"Related is the idea of structural stability and certain variations. This kind of is a property of a dynamical system itself (not of a or orbit) and asserts that nearby dynamical systems have the same structure. The 'same structure' can be defined in several interesting ways, but the basic idea is that two dynamical systems have 'the same structure' if they have the same gross behavior, or the same qualitative behavior. For example, the original definition of 'same structure' of two dynamical systems was that there was an orbit preserving continuous transformation between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not stable, and have good but weaker stability properties." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)
"For a long time the aim of combinatorial analysis was to count the different ways of arranging objects under given circumstances. Hence, many of the traditional problems of analysis or geometry which are concerned at a certain moment with finite structures, have a combinatorial character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathematics, but exclusively for finite sets." (Louis Comtet, "Advanced Combinatorics", 1974)
"For the mathematician, the physical way of thinking is merely the starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)
"Combinatorial analysis, or combinatorics, is the study of how things can be arranged. In slightly less general terms, combinatorial analysis embodies the study of the ways in which elements can be grouped into sets subject to various specified rules, and the properties of those groupings. […] Combinatorial analysis often asks for the total number of different ways that certain things can be combined according to certain rules." (Martin Gardner, "Aha! Insight", 1978)
"Graph theory is the study of sets of points that are joined by lines." (Martin Gardner,"Aha! Insight", 1978)
"The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. [...] Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa." (C West Churchman et al, "Thinking for Decisions Deduction Quantitative Methods", 1975)
"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)
"Having vegetated on the fringes of mathematical science for centuries, combinatorics has now burgeoned into one of the fastest growing branches of mathematics – undoubtedly so if we consider the number of publications in this field, its applications in other branches of mathematics and in other sciences, and also the interest of scientists, economists and engineers in combinatorial structures. The mathematical world was attracted by the successes of algebra and analysis and only in recent years has it become clear, due largely to problems arising from economics, statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty." (László Lovász, "Combinatorial Problems and Exercises", 1979)
"If we gather more and more data and establish more and more associations, however, we will not finally find that we know something. We will simply end up having more and more data and larger sets of correlations." (Kenneth N Waltz, "Theory of International Politics Source: Theory of International Politics", 1979)
"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements." (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)
"It is now generally recognized that the field of combinatorics has, over the past years, evolved into a fully-fledged branch of discrete mathematics whose potential with respect to computers and the natural sciences is only beginning to be realized. Still, two points seem to bother most authors: The apparent difficulty in defining the scope of combinatorics and the fact that combinatorics seems to consist of a vast variety of more or less unrelated methods and results. As to the scope of the field, there appears to be a growing consensus that combinatorics should be divided into three large parts: (i) Enumeration, including generating functions, inversion, and calculus of finite differences; (ii) Order Theory, including finite sets and lattices, matroids, and existence results such as Hall's and Ramsey's; (iii) Configurations, including designs, permutation groups, and coding theory." (Martin Aigner, "Combinatorial Theory", 1979)
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