"The fuzzy set theory is taking the same logical approach as what people have been doing with the classical set theory: in the classical set theory, as soon as the two-valued characteristic function has been defined and adopted, rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued characteristic function" (the membership function) has been chosen and fixed, a rigorous mathematical theory can be fully developed." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)
"An organizing frame provides a topology for the space it organizes; that is, it provides a set of organizing relations among the elements in space. When two spaces share the same organizing frame, they share the corresponding topology and so can easily be put into correspondence. Establishing a cross-space mapping between inputs becomes straightforward." (Gilles Fauconnier, "The Way We Think: Conceptual Blending and The Mind's Hidden Complexities", 2002)
"Fuzzy relations are developed by allowing the relationship between elements of two or more sets to take on an infinite number of degrees of relationship between the extremes of 'completely related' and 'not related', which are the only degrees of relationship possible in crisp relations. In this sense, fuzzy relations are to crisp relations as fuzzy sets are to crisp sets; crisp sets and relations are more constrained realizations of fuzzy sets and relations. " (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)
"Systems-people everywhere share certain attributes, but each specific System tends to attract people with specific sets of traits. […] Systems attract not only Systems-people who have qualities making for success within the System; they also attract individuals who possess specialized traits adapted to allow them to thrive at the expense of the System; i.e., persons who parasitize the System." (John Gall, "Systemantics: The Systems Bible", 2002)
"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)
"Combinatorics could be described as the study of arrangements of objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and, if so, in how many different ways it can be done. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)
"Essentially, the theory of finite state machines is that of computation. It postulates two finite sets of external states called 'input states' and 'output states', one finite set of 'internal states', and two explicitly defined operations" (computations) which determine the instantaneous and temporal relations between these states." (Heinz von Foerster, "Understanding Understanding: Essays on Cybernetics and Cognition", 2003)
"Knowledge is encoded in models. Models are synthetic sets of rules, and pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Didier Sornette, "Why Stock Markets Crash - Critical Events in Complex Systems", 2003)
"A symmetry is a set of transformations applied to a structure, such that the transformations preserve the properties of the structure." (Philip Dorrell, "What is Music?: Solving a Scientific Mystery", 2004)
"A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists" (and the greatest possibility of using different consistent axiom systems)." (Paul Cohen, "Skolem and pessimism about proof in mathematics". Philosophical Transactions of the Royal Society A 363 (1835), 2005)
"Does set theory, once we get beyond the integers, refer to an existing reality, or must it be regarded, as formalists would regard it, as an interesting formal game? [...] A typical argument for the objective reality of set theory is that it is obtained by extrapolation from our intuitions of finite objects, and people see no reason why this has less validity. Moreover, set theory has been studied for a long time with no hint of a contradiction. It is suggested that this cannot be an accident, and thus set theory reflects an existing reality. In particular, the Continuum Hypothesis and related statements are true or false, and our task is to resolve them." (Paul Cohen, "Skolem and pessimism about proof in mathematics", Philosophical Transactions of the Royal Society A 363 (1835), 2005)
"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)
"The Nash equilibrium is often rationalized using a story about how people think and how their behaviour is related to their thoughts. Economists generally assume that, from a set of alternatives, a player will actively choose the one he likes best. This is the assumption of economic rationality, one of the core assumptions of standard game theory. Rationality alone will not predict behaviour in a game, but it leads us to single out the member of any inducement correspondence that yields the greatest payoff for the player that is making the choice. The resulting behaviour is sometimes described as 'myopic' because it fails to take into account how other players might respond to a given choice." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"One important and standardized block of information in the formal descriptions used by game theorists is called a game form. A form specifies the payoffs associated with every possible combination of decisions. There are several widely used forms, including the strate-gic form, typically presented in a matrix, the extensive form, which is usually represented as a tree, and the characteristic function form, expressed as a function on subsets of players." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Set theory is unusual in that it deals with remarkably simple but apparently ineffable objects. A set is a collection, a class, an ensemble, a batch, a bunch, a lot, a troop, a tribe. To anyone incapable of grasping the concept of a set, these verbal digressions are apt to be of little help. […] A set may contain finitely many or infinitely many members. For that matter, a set such as {} may contain no members whatsoever, its parentheses vibrating around a mathematical black hole. To the empty set is reserved the symbol Ø, the figure now in use in daily life to signify access denied or don’t go, symbolic spillovers, I suppose, from its original suggestion of a canceled eye." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"A group is a set along with a rule that tells how to combine any two elements in the set to get another element in the set. We usually use the word composition to describe the act of combining two elements of the group to get a third." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)
"Each fuzzy set is uniquely defined by a membership function. […] There are two approaches to determining a membership function. The first approach is to use the knowledge of human experts. Because fuzzy sets are often used to formulate human knowledge, membership functions represent a part of human knowledge. Usually, this approach can only give a rough formula of the membership function and fine-tuning is required. The second approach is to use data collected from various sensors to determine the membership function. Specifically, we first specify the structure of membership function and then fine-tune the parameters of membership function based on the data." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)
"The set of complex numbers is another example of a field. It is handy because every polynomial in one variable with integer coefficients can be factored into linear factors if we use complex numbers. Equivalently, every such polynomial has a complex root. This gives us a standard place to keep track of the solutions to polynomial equations." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)
"A little ingenuity is involved, but once a couple of tricks are learnt, it is not hard to show many sets of numbers are countable, which is the term we use to mean that the set can be listed in the same fashion as the counting numbers. Otherwise a set is called uncountable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)
"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I. Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)
"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)
"One of the nice features of the metric space setting is that all topological notions can be formulated in terms of sequences. Such is not the case in an arbitrary topological space. [...] Whereas a sequence is modeled on the natural numbers, a net is modeled on a more general object called a directed set. The general feel of the subject is similar to that for sequences, but it is rather more abstract." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)
"The Continuum Hypothesis is the assertion that there are no cardinalities strictly between the cardinality of the integers and the cardinality of the continuum (the cardinality of the reals). [...] In logical terms, we say that the Continuum Hypothesis is independent from the other axioms of set theory, in particular it is independent from the Axiom of Choice." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)
"The most fundamental tool in the subject of point-set topology is the homeomorphism. This is the device by means of which we measure the equivalence of topological spaces." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)
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