- 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
The organic unity of mathematics is inherent in the nature of this
science, for mathematics is the foundation of all exact knowledge of
-David Hilbert (1900 Paris Lecture)
Technical skill is mastery of complexity while creativity is mastery of simplicity.
(note that this quote applies equally well to music and mathematics).
-E. C. Zeeman
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.
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.
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.
The value of a mathematical discipline is to be determined by its applicability
to the empirical sciences. Doctoral Dissertation, Berlin, April 23, 1880.
The value of a mathematical discipline cannot be measured by its applicability to
the empirical sciences. Doctoral Dissertation, Berlin, April 23, 1880.
- Problems worthy of attack prove their worth by fighting back.
In brief, the flight into abstract generality must start from and
return to the concrete and the specific.
Nonlinear constitutive modeling is a jungle.
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".