Principles of Mathematical Logic

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Author: David Hilbert,Wilhelm Ackermann,Robert E. Luce

Publisher: American Mathematical Soc.

ISBN: 0821820249

Category: Mathematics

Page: 172

View: 5056

David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. This translation is based on the second German edition and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Godel's completeness proof for the predicate calculus has been updated. In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.

Mathematical Grammar of Biology

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Author: Michel Eduardo Beleza Yamagishi

Publisher: Springer

ISBN: 3319626892

Category: Mathematics

Page: 82

View: 9539

This seminal, multidisciplinary book shows how mathematics can be used to study the first principles of DNA. Most importantly, it enriches the so-called “Chargaff’s grammar of biology” by providing the conceptual theoretical framework necessary to generalize Chargaff’s rules. Starting with a simple example of DNA mathematical modeling where human nucleotide frequencies are associated to the Fibonacci sequence and the Golden Ratio through an optimization problem, its breakthrough is showing that the reverse, complement and reverse-complement operators defined over oligonucleotides induce a natural set partition of DNA words of fixed-size. These equivalence classes, when organized into a matrix form, reveal hidden patterns within the DNA sequence of every living organism. Intended for undergraduate and graduate students both in mathematics and in life sciences, it is also a valuable resource for researchers interested in studying invariant genomic properties.

A Computable Universe

Understanding and Exploring Nature as Computation

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Author: Hector Zenil

Publisher: World Scientific

ISBN: 9814374296

Category: Mathematics

Page: 810

View: 5607

This volume discusses the foundations of computation in relation to nature. It focuses on two main questions: What is computation? and How does nature compute?

Levels of Infinity

Selected Writings on Mathematics and Philosophy

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Author: Hermann Weyl,Peter Pesic

Publisher: Courier Corporation

ISBN: 0486489035

Category: Mathematics

Page: 240

View: 1332

This original anthology collects 10 of Weyl's less-technical writings that address the broader scope and implications of mathematics. Most have been long unavailable or not previously published in book form. Subjects include logic, topology, abstract algebra, relativity theory, and reflections on the work of Weyl's mentor, David Hilbert. 2012 edition.

Grundzüge der theoretischen Logik

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Author: David Hilbert,Wilhelm Ackermann

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: 155

View: 2722

The Principles of Inductive Logic

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Author: John Venn

Publisher: Taylor & Francis US

ISBN: 9780828402651

Category: Mathematics

Page: 604

View: 423

Venn, best known for his diagrams for set theory, primarily studied logic and probability theory. The present book is a study of the principles of logic, with special emphasis on inference and induction. From the Preface to the First Edition (1889): ``As many readers will probably perceive, the main original guiding influence with me--as with most of those of the middle generation, and especially with most of those who approached logic with previous mathematical or scientific training--was that of Mill ... I still continue to regard the general attitude towards phenomena, which Mill took up as a logician, to be the soundest and most useful for scientific study ... '' From the Preface to the Second Edition (1907): ``Though thus leaving the main outlines unaltered I have done what I could to improve the work, and to try to bring it up to date ... A number of paragraphs have been altered, others have been re-written, and many hundreds of minor alterations, additions and corrections inserted ... ''

Symbolic Logic

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Author: John Venn

Publisher: American Mathematical Soc.

ISBN: 9780821841990

Category: Mathematics

Page: 540

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Venn's style is to take his readers very much into his confidence: as he builds the theory, he carefully points out the alternative paths he might have taken, the alternative definitions he might have used, he shows what the implications of these alternatives are, and justifies his choice on the broadest possible grounds. What is distinctive about this work may be given in part in Venn's own words: ``The thorough examination of symbolic logic as a whole, that is, in its relation to ordinary logic and ordinary thought and language; the establishment of every general symbolic expression and rule on purely logical principles, instead of looking mainly to its formal justification; and the invention and employment of a scheme of diagrammatic notation which shall be in true harmony with our generalizations.''

Geometry and the Imagination

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Author: David Hilbert,Stephan Cohn-Vossen

Publisher: American Mathematical Soc.

ISBN: 0821819984

Category: Mathematics

Page: 357

View: 757

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer - even after more than half a century has passed! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. 'Hilbert and Cohn-Vossen' is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces.The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 1/5 - 1/7 - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.One of the most remarkable chapters is 'Projective Configurations'. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Gottingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the 'pantheon' of great mathematics books.

Grundzüge der Mengenlehre

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Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780828400619

Category: Mathematics

Page: 476

View: 6945

This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.

Hausdorff on Ordered Sets

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Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 0821837885

Category: Mathematics

Page: 322

View: 8314

Georg Cantor, the founder of set theory, published his last paper on sets in 1897. In 1900, David Hilbert made Cantor's Continuum Problem and the challenge of well-ordering the real numbers the first problem in his famous Paris lecture. It was time for the appearance of the second generation of Cantorians. They emerged in the decade 1900-1909, and foremost among them were Ernst Zermelo and Felix Hausdorff. Zermelo isolated the Choice Principle, proved that every set could be well-ordered, and axiomatized the concept of set. He became the father of abstract set theory. Hausdorff eschewed foundations and pursued set theory as part of the mathematical arsenal. He was recognized as the era's leading Cantorian. From 1901-1909, Hausdorff published seven articles in which he created a representation theory for ordered sets and investigated sets of real sequences partially ordered by eventual dominance, together with their maximally ordered subsets. These papers are translated and appear in this volume. Each is accompanied by an introductory essay. These highly accessible works are of historical significance, not only for set theory, but also for model theory, analysis and algebra.

The Mathematical Theory of Huygens' Principle

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Author: Bevan B. Baker,Edward Thomas Copson

Publisher: Taylor & Francis US

ISBN: 9780821834787

Category: Mathematics

Page: 193

View: 1279

From the Preface: 'The present monograph deals with the mathematical theory of Huygens' principle in optics and its application to the theory of diffraction. No attempt has been made to give a complete account of the various methods of solving special diffraction problems. [The authors] are concerned only with the general theory of the solution of the partial differential equations governing the propagation of light and [they] discuss some of the simpler diffraction problems merely as illustrative examples'.

The Conceptual Foundations of Quantum Mechanics

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Author: Leonard Eisenbud

Publisher: American Mathematical Soc.

ISBN: 0821841793

Category: Science

Page: 148

View: 7914

This book provides a clear and logical path to understanding what quantum mechanics is about. It will be accessible to undergraduates with minimal mathematical preparation: all that is required is an open mind, a little algebra, and a first course in undergraduate physics. Quantum mechanics is arguably the most successful physical theory. It makes predictions of incredible accuracy. It provides the structure underlying all of our electronic technology, and much of our mastery over materials. But compared with Newtonian mechanics, or even relativity, its teachings seem obscure--they have no counterpart in everyday experience, and they sometimes contradict our simplest notions of how the world works. A full understanding of the theory requires prior mastery of very advanced mathematics. This book aims at a different goal: to teach the reader, step by step, how the theory came to be and what, fundamentally, it is about. Most students learn physics by learning techniques and formulas. This is especially true in a field like quantum mechanics, whose content often contradicts our common sense, and where it's tempting to retreat into mathematical formalism. This book goes behind the formalism to explain in direct language the conceptual content and foundations of quantum mechanics: the experiments that forced physicists to construct such a strange theory, and the essential elements of its strangeness.