A Calculus of Ideas

Author: Ulf Grenander
Publisher: World Scientific
ISBN: 981438318X
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
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

How Humans Learn to Think Mathematically

Author: David Tall
Publisher: Cambridge University Press
ISBN: 1107035708
Format: PDF
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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.

Where Mathematics Comes from

Author: George Lakoff
Publisher: Basic Books (AZ)
ISBN:
Format: PDF, ePub, 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.

How Mathematicians Think

Author: William Byers
Publisher: Princeton University Press
ISBN: 9780691145990
Format: PDF
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To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

What is Mathematics

Author: Richard Courant
Publisher: Oxford University Press, USA
ISBN: 9780195105193
Format: PDF, Kindle
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A discussion of fundamental mathematical principles from algebra to elementary calculus designed to promote constructive mathematical reasoning.

Simplicity

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

Infinitesimal

Author: Amir Alexander
Publisher: Oneworld Publications
ISBN: 1780745338
Format: PDF, ePub, Docs
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On August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world. Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal. Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.

Infinity and the Mind

Author: Rudy Rucker
Publisher: Princeton University Press
ISBN: 1400849047
Format: PDF, Mobi
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In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

Mathematics and the Physical World

Author: Morris Kline
Publisher: Courier Corporation
ISBN: 0486136310
Format: PDF, ePub, Docs
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Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non-Euclidean geometries. Also describes how math is used in optics, astronomy, and other phenomena.