Quotes About Mathematics

Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk. The dear God has made the whole numbers, all the rest is man's work.

I was very much on the mathematical side, where you probably do your best work before you're forty-five. Having passed that significant date, I thought I would do something else.

I use geometric and mathematical ideas to organize material, but those are tools. The purpose of the work is not to expose that at all, but to arrive at some kind of expressiveness.

The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished.

Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.

It's fine to work on any problem, so long as it generates interesting mathematics along the way - even if you don't solve it at the end of the day.

The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another.

Math has a lot of negative stereotypes, but it can actually be fun and incredibly empowering.

Nature doesn't feel compelled to stick to a mathematically precise algorithm; in fact, nature probably can't stick to an algorithm.

At school, I wasn't as interested in mathematics. I did OK, but at the earliest point I could stop doing math, I stopped.

I won some genetic lottery. I always happened to be strangely good at mathematics in my head. I just popped out weird.

Manuel Blum, who many people consider the father of cryptography [encryption, etc.].

Roses belong to the set R

Topology allows the possibility of making qualitative predictions when quantitative ones are impossible.

a continuous curve that goes from the lower half-plane to the upper half-plane must cross the horizontal axis at some point.

Compactness is a powerful property of spaces, and it is used in many ways in many different areas of mathematics. One is via appeal to local-to-global principles: one establishes local control on a function, or on some other quantity, and then uses compactness to boost the local control to global control.

differentiation is the adjoint of the boundary operation.

differential equations that can be solved in "closed form," that is, by means of a formula for the unknown function f, are the exception rather than the rule

there certainly are philosophers who take seriously the question of whether numbers exist, and this distinguishes them from mathematicians, who either find it obvious that numbers exist or do not understand what is being asked.

One can also use compactifications to view the continuous as the limit of the discrete: for instance, it is possible to compactify the sequence / 2,/ 3,/ 4, . . . of cyclic groups in such a way that their limit is the circle group = /.

if X and Y are two topological spaces and if f : X ? Y is a function between them, then we simply define f to be continuous if f-1 (U) is open for every open set U ? Y. Remarkably, we have found a useful definition of continuity that does not rely on a notion of distance.

The curious switch, from initially perceiving an obstruction to a problem to eventually embodying this obstruction as a number or an algebraic object of some sort that we can effectively study, is repeated over and over again, in different contexts, throughout mathematics. Much later, complex quadratic irrationalities also made their appearance. Again these were not at first regarded as "numbers as such," but rather as obstructions to the solution of problems.

we do not have anything directly comparable to continued-fraction expansions for a complex quadratic irrationality. In fact, the simple, but true, answer to the problem of how to find an infinite number of rational numbers that converge to such an irrationality is that you cannot! Correspondingly, the analogue of the Pell equation has only finitely many solutions.

the kernel of a homomorphism is closed under addition, and also under multiplication by any element of the ring. These two properties define the notion of an ideal.