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).