannot have both. I mean a fish having a meeting one billion years ago under the water and saying: "We are fish, we have all this power, now it is time to conquer the land". But you cannot conquer the land while remaining fish. You can't go to the real world remaining mathematicians; that's absurd. Either you study real problems in the real world - it's a remarkable intellectual challenge, or you remain a mathematician.
(M. Gromov, Dead Sea discussions, in: GAFA 2000. Visions in Mathematics)

Other interesting opinions:

Hironaka and Courant on Analysis
Hilbert and Gromov on analytic functions and complex numbers
Weil on modern analysis
d'Ambra and Gromov on Mobius transformations
On J. E. Littlewood
Lyapunov, comparing Chebyshev and Riemann (in Russian, of course).

French trnslation of this page