"[…] a conceptual model is a diagram connecting variables and constructs based on theory and logic that displays the hypotheses to be tested." (Mary Wolfinbarger Celsi et al, "Essentials of Business Research Methods", 2011)
"A proof in logic and mathematics is, traditionally, a deductive argument from some given assumptions to a conclusion. Proofs are meant to present conclusive evidence in the sense that the truth of the conclusion should follow necessarily from the truth of the assumptions. Proofs must be, in principle, communicable in every detail, so that their correctness can be checked." (Sara Negri & Jan von Plato, "Proof Analysis", 2011)
"Like classical logic, fuzzy logic uses formulas to formally represent statements about the world. Given an appropriate semantic structure (such as an evaluation of propositional symbols in the case of propositional logic, or a relational structure in the case of predicate logic), a truth degree of formula ? is denoted by ||?||. It is significant that the truth degree ||?|| of ? may in general be any element of the set of truth degrees. That is, formulas in fuzzy logic are true to degrees , not just true or false as in the case of classical logic." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
"Nevertheless, the use of fuzzy logic is supported by at least the following three arguments. First, fuzzy logic is rooted in the intuitively appealing idea that the truths of propositions used by humans are a matter of degree. An important consequence is that the basic principles and concepts of fuzzy logic are easily understood. Second, fuzzy logic has led to many successful applications, including many commercial products, in which the crucial part relies on representing and dealing with statements in natural language that involve vague terms. Third, fuzzy logic is a proper generalization of classical logic, follows an agenda similar to that of classical logic, and has already been highly developed. An important consequence is that fuzzy logic extends the rich realm of applications of classical logic to applications in which the bivalent character of classical logic is a limiting factor." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
"The principal idea employed by fuzzy logic is to allow for a partially ordered scale of truth-values, called also truth degrees, which contains the values representing false and true , but also some additional, intermediary truth degrees. That is, the set {0,1} of truth-values of classical logic, where 0 and 1 represent false and true , respectively, is replaced in fuzzy logic by a partially ordered scale of truth degrees with the smallest degree being 0 and the largest one being 1." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand." (Peter Rowlett, "The Unplanned Impact of Mathematics", Nature Vol. 475 (7355), 2011)
"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)
"Mathematical intuition is the mind’s ability to sense form and structure, to detect patterns that we cannot consciously perceive. Intuition lacks the crystal clarity of conscious logic, but it makes up for that by drawing attention to things we would never have consciously considered." (Ian Stewart, "Visions of Infinity", 2013)
"Proof, in fact, is the requirement that makes great problems problematic. Anyone moderately competent can carry out a few calculations, spot an apparent pattern, and distil its essence into a pithy statement. Mathematicians demand more evidence than that: they insist on a complete, logically impeccable proof. Or, if the answer turns out to be negative, a disproof. It isn’t really possible to appreciate the seductive allure of a great problem without appreciating the vital role of proof in the mathematical enterprise. Anyone can make an educated guess. What’s hard is to prove it’s right. Or wrong." (Ian Stewart, "Visions of Infinity", 2013)
"Abstraction is an essential knowledge process, the process (or, to some, the alleged process) by which we form concepts. It consists in recognizing one or several common features or attributes (properties, predicates) in individuals, and on that basis stating a concept subsuming those common features or attributes. Concept is an idea, associated with a word expressing a property or a collection of properties inferred or derived from different samples. Subsumption is the logical technique to get generality from particulars." (Hourya B Sinaceur," Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)
"Mathematical abstraction is the process of considering and manipulating operations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content connected to particular applications. […] abstraction is the process of passing from things to ideas, properties and relations, to properties of relations and relations of properties, to properties of relations between properties, etc. Being a fundamental thinking process, abstraction has two faces: a logical face and evidently a psychological aspect that is the target of cognitive sciences." (Hourya B Sinaceur,"Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)
"A conceptual model is a framework that is initially used in research to outline the possible courses of action or to present an idea or thought. When a conceptual model is developed in a logical manner, it will provide a rigor to the research process." (N Elangovan & R Rajendran, "Conceptual Model: A Framework for Institutionalizing the Vigor in Business Research", 2015)
"One of the most powerful transformational catalysts is knowledge, new information, or logic that defies old mental models and ways of thinking. […] The key to transforming mental models is to interrupt the automatic responses that are driven by the old model and respond differently based on the new model. Each time you are able to do this, you are actually loosening the old circuit and creating new neural connections in your brain, often referred to as self-directed neuroplasticity." (Elizabeth Thornton, "The Objective Leader", 2015)
"Science, at its core, is simply a method of practical logic that tests hypotheses against experience. Scientism, by contrast, is the worldview and value system that insists that the questions the scientific method can answer are the most important questions human beings can ask, and that the picture of the world yielded by science is a better approximation to reality than any other." (John M Greer, "After Progress: Reason and Religion at the End of the Industrial Age", 2015)
"Cybernetics is an interdisciplinary science. It originated ‘at the junction’ of mathematics, logic, semiotics, physiology, biology and sociology. Among its inherent features, we mention analysis and revelation of general principles and approaches in scientific cognition. Control theory, communication theory, operations research and others represent most weighty theories within cybernetics 1.0." (Dmitry A Novikov, "Cybernetics 2.0", 2016)
"Mathematical rigour is the thing that enables mathematicians to agree with one another about what is and isn’t correct, rather than just having arguments about competing theories and never coming to a conclusion. Mathematics is based on the rules of logic, the idea being that if you only use objects that behave strictly according to the rules of logic, then as long as you only strictly apply the rules of logic, no disagreements can ever arise."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematics starts with the process of stripping away the ambiguities and leaving only things that can be unambiguously manipulated according to logic. It continues by then manipulating those things according to logic to see what happens." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"The crucial concept that brings all of this together is one that is perhaps as rich and suggestive as that of a paradigm: the concept of a model. Some models are concrete, others are abstract. Certain models are fairly rigid; others are left somewhat unspecified. Some models are fully integrated into larger theories; others, or so the story goes, have a life of their own. Models of experiment, models of data, models in simulations, archeological modeling, diagrammatic reasoning, abductive inferences; it is difficult to imagine an area of scientific investigation, or established strategies of research, in which models are not present in some form or another. However, models are ultimately understood, there is no doubt that they play key roles in multiple areas of the sciences, engineering, and mathematics, just as models are central to our understanding of the practices of these fields, their history and the plethora of philosophical, conceptual, logical, and cognitive issues they raise." (Otávio Bueno, [in" Springer Handbook of Model-Based Science", Ed. by Lorenzo Magnani & Tommaso Bertolotti, 2017])
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