A Calculus of Ideas

Author: Ulf Grenander
Publisher: World Scientific
ISBN: 9814383198
Format: PDF, Kindle
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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

Probabilities on Algebraic Structures

Author: Ulf Grenander
Publisher: Courier Corporation
ISBN: 0486462870
Format: PDF, ePub, Docs
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This systematic approach covers semi-groups, groups, linear vector spaces, and algebra. It states and studies fundamental probabilistic problems for these spaces, focusing on concrete results. 1963 edition.

Pattern Theory

Author: Ulf Grenander
Publisher: Oxford University Press
ISBN: 0198505701
Format: PDF, Docs
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'Pattern Theory' provides a comprehensice & accessible overview of the modern challenges in signal, data & pattern analysis in speech recognition, computational linguistics, image analysis & computer vision. Aimed at graduate students the text includes numerous exercises & an extensive bibliography.

How Humans Learn to Think Mathematically

Author: David Tall
Publisher: Cambridge University Press
ISBN: 1107035708
Format: PDF, ePub
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How Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively affects subsequent understanding of fractions, negative numbers, algebra, and the introduction of definitions and proof. Tall's explanations for these developments are accessible to a general audience while encouraging specialists to relate their areas of expertise to the full range of mathematical thinking. The book offers a comprehensive framework for understanding mathematical growth, from practical beginnings through theoretical developments, to the continuing evolution of mathematical thinking at the highest level.

Simplicity

Author: Craig Dilworth
Publisher: Rowman & Littlefield
ISBN: 0739177230
Format: PDF, Docs
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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.

Elements of Pattern Theory

Author: Ulf Grenander
Publisher: JHU Press
ISBN: 9780801851889
Format: PDF
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"A dazzling tour de force on patterns. It is a substantial, original contribution by a leader-indeed, originator-in the field, and has the potential for significant impact on the direction of future research." -- Alan F. Karr, National Institute of Statistical Sciences

Where Mathematics Comes from

Author: George Lakoff
Publisher: Basic Books (AZ)
ISBN:
Format: PDF, Docs
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Provides an in-depth analysis of the cognitive science of mathematical ideas that argues that conceptual metaphor plays a definitive role in mathematical ideas, exploring such concepts as arithmetic, algebra, sets, logic, and infinity. 20,000 first printing.

Infinitesimal How a Dangerous Mathematical Theory Shaped the Modern World

Author: Amir Alexander
Publisher: Scientific American / Farrar, Straus and Giroux
ISBN: 1429953926
Format: PDF
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Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement-and it will forever change the way you look at a simple line. On August 10, 1632, five men in flowing black robes convened in a somber Roman palazzo to pass judgment on a deceptively simple proposition: that a continuous line is composed of distinct and infinitely tiny parts. With the stroke of a pen the Jesuit fathers banned the doctrine of infinitesimals, announcing that it could never be taught or even mentioned. The concept was deemed dangerous and subversive, a threat to the belief that the world was an orderly place, governed by a strict and unchanging set of rules. If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos. In Infinitesimal, the award-winning historian Amir Alexander exposes the deep-seated reasons behind the rulings of the Jesuits and shows how the doctrine persisted, becoming the foundation of calculus and much of modern mathematics and technology. Indeed, not everyone agreed with the Jesuits. Philosophers, scientists, and mathematicians across Europe embraced infinitesimals as the key to scientific progress, freedom of thought, and a more tolerant society. As Alexander reveals, it wasn't long before the two camps set off on a war that pitted Europe's forces of hierarchy and order against those of pluralism and change. The story takes us from the bloody battlefields of Europe's religious wars and the English Civil War and into the lives of the greatest mathematicians and philosophers of the day, including Galileo and Isaac Newton, Cardinal Bellarmine and Thomas Hobbes, and Christopher Clavius and John Wallis. In Italy, the defeat of the infinitely small signaled an end to that land's reign as the cultural heart of Europe, and in England, the triumph of infinitesimals helped launch the island nation on a course that would make it the world's first modern state. From the imperial cities of Germany to the green hills of Surrey, from the papal palace in Rome to the halls of the Royal Society of London, Alexander demonstrates how a disagreement over a mathematical concept became a contest over the heavens and the earth. The legitimacy of popes and kings, as well as our beliefs in human liberty and progressive science, were at stake-the soul of the modern world hinged on the infinitesimal.

Ernst Zermelo Collected Works Gesammelte Werke II

Author: Ernst Zermelo
Publisher: Springer Science & Business Media
ISBN: 3540708561
Format: PDF, Mobi
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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.