"Optimization by individual agents, often used to
derive competitive equilibria, are unnecessary for an actual economy to
approximately attain such equilibria. From the failure of humans to optimize in
complex tasks, one need not conclude that the equilibria derived from the
competitive model are descriptively irrelevant. We show that even in complex
economic systems, such equilibria can be attained under a range of surprisingly
weak assumptions about agent behavior." (Antoni Bosch-Domènech & Shyam
Sunder, "Tracking the Invisible Hand", 2000)
"Conventional wisdom, fooled by our misleading
'physical intuition', is that the real world is continuous, and that discrete
models are necessary evils for approximating the 'real' world, due to the
innate discreteness of the digital computer." (Doron Zeilberger,
"'Real' Analysis is a Degenerate Case of Discrete Analysis", 2001)
"Most physical systems, particularly those complex ones, are extremely difficult to model by an accurate and precise mathematical formula or equation due to the complexity of the system structure, nonlinearity, uncertainty, randomness, etc. Therefore, approximate modeling is often necessary and practical in real-world applications. Intuitively, approximate modeling is always possible. However, the key questions are what kind of approximation is good, where the sense of 'goodness' has to be first defined, of course, and how to formulate such a good approximation in modeling a system such that it is mathematically rigorous and can produce satisfactory results in both theory and applications." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)
"But linearity is often an approximation to a more complicated reality. Most systems behave linearly only when they are close to equilibrium, and only when we don't push them too hard."
"Mathematical modeling is as much ‘art’ as ‘science’: it requires the practitioner to (i) identify a so-called ‘real world’ problem (whatever the context may be); (ii) formulate it in mathematical terms (the ‘word problem’ so beloved of undergraduates); (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable); and (iv) interpret the solution in the context of the original problem." (John A Adam, "Mathematics in Nature", 2003)
"Fuzzy models can provide good numerical approximation of functions as well as linguistic information over the behavior of the functions. […] Fuzzy models with embedded linguistic interpretability are useful to extract knowledge from data. This knowledge is represented as a set of IF–THEN rules where the antecedents and the consequences are semantically meaningful." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)
"Logic is the study of methods and principles of
reasoning, where reasoning means obtaining new propositions from existing
propositions. In classical logic, propositions are required to be either true
or false; that is, the truth value of a proposition is either 0 or 1. Fuzzy
logic generalizes classical two-value logic by allowing the truth values of a
proposition to be any numbers in [0, 1]. This generalization allows us to
perform fuzzy reasoning, also called approximate reasoning; that is, deducing
imprecise conclusions (fuzzy propositions) from a collection of imprecise
premises (fuzzy propositions). In this section, we first introduce some basic concepts
and principles in classical logic and then study their generalizations to fuzzy
logic." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy
Control", 2006)
"All models are approximations. Essentially, all
models are wrong, but some are useful. However, the approximate nature of the
model must always be borne in mind." (George E P Box & Norman R
Draper, "Response Surfaces, Mixtures, and Ridge Analyses", 2007)
"Science is the art of the appropriate approximation. While the flat earth model is usually spoken of with derision it is still widely used. Flat maps, either in atlases or road maps, use the flat earth model as an approximation to the more complicated shape." (Byron K. Jennings, "On the Nature of Science", Physics in Canada Vol. 63 (1), 2007)
"Approximate symmetry is a softening of the hard
dichotomy between symmetry and asymmetry. The extent of deviation from exact
symmetry that can still be considered approximate symmetry will depend on the
context and the application and could very well be a matter of personal
taste." (Joe Rosen, "Symmetry Rules: How Science and Nature Are
Founded on Symmetry", 2008)
"From the historical point of view, since
inequalities are associated with order, they arose as soon as people started
using numbers, making measurements, and later, finding approximations and
bounds. Thus inequalities have a long and distinguished role in the evolution
of mathematics." (Claudi Alsina & Roger B Nelsen, "When Less is
More: Visualizing Basic Inequalities", 2009)
"Fuzzy logic is an application area of fuzzy set theory dealing with uncertainty in reasoning. It utilizes concepts, principles, and methods developed within fuzzy set theory for formulating various forms of sound approximate reasoning. Fuzzy logic allows for set membership values to range (inclusively) between 0 and 1, and in its linguistic form, imprecise concepts like 'slightly', 'quite' and 'very'. Specifically, it allows partial membership in a set." (Larbi Esmahi et al, Adaptive Neuro-Fuzzy Systems, 2009)
"Stated loosely, models are simplified, idealized
and approximate representations of the structure, mechanism and behavior of
real-world systems. From the standpoint of set-theoretic model theory, a
mathematical model of a target system is specified by a nonempty set - called
the model’s domain, endowed with some operations and relations, delineated by
suitable axioms and intended empirical interpretation." (Zoltan Domotor,
"Mathematical Models in Philosophy of Science" [Mathematics of
Complexity and Dynamical Systems, 2012])
"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)
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