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Mathematics and logic as information compression by multiple alignment, unification and search
"... This article introduces the conjecture that mathematics, logic and related disciplines may usefully be understood as information compression by ‘multiple alignment’, ‘unification ’ and ‘search’. As a preparation for the two main sections of the article, concepts of HartleyShannon information theory ..."
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This article introduces the conjecture that mathematics, logic and related disciplines may usefully be understood as information compression by ‘multiple alignment’, ‘unification ’ and ‘search’. As a preparation for the two main sections of the article, concepts of HartleyShannon information theory and Algorithmic Information Theory are briefly reviewed together with a summary of the essentials of the simpler ‘standard ’ methods for information compression. Related areas of research are briefly described: philosophical connections, information compression (IC) in brains and nervous systems, and IC in relation to inductive inference, Minimum Length Encoding and probabilistic reasoning. Then the concepts of information compression by ‘multiple alignment’, ‘unification ’ and ‘search ’ (ICMAUS) are outlined together with a brief description of the SP61 computer model that is a partial realisation of the ICMAUS framework. The first of the two main sections describes how many of the commonlyused forms and structures in mathematics, logic and related disciplines (such as theoretical linguistics and computer programming) may be seen as devices for IC. In some cases, these forms and structures may be interpreted in terms of the ICMAUS framework. The second main section describes a selection of examples where processes of calculation and inference in mathematics, logic and related disciplines may be understood as IC. In many cases, these examples may be understood more specifically in terms of the ICMAUS concepts and illustrated with output from the SP61 model. 1 Associated issues are briefly discussed. Abbreviations used in the article: IC—information compression; ICMAUS—information compression by multiple alignment, unification and search; MA—multiple alignment; ML—mathematics and logic;
C. Roy Keys Inc.
"... this paper shows (Section 2) that Fibonacci series underlies fractal fluctuations on all spacetime scales ..."
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this paper shows (Section 2) that Fibonacci series underlies fractal fluctuations on all spacetime scales
ARISTOTELIAN REALISM
"... Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as rat ..."
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Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,
Using Recent History in the Teaching of Mathematics How to Use History to Clarify Common Confusions in Geometry1
"... Most people judge the size of cities simply from their circumference. So that when one says that Megalopolis is fifty stades in contour and Sparta fortyeight, but that Sparta is twice as large as Megalopolis, what is said seems unbelievable to them. And when in order to puzzle them even more, one t ..."
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Most people judge the size of cities simply from their circumference. So that when one says that Megalopolis is fifty stades in contour and Sparta fortyeight, but that Sparta is twice as large as Megalopolis, what is said seems unbelievable to them. And when in order to puzzle them even more, one tells them that a city or camp with the circumference of forty stades may be twice as large as one of the circumference of which is one hundred stades, what is said seems to them absolutely astounding. The reason of this is that we have forgotten the lessons in geometry we learnt as children.
ACKNOWLEDGEMENTS
"... There are several people I want to recognize for their involvement and support of ..."
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There are several people I want to recognize for their involvement and support of
UNIVERSITY OF MIAMI Sound Source Separation using 3D Correlogram, Fuzzy Logic, and Neural Networks
, 2002
"... With the advance of neuroscience, new mathematical tools were developed to help engineers build computational intelligent systems that simulate the way the human brain processes external information. The problem of separating one sound signal from a sound mixture is based on how we extract the cues ..."
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With the advance of neuroscience, new mathematical tools were developed to help engineers build computational intelligent systems that simulate the way the human brain processes external information. The problem of separating one sound signal from a sound mixture is based on how we extract the cues necessary to distinguish that particular signal. When two related nonlinear sound signals are mixed together (composition) it is impossible to separate (decomposition) them with traditional linear mathematical methods. The human brain can easily distinguish two different persons talking at the same time. It uses its previous knowledge of at least one of the sound signals. In this project an Artificial FuzzyNeural System will be used as the main tool to detect and classify a sound library database. A database will be used for sound search and analysis in order to separate two or more sound signals from a mixture. To preprocess the sound mixture a 3D Correlogram will be used as the auditory scene analysis tool. The final approach of this research is to design a system that learns every new sound signal that will be used with the next sound separation.
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"... Abstract: In this work, without assuming the fifth Euclidean postulate, the author established the following theorems: 1.There exists a spherical quadrilateral whose interior angle sum is equal to 360 degrees. 2.The sum of the interior angles of a triangle is a straight angle. ..."
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Abstract: In this work, without assuming the fifth Euclidean postulate, the author established the following theorems: 1.There exists a spherical quadrilateral whose interior angle sum is equal to 360 degrees. 2.The sum of the interior angles of a triangle is a straight angle.
unknown title
"... Abstract: In this work, without assuming the fifth Euclidean postulate, the author established the following theorems: 1.There exists a spherical quadrilateral whose interior angle sum is equal to 360 degrees. 2.The sum of the interior angles of a triangle is a straight angle. ..."
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Abstract: In this work, without assuming the fifth Euclidean postulate, the author established the following theorems: 1.There exists a spherical quadrilateral whose interior angle sum is equal to 360 degrees. 2.The sum of the interior angles of a triangle is a straight angle.