• Each progress in mathematics is based on the discovery of stronger tools and easier methods, which at the time makes it easier to understand earlier methods. By making these stronger tools and easier methods his own, it is possible for the individual researcher to orientate himself in the different branches of mathematics.
  The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena.
  -David Hilbert (1900 Paris Lecture)



• Technical skill is mastery of complexity while creativity is mastery of simplicity.
  -E. C. Zeeman
(note that this quote applies equally well to music and mathematics).


• Although it may be fashionable to acknowledge that everything is connected to everything else in principle, some things are more tightly connected to each other than to all the rest. Such a little knot of causal interactions goes by the name of a system.
  -Art Winfree



• Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation and guesswork.
  -Paul Halmos



• Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.
  -Niels Abel



• The value of a mathematical discipline is to be determined by its applicability to the empirical sciences.
  -Carle Runge
Doctoral Dissertation, Berlin, April 23, 1880.


• The value of a mathematical discipline cannot be measured by its applicability to the empirical sciences.
  -Ferdinand Rudio
Doctoral Dissertation, Berlin, April 23, 1880.


• Problems worthy of attack prove their worth by fighting back.
  -Paul Erdos
  
  
• In brief, the flight into abstract generality must start from and return to the concrete and the specific.
  -Richard Courant



• Nonlinear constitutive modeling is a jungle.
  -Dan Joseph



• On duality:My impression is that duality appeared first in projective geometry where one interchanges the role of points and lines (this evolved slowly from Apollonius' [c. 262-190 BC] use of "pole" and "polar" through 1850). Lagrange introduced the adjoint of a differential operator in the eighteenth century (this is the essence of Lgarange's identity for linear second order ordinary differential operators) while the adjoint of a matrix seems to have been used significantly only in the ninteenth century. Green's second identity (1828) asserts that the Laplacian is formally self-adjoint. Lagrangian and Hamiltonian mechanics are dual objects: Lagrangian living on the tangent bundle, Hamiltonian on the cotangent bundle. [...] Duality is even a standard device in rhetoric: "Do unto others as you would want other do unto you," and J.F. Kennedy's "...ask not what your country can do for you, ask what you can do for your country".
  -Jerry Kazdan