Miles Reid - Collected Quotes

"A group G is a topological group if it has a topology defined on it so that multiplication and inverse are continuous. In more detail, a topological group is an object G having two quite different structures: a collection of open subsets satisfying the axioms for a topology, and a multiplication map with identity and inverse satisfying the groupaxioms." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"[...] a topological space is a ‘metric space without a metric’. In analysis, this idea leads to a fairly minor generalisation of the definition of metric space, but the definition of topology has applications in other areas of math, where it turns out to be logical or algebraic in content. [...] Topology has lots of advantages even when the only spaces of interest are metric spaces. It provides, in particular, a simple rigorous language for ‘sufficiently near’ without epsilons and deltas." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Descartes’ idea to use numbers to describe points in space involves the choice of a coordinate system or coordinate frame: an origin, together with axes and units of length along the axes. A recurring theme of all the different geometries [...] is the question of what a coordinate frame is, and what I can get out of it. While coordinates provide a convenient framework to discuss points, lines, and so on, it is a basic requirement that any meaningful statement in geometry is independent of the choice of coordinates. That is, coordinate frames are a humble technical aid in determining the truth, and are not allowed the dignity of having their own meaning." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Descartes’ invention of coordinate geometry is another key ingredient in modern science. It is scarcely an accident that calculus was discovered by Leibnitz and Newton (independently, alphabetical order) in the fifty years following the dissemination of Descartes’ ideas. Interactions between the axiomatic and the coordinate-based points of view go in both ways: coordinate geometry gives models of axiomatic geometries, and conversely, axiomatic geometries allow the introduction of number systems and coordinates." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches of mathematics and physics." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"More generally an axiomatic projective space has a lattice of projective linear subspaces, the incidence relation ⊂, intersection and linear span, and suitable axioms. It is best not to insist a priori that the dimension of the space or its projective linear subspaces is specified. The most important case is the infinite dimensional case, which von Neumann used to give axiomatic foundations to quantum mechanics, when dimensions of projective linear subspaces can take values in R or the value ∞." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Emmy Noether’s principle of conserved quantities says that for a physical system with a symmetry group, there are as many conserved quantities (constants of the system unchanged as a function of time) as parameters for the group." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"One may consider the fundamental objects in math to be numbers of various kinds; the basic operations on them are then addition and multiplication (together with subtraction, division, taking roots, etc., which are in some sense the inverses of the basic operations). There would be no point in having numbers if you could not calculate with them. The reason that we use numbers to model the real world is precisely that it is easier to perform operations on numbers than make the corresponding constructions on objects out there in the wild." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"The action of a transformation group on a space is another way of saying symmetry. To say that an object has symmetry means that it is taken into itself by a group action: rotational symmetry means symmetry under the group of rotations about an axis." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"The idea of a topological space is a natural abstraction and generalisation of the idea of a metric space. When going from a metric space (X, d) to the corresponding topological space, we forget the metric, and keep only the notion of neighbourhoods, or equivalently open sets. There are several advantages. In the context of metric spaces, closeness means that the distance d(x, y) is small. But just as some things in life have a value that cannot be expressed as a sum of money, in some contexts closeness cannot always be expressed as a distance measured as a real number." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"The statements on asymptotes are qualitative views of what happens to the curves when x or y is large (quite vague, even arguable for those in quotes). But we have not so far said what asymptotic directions or points at infinity actually are, which is a disadvantage in discussing asymptotes formally or in calculating with them. Making sense of asymptotes (of algebraic plane curves), and providing a simple framework for calculating with them is one thing that projective geometry does very well." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Together with plane Euclidean geometry, spherical and hyperbolic geometry are 2-dimensional geometries with the following properties: (1) distance, lines and angles are defined and invariant under motions; (2) the motions act transitively on points and directions at a point; (3) locally, incidence properties are as in plane Euclidean geometry." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)