Quotes from Timothy Gowers

This attitude [the abstract method in mathematics] can be encapsulated in the following slogan: a mathematical object is what it does.

... the atlas is a manifold. This is a typical mathematician's use of the word "is", and should not be confused with the normal use.

Here is something Category-Theorists like: it is trivial, but not trivially trivial.

Modules are to rings as vector spaces are to fields. In other words, they are algebraic structures where the basic operations are addition and scalar multiplication, but now the scalars are allowed to come from a ring rather than a field.

Can we have genuine knowledge of space without ever leaving our armchairs?

It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using.

Roses belong to the set R

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

the simple algebraic equation ?+k3 = 0. This is called the dispersion relation of (1): with the help of the Fourier transform it is not hard to show that every solution is a superposition of solutions of the form ei(kx-?t), and the dispersion relation tells us how the "wave number" k is related to the "angular frequency" ? in each of these elementary solutions.

The function ei(kx-?t) represents a wave that travels at a speed of ?/k, which we have just shown to be equal to -k2. Therefore, the different plane-wave components of the solution travel at different speeds: the higher the angular frequency, the greater the speed. For this reason, the equation (1) is called dispersive.

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.

starting from any topological space, we construct an algebraic object, in this case a group. If two spaces are homeomorphic, then their fundamental groups (and higher homotopy groups) must be isomorphic. This is richer than the original idea of just measuring the number of holes, since a group contains more information than just a number.

ut = -uux - uxxx. Why is it that this equation gives rise to the remarkable stability of the solutions that was observed experimentally by Russell? Intuitively, the reason is that there is a balance between the dispersing effect of the uxxx term and the shock-forming effect of the uux term.

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.

A typical quotient construction for an algebraic structure A will identify some substructure B and regard two elements of A as "equivalent if they "differ by an element of B.

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.

An algebraic integer of degree two is simply a root of a quadratic polynomial of the form X2 + aX + b with a, b ordinary integers.