A Calculus of Ideas

Author: Ulf Grenander
Publisher: World Scientific
ISBN: 981438318X
Format: PDF, Mobi
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This monograph reports a thought experiment with a mathematical structure intended to illustrate the workings of a mind. It presents a mathematical theory of human thought based on pattern theory with a graph-based approach to thinking. The method illustrated and produced by extensive computer simulations is related to neural networks. Based mainly on introspection, it is speculative rather than empirical such that it differs radically in attitude from the conventional wisdom of current cognitive science.

A Calculus of Ideas

Author: Ulf Grenander
Publisher: World Scientific
ISBN: 9814439118
Format: PDF, Mobi
Download Now
This monograph reports a thought experiment with a mathematical structure intended to illustrate the workings of a mind. It presents a mathematical theory of human thought based on pattern theory with a graph-based approach to thinking. The method illustrated and produced by extensive computer simulations is related to neural networks. Based mainly on introspection, it is speculative rather than empirical such that it differs radically in attitude from the conventional wisdom of current cognitive science. Contents:An Architecture for the Mind:IntroductionCreating Thoughts: Algebra of ThinkingPersonality of a Mind:A Mind EquationTwo Personalities:Building a GOLEM, a Thinking MachineAs Thinking DevelopsRealizing a Mind:MIND and BrainReflectionsDoubts and CertaintiesAppendics:Some Famous Mind TheoriesConsistency of Probability MeasureA Modality LatticeDynamics of Random Process of ThinkingSoftware for GOLEM Readership: Graduate students in applied mathematics, statistics and cognitive science as well as postgraduates and researchers in these fields. Keywords:Thinking;Pattern Theory;Language;SoftwareKey Features:Unconventional speculative approach using thought experimentsQuite different from other studies of human thoughtWritten by the inventor of pattern theoryDue to the controversial nature of the theory presented, it will appeal to the adventure-minded researchers in applied mathematics, statistics and cognitive function

The Tangled Origins of the Leibnizian Calculus

Author: Richard C Brown
Publisher: World Scientific
ISBN: 9814401617
Format: PDF, Mobi
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This book is a detailed study of Gottfried Wilhelm Leibniz's creation of calculus from 1673 to the 1680s. We examine and analyze the mathematics in several of his early manuscripts as well as various articles published in the Acta Eruditorum. It studies some of the other lesser known “calculi” Leibniz created such as the Analysis Situs, delves into aspects of his logic, and gives an overview of his efforts to construct a Universal Characteristic, a goal that has its distant origin in the Ars Magna of the 13th century Catalan philosopher Raymond Llull, whose work enjoyed a renewed popularity in the century and a half prior to Leibniz. This book also touches upon a new look at the priority controversy with Newton and a Kuhnian interpretation of the nature of mathematical change. This book may be the only integrated treatment based on recent research and should be a thought-provoking contribution to the history of mathematics for scholars and students, interested in either Leibniz's mathematical achievement or general issues in the field. Contents:Evolution or Revolution in MathematicsIssues in Seventeenth Century MathematicsIsaac Barrow: A Foil to LeibnizA Young Central European PolymathFirst Steps in MathematicsThe Creation of CalculusLogicThe Universal CharacteristicThe Baroque Cultural ContextEpilogueSome Concluding Remarks on Mathematical ChangeAppendices:A: A Transmutation Theorem of LeibnizB: Leibniz's Series Quadrature of a ConicC: Syllogistic LogicD: The Vis Viva DisputeE: Some Applications of Curves and Neusis in Greek GeometryF: InfinitesimalsA Note on the Author Readership: Advanced undergraduate students, graduate students and researchers in mathematics, history of mathematics or history of science. Keywords:Leibniz;Calculus;Geometry;17th Century MathematicsKey Features:The thoroughness and comprehensiveness of the treatment of this book are based on recent researchTechnical details of the mathematics are carefully dealt with instead of just being summarized for the general readerNo other work on the development of calculus includes a description and analysis of the Baroque/Renaissance atmosphere of fascination with symbols, emblems, Real Characters and philosophical languages which motivated both Leibniz's mathematics and his search for the Universal Characteristic

Simplicity

Author: Craig Dilworth
Publisher: Rowman & Littlefield
ISBN: 0739177230
Format: PDF, ePub
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Simplicity provides a new logic with which to approach intellectual situations. Using the simplicity way of thinking as a tool helps clarify intellectual standpoints and conceptually problematic situations in philosophy, mathematics and physics.

Ernst Zermelo Collected Works Gesammelte Werke II

Author: Ernst Zermelo
Publisher: Springer Science & Business Media
ISBN: 3540708561
Format: PDF
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Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and is best-known for the first formulation of the axiom of choice. However, his papers also include pioneering work in applied mathematics and mathematical physics. This edition of his collected papers consists of two volumes. The present Volume II covers Ernst Zermelo’s work on the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field who comments on the historical background, motivation, accomplishments, and influence.

Exact Methods in the Study of Language and Text

Author: Peter Grzybek
Publisher: Walter de Gruyter
ISBN: 3110894211
Format: PDF, ePub, Mobi
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The collection contains more than 60 original papers and reflects current research topics in linguistics and text analysis. Most of the papers present recent results of empirical quantitative investigations; others focus on methodological issues, whereas some of them are of a more theoretical, systems-theoretical/semiotic character. Finally, a number of contributions form typical integrative deductive-inductive studies. The volume is a valuable source of information about the current state-of-the-art in quantitative linguistic research, presented by renowned representatives of the field.

A Century of Mathematics

Author: John Ewing
Publisher: Cambridge University Press
ISBN: 9780883854570
Format: PDF, Mobi
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This is the story of American mathematics during the past century. It contains articles and excerpts from a century of the American Mathematical Monthly, giving the reader an opportunity to skim all one hundred volumes of this popular mathematics magazine without actually opening them. It samples mathematics year by year and decade by decade. The reader can glimpse the mathematical community at the turn of the century, the controversy about Einstein and relativity, the debates about formalism in logic, the immigration of mathematicians from Europe, and the frantic effort to organize as the war began. More recent articles deal with the advent of computers and the changes they brought, and with some of the triumphs of modern research.

Prime Obsession

Author: John Derbyshire
Publisher: Joseph Henry Press
ISBN: 0309141257
Format: PDF, ePub, Mobi
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In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark â€" a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic â€" defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark â€" the Riemann Hypothesis â€" that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows â€" subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many â€" the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof â€" and those who have been consumed by it.