Timothy Gowers > Quotes

“It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using.”
― Timothy Gowers, The Princeton Companion to Mathematics

“Can we have genuine knowledge of space without ever leaving our armchairs?”
― Timothy Gowers, The Princeton Companion to Mathematics

“Roses belong to the set R”
― Timothy Gowers, The Princeton Companion to Mathematics

“A3 0 is an additive identity: 0 + a = a for any number a. That is all you need to know about 0. Not what it means – just a little rule that tells you what it does.”
― Timothy Gowers, Mathematics: A Very Short Introduction

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“Fundamental to understanding the arithmetic of Rd is the following question: which ordinary prime numbers p are irreducible elements of Rd and which ones factorize as products of irreducible elements in Rd? We will see shortly that if a prime number does factorize in Rd, it must be expressible as the product of precisely two irreducible factors.”
― Timothy Gowers, The Princeton Companion to Mathematics

“if, for each t, we write u(t) for the function from to that takes x to u(x, t), then it describes how the function u(t) “evolves” over time. The Cauchy problem for an evolution equation is the problem of determining this evolution from knowledge of its initial value u(0).”
― Timothy Gowers, The Princeton Companion to Mathematics

“differentiation is the adjoint of the boundary operation.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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 = /.”
― Timothy Gowers, The Princeton Companion to Mathematics

“The function ei(kx-ωt) represents a wave that travels at a speed of ω/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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, Mathematics: A Very Short Introduction

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“There is a very important “symmetry,” or ALRROMORPHISM”
― Timothy Gowers, The Princeton Companion to Mathematics

“We have seen that integration is a duality pairing between manifolds and forms. Since manifolds push forward under Φ from X to Y, we expect forms to pull back from y to X. Indeed, given any k-form ω on Y, we can define the pullback Φ* ω as the unique k-form on X such that we have the change-of-variables formula ∫ Φ(s) ω = ∫s Φ*(ω). In the case of 0-forms (i.e., scalar functions), the pullback Φ* f : x → of a scalar function f: y → is given explicitly by Φ* f (x) = f (Φ(x)), while the pullback of a 1-form ω is given explicitly by the formula (Φ* ω) x(v) = ωΦ(x) (Φ*v).”
― Timothy Gowers, The Princeton Companion to Mathematics

“a functor F from a category X to a category Y takes the objects of X to the objects of Y and the morphisms of X to the morphisms of Y in such a way that the identity of a is taken to the identity of Fa and the composite of f and g is taken to the composite of Ff and Fg. An important example of a functor is the one that takes a topological space S with a “marked point” s to its fundamental group π(S, s): it is one of the basic theorems of algebraic topology that a continuous map between two topological spaces (that takes marked point to marked point) gives rise to a homomorphism between their fundamental groups.”
― Timothy Gowers, The Princeton Companion to Mathematics

“The description given earlier of the relationship between integrating a 2-form over the surface of a sphere and integrating its derivative over the solid sphere can be thought of as a generalization of the fundamental theorem of calculus, and can itself be generalized considerably: Stokes’s theorem is the assertion that for any oriented manifold S and form ω, where ∂ S is the oriented boundary of S (which we will not define here). Indeed one can view this theorem as a definition of the derivative operation ω → dω; thus, differentiation is the adjoint of the boundary operation. (For instance, the identity (11) is dual to the geometric observation that the boundary ∂s of an oriented manifold itself has no boundary: ∂(∂S) = ∅.) As a particular case of Stokes’s theorem, we see that ∫s dω = 0 whenever S is a closed manifold, i.e., one with no boundary. This observation lets one extend the notions of closed and exact forms to general differential forms, which (together with (11)) allows one to fully set up de Rham cohomology.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“a continuous curve that goes from the lower half-plane to the upper half-plane must cross the horizontal axis at some point.”
― Timothy Gowers, The Princeton Companion to Mathematics

“the simple algebraic equation ω+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-ωt), and the dispersion relation tells us how the “wave number” k is related to the “angular frequency” ω in each of these elementary solutions.”
― Timothy Gowers, The Princeton Companion to Mathematics

“the second derivative conveys just the idea we want—a comparison between the value at x and the average value near x. It is worth noting that if f is linear, then the average of f(x - h) and f(x + h) will be equal to f(x), which fits with the familiar fact that the second derivative of a linear function f is zero.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“there are only four units in the ring R-1 of Gaussian integers, namely ±1 and ±i; multiplication by any of these units effects a symmetry of the infinite square tiling”
― Timothy Gowers, The Princeton Companion to Mathematics

“Topology allows the possibility of making qualitative predictions when quantitative ones are impossible.”
― Timothy Gowers, The Princeton Companion to Mathematics

“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.”
― Timothy Gowers, The Princeton Companion to Mathematics

“The collection of all real or complex numbers that are integral linear combinations of 1 and τd is closed under addition, subtraction, and multiplication, and is therefore a ring, which we denote by Rd. That is, Rd is the set of all numbers of the form a + bτd where a and b are ordinary integers. These rings Rd are our first, basic, examples of rings of algebraic integers beyond that prototype, , and they are the most important rings that are receptacles for quadratic irrationalities. Every quadratic irrational algebraic integer is contained in exactly one Rd.”
― Timothy Gowers, The Princeton Companion to Mathematics