Plato's Ghost: The Modernist Transformation of Mathematics

Glossary

Glossary The purpose of this glossary is to give a more accurate impression of the meaning of some of the mathematical terms used in this book. When it was not possible to give precise, meaningful definitions, I have to try for intelligibility rather than precision. Alephs The cardinality of a transfinite initial ordinal number is denoted by a symbol called an aleph (the least ordinal number of a given cardinality is called the corresponding initial ordinal). The axiom of choice implies that every infinite cardinal number is an aleph. Algebraic integers An algebraic integer is a root of a monic polynomial whose coefficients are integers. (A monic polynomial is one where the coefficient of the highest power of the variable is 1.) Examples: ffiffiffi 2 p , which is a root of x2 2 ¼ 0, and o, a root other than 1 of x3 1 ¼ 0. Archimedean, non-Archimedean axioms Given a set of elements X that can be added together and compared in size, so, for any two elements a, b [ X, exactly one of these three statements is true: a 0 there is a d > 0 such that |x t| 0 there is a d > 0 such that |x t| 0, where the e may depend on d but not on t. A sequence un converges to a value u if and only if for every e > 0 there is a N > 0 such that n > N Þ |un u| 0 there is a N > 0 such that n > N Þ |un u| 0 there is a d > 0 such that jx tj\ ) j f(x) f(t) x t j\. Dimension in topology. A topological space is a set together with a distinguished collection T of its subsets that has the properties that arbitrary unions and arbitrary finite intersections of sets in T are again in T. For example, the open intervals on the real line form a topology for the real line. Roughly speaking, the dimension of a topological space is the smallest integer n such that any collection of n þ 2 open sets whose union is the whole space may contain a disjoint pair of open sets, but no collection of n þ 1 open sets whose union is the whole space contains a disjoint pair of open sets. On this definition, the real line has dimension 1, as it should. Dirichlet principle and Dirichlet problem See 2.2.4. A field F is a set of elements satisfying three families of conditions. First, it forms a commutative group with a composition ‘‘þ’’ called addition. We denote the identity element for ‘‘þ’’ by 0. Second, the set F with the element 0 removed forms a commutative group with a composition ‘‘.’’ called multiplication. Third, there are laws connecting the addition and the multiplication, such as (for elements a, b, c of F): a.(b þ c) ¼ a.b þ a.c. Finite geometry (§4.1.1). A plane geometry is finite if it consists of a finite number of points and lines; analogous definitions hold in higher dimensions. Fourier series, coefficients of, convergence of See §2.2.1. Gaussian curvature A measure of how curved a surface in space, S, is at any point P on it. If the surface looks like a sphere in a small neighborhood of P, then the Gaussian curvature gives the size of the closest spherical approximation to the surface at P; and if the surface is saddle shaped in a small neighborhood of P, then the Gaussian curvature gives the size of the closest saddle-shaped approximation to the surface at P. Geometry: Synthetic v. analytic geometry v. algebraic geometry Geometry is synthetic if the fundamental (undefined) terms are point, line, plane, etc., and there are (undefined) concepts of a point lying on a line, two lines meeting in a point, etc. Geometry is analytic or algebraic (the terms are treated as synonymous in this book) if the terms point, line, plane, etc. are defined by equations, and the incidence relations are said to hold when certain equations are satisfied. Grelling’s paradox See §4.7.3. 468  GLOSSARY Group, group theory. A set X is a group when it is closed under a binary operation ‘‘.’’ (so if a, b [ X then a.b [ X) with the properties that  there is an e [ X such that a.e ¼ e.a ¼ a for every a [ X  for every a [ X there is a a* [ X such that...