George F Simmons - Collected Quotes

"A special role is played in the theory of metric spaces by the class of open spheres within the class of all open sets. The main feature of their relationship is that the open sets coincide with all unions of open spheres, and it follows from this that the continuity of a mapping can be expressed either in terms of open spheres or in terms of open sets, at our convenience." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"A topological space can be thought of as a set from which has been swept away all structure irrelevant to the continuity of functions defined on it." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Analysis is primarily concerned with limit processes and continuity, so it is not surprising that mathematicians thinking along these lines soon found themselves studying (and generalizing) two elementary concepts: that of a convergent sequence of real or complex numbers, and that of a continuous function of a real or complex variable." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Determinants are often advertised to students of elementary mathematics as a computational device of great value and efficiency for solving numerical problems involving systems of linear equations. This is somewhat misleading, for their value in problems of this kind is very limited. On the other hand, they do have definite importance as a theoretical tool. Briefly, they provide a numerical means of distinguishing between singular and non-singular matrices (and operators)." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Historically speaking, topology has followed two principal lines of development. In homology theory, dimension theory, and the study of manifolds, the basic motivation appears to have come from geometry. In these fields, topological spaces are looked upon as generalized geometric configurations, and the emphasis is placed on the structure of the spaces themselves. In the other direction, the main stimulus has been analysis. Continuous functions are the chief objects of interest here, and topological spaces are regarded primarily as carriers of such functions and as domains over which they can be integrated. These ideas lead naturally into the theory of Banach and Hilbert spaces and Banach algebras, the modern theory of integration, and abstract harmonic analysis on locally compact groups." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"In all candor, we must admit that the intuitive meaning of compactness for topological spaces is somewhat elusive. This concept, however, is so vitally important throughout topology […]" (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"In many branches of mathematics - in geometry as well as analysis - it has been found extremely convenient to have available a notion of distance which is applicable to the elements of abstract sets. A metric space (as we define it below) is nothing more than a non-empty set equipped with a concept of distance which is suitable for the treatment of convergent sequences in the set and continuous functions defined on the set." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963) 

"It is extremely helpful to the imagination to have a geometric picture available in terms of which we can visualize sets and operations on sets. […] Diagrammatic thought of this kind is admittedly loose and imprecise; nevertheless, the reader will find it invaluable. No mathematics, however abstract it may appear, is ever carried on without the help of mental images of some kind, and these are often nebulous, personal, and difficult to describe." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"It is sometimes said that mathematics is the study of sets and functions. Naturally, this oversimplifies matters; but it does come as close to the truth as an aphorism can." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"It seems to me that a worthwhile distinction can be drawn between two types of pure mathematics. The first - which unfortunately is somewhat out of style at present - centers attention on particular functions and theorems which are rich in meaning and history, like the gamma function and the prime number theorem, or on juicy individual facts […] The second is concerned primarily with form and structure." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The essence of analytic geometry lies in the possibility of exploiting this identification by using algebraic tools in geometric arguments and giving geometric interpretations to algebraic calculations." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up, onto the highlands of mathematics itself, where these concepts are indispensable in almost all of pure mathematics as it is today." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)