Logic And Mathematic Proofs Pdf

This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.

Author: Robert S. Wolf

Publisher: W. H. Freeman

ISBN: 0716730502

Category: Mathematics

Page: 4

View: 194

This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.

Author: Robert S Wolf

Publisher: W.H. Freeman

ISBN: 0716732475

Category:

Page:

View: 390

This influential book discusses the nature of mathematical discovery, development, methodology and practice, forming Imre Lakatos's theory of 'proofs and refutations'.

Author: Imre Lakatos

Publisher: Cambridge University Press

ISBN: 9781107113466

Category: Mathematics

Page: 196

View: 865

This influential book discusses the nature of mathematical discovery, development, methodology and practice, forming Imre Lakatos's theory of 'proofs and refutations'.

Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory.This book contains 'solutions' to some of the most noteworthy mathematical proofs (QED) and is designed to be a reference and ...

Author: Edited by Paul F. Kisak

Publisher: Createspace Independent Publishing Platform

ISBN: 1519464339

Category:

Page: 256

View: 536

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture. Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. This book contains 'solutions' to some of the most noteworthy mathematical proofs (QED).

The Science of Conjecture provides a history of rational methods of dealing with uncertainty and explores the coming to consciousness of the human understanding of risk.

Author: James Franklin

Publisher: JHU Press

ISBN: 9781421418803

Category: History

Page: 520

View: 725

How did we make reliable predictions before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? In The Science of Conjecture, James Franklin examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. The Science of Conjecture provides a history of rational methods of dealing with uncertainty and explores the coming to consciousness of the human understanding of risk. "A remarkable book. Mr. Franklin writes clearly and exhibits a wry wit. But he also ranges knowledgeably across many disciplines and over many centuries."— Wall Street Journal " The Science of Conjecture opens an old chest of human attempts to draw order from havoc and wipes clean the rust from some cast-off classical tools that can now be reused to help build a framework for the unpredictable future."— Science "Franklin's style is clear and fluent, with an occasional sly Gibbonian aside to make the reader chuckle."— New Criterion James Franklin is a professor in the School of Mathematics and Statistics at the University of New South Wales.

We also suspect that, especially for those who enjoy solving puzzles and unraveling the mysteries of sciences, the nature of the methodologies will provide substantial stimulation.This volume introduce some readers to the excitement of ...

Author: Larry Wos

Publisher: Rinton PressInc

ISBN: 1589490231

Category: Mathematics

Page: 372

View: 217

Most appealing - and sometimes even stirring - is a well-constructed case showing that, without doubt, some given assertion holds. Typically, such a case is based on logical and flawless reasoning, on a sequence of steps that follow inevitably from the hypotheses used to deduce each. In other words, a proof is given establishing that the assertion under consideration indeed holds. Such proofs are clearly crucial to logic and to mathematics. Not so obvious, but true, proofs are crucial to circuit design, program writing, and, more generally, to various activities in which reasoning plays a vital role. Indeed, most desirable is the case in which no doubt exists regarding the absence of flaws in the design of a chip, in the structure of a computer program, in the argument on which an important decision is based. Such careful reasoning is even the key factor in games that include chess and poker. This book features one example after another of flawless logical reasoning the context is that of finding proofs absent from the literature. The means for finding the missing proofs is reliance on a single computer program, William McCune's automated reasoning program OTTER. One motivating force for writing this book is to interest others in automated reasoning, logic and mathematics. As the text strongly indicates, we delight in using OTTER equally in two quite distinct activities: finding a proof where none is offered by the literature, and finding a proof far more appealing than any the literature provides. We believe that the challenge offered by the type of problem featured in this book can be as engrossing as solving puzzles and playing various games that appeal to the mind. Indeed,sometimes, inexpressible is the excitement engendered when seeking a proof with fewer steps than was found by one of the great minds of the twentieth century. A second motivating force resets with our obvious enjoyment of the type of research featured in this book. Like the fancier of fine wines, we continually seek new open questions to attack, whether (at one end of the spectrum) they concern the settling of a conjecture or (at the other end) the focus is on proof betterment. We encourage readers to send us additional open questions and challenging problems. Another factor that motivated us was our wish to collect in a single volume a surprisingly large number of proofs, most of which were previously absent from the literature. In some cases, no proof was offered of any type; in some cases, the proof that was offered was far from axiomatic. None of the proofs rely on induction, or on metal argument, or on higher-order logic. In one sense, the book can serve as an encyclopedia of proofs -- many new and many improved - a work that sometimes extends, sometimes replaces, and sometimes supplements the research of more than a century. These proofs offer the implicit challenge of finding others that are further improvements. In a rather different sense, the book may serve as the key to eventually answering one open question after another, whether the context is logic, mathematics, design, synthesis, or some other area relying on sound reasoning. In that regards, we include in details numerous diverse methodologies are themselves intriguing. For an example, one methodology asks for two independent paths that lead to success and, rather than emphasizing what is common to both (theirintersection), instead heavily focuses on what is not shared (their symmetric difference). Although the emphasis here is on their use in the context of logic and mathematics, we conjecture that the methodologies we offer will prove most useful in a far wider context. We also suspect that, especially for those who enjoy solving puzzles and unraveling the mysteries of sciences, the nature of the methodologies will provide substantial stimulation. This volume introduce some readers to the excitement of discovering new results, increase the intrigue of those already familiar with such excitement, and (for the expert) add to the arsenal of weapons for attacking deep questions and hard problems.

This book explores when and why the rudiments of mathematical capability first appeared among human beings, what its fundamental concepts are, and how and why it has grown into the richly branching complex of specialties that it is today.

Author: Raymond Nickerson

Publisher: Taylor & Francis

ISBN: 9781136945397

Category: Psychology

Page: 595

View: 631

The development of mathematical competence -- both by humans as a species over millennia and by individuals over their lifetimes -- is a fascinating aspect of human cognition. This book explores a vast range of psychological questions related to mathematical cognition, and provides fascinating insights for researchers and students of cognition and instructors of mathematics.

Naive conjectures and concepts must pass through the crucible of proofs and refutations. The results are improved conjectures (theorems) and improved (proof-generated or theoretical) concepts. The logic of mathematical discovery "is ...

Author: Solomon Feferman

Publisher: Oxford University Press on Demand

ISBN: 9780195080308

Category: Philosophy

Page: 340

View: 636

Solomon Feferman is one of the leading figures in logic and the foundations of mathematics. This volume brings together a selection of his most important essays dealing with the light which results in modern logic cast on significant problems in the foundations of mathematics. It is essential reading for anyone interested in these subjects. Feferman presents key issues in the work of Cantor, Hilbert, Weyl, and Godel among others, and explains how they are dealtwith by proof theory and other parts of logic. A number of the papers appeared originally in obscure places and are not well-known, and others are published here for the first time. All of the material has been revised and annotated to bring it up to date.

This book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras.

Author: Alain Valette

Publisher: Birkhäuser

ISBN: 9783034881876

Category: Mathematics

Page: 104

View: 102

The Baum-Connes conjecture is part of A. Connes' non-commutative geometry programme. It can be viewed as a conjectural generalisation of the Atiyah-Singer index theorem, to the equivariant setting (the ambient manifold is not compact, but some compactness is restored by means of a proper, co-compact action of a group "gamma"). Like the Atiyah-Singer theorem, the Baum-Connes conjecture states that a purely topological object coincides with a purely analytical one. For a given group "gamma", the topological object is the equivariant K-homology of the classifying space for proper actions of "gamma", while the analytical object is the K-theory of the C*-algebra associated with "gamma" in its regular representation. The Baum-Connes conjecture implies several other classical conjectures, ranging from differential topology to pure algebra. It has also strong connections with geometric group theory, as the proof of the conjecture for a given group "gamma" usually depends heavily on geometric properties of "gamma". This book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras. It presents, for the first time in book form, an introduction to the Baum-Connes conjecture. It starts by defining carefully the objects in both sides of the conjecture, then the assembly map which connects them. Thereafter it illustrates the main tool to attack the conjecture (Kasparov's theory), and it concludes with a rough sketch of V. Lafforgue's proof of the conjecture for co-compact lattices in in Spn1, SL(3R), and SL(3C).

Popper, when (in fact in 1934) dividing the aspects of discovery between psychology and logic in such a way that no ... and Descartes were both mistaken; (b) the logic of scientific discovery is the logic of conjectures and refutations.

Author: Imre Lakatos

Publisher: Cambridge University Press

ISBN: 9781107268104

Category: Science

Page:

View: 852

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

For higher-order logic the conjecture was verified by Prawitz (1969) and Takahashi (1967). In these papers Schütte's semantical equivalent to cut-elimination is proved non-constructively. 1.4 The Relation Between Proofs and Meaning ...

Author: Heinrich Wansing

Publisher: Springer

ISBN: 9783319110417

Category: Philosophy

Page: 458

View: 471

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three "dogmas of proof-theoretic semantics" are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of p roofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.

This volume is a translation of the book GAdel, written in Japanese by Gaisi Takeuti, a distinguished proof theorist.

Author: Gaisi Takeuti

Publisher: World Scientific

ISBN: 9789812795359

Category: Electronic books

Page: 135

View: 145

This volume is a translation of the book GAdel, written in Japanese by Gaisi Takeuti, a distinguished proof theorist. The core of the book comprises a memoir of K GAdel, Takeuti's personal recollections, and his interpretation of GAdel's attitudes towards mathematical logic. It also contains Takeuti's recollection of association with some other famous logicians. Everything in the book is original, as the author adheres to his own experiences and interpretations. There is also an article on Hilbert's second problem as well as on the author's fundamental conjecture about second order logic. Contents: On GAdel; Work of Paul Bernays and Kurt GAdel; Hilbert and GAdel; Short Biographies of Logicians; Set Theory and Related Topics; From Hilbert to GAdel; Axioms of Arithmetic and Consistency OCo The Second Problem of Hilbert; A Report from GAdel '96; Having Read OC GAdel RememberedOCO A Tribute to the Memory of Professor GAdel' Appendices: On GAdel's Continuum Hypothesis; Birth of Second Order Proof Theory by the Fundamental Conjecture on GLC. Readership: Those interested in mathematics, especially logic or the history of mathematics."

In a normal formula all proof polynomials occurring negatively are atomic, which is always the case for combinatory terms in combinatory logic. This prompted Goris to formulate a natural conjecture about proof polynomials that every ...

Author: Sergei Artemov

Publisher: Springer

ISBN: 9783540727347

Category: Computers

Page: 516

View: 822

This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2007, held in New York, NY, USA in June 2007. The volume presents 36 revised refereed papers that address all current aspects of logic in computer science.

For automatic theorem provers it is as important to disprove false conjectures as it is to prove true ones, ... This paper describes an abstraction mechanism for first-order logic over an arbitrary but fixed term algebra to second-order ...

Author: Moshe Vardi

Publisher: Springer Science & Business Media

ISBN: 9783540201014

Category: Computers

Page: 436

View: 520

This book constitutes the refereed proceedings of the 10th International Conference on Logic Programming, Artificial Intelligence, and Reasoning, LPAR 2003, held in Almaty, Kazakhstan in September 2003. The 27 revised full papers presented together with 3 invited papers were carefully reviewed and selected from 65 submissions. The papers address all current issues in logic programming, automated reasoning, and AI logics in particular description logics, proof theory, logic calculi, formal verification, model theory, game theory, automata, proof search, constraint systems, model checking, and proof construction.

This volume is a collection of papers based on the Symposium on Advances in Mathematical Logic 2018. The symposium was held September 18–20, 2018, at Kobe University, Japan, and was dedicated to the memory of Professor Gaisi Takeuti.

Author: Toshiyasu Arai

Publisher: Springer

ISBN: 9811641722

Category: Mathematics

Page: 240

View: 709

​Gaisi Takeuti was one of the most brilliant, genius, and influential logicians of the 20th century. He was a long-time professor and professor emeritus of mathematics at the University of Illinois at Urbana-Champaign, USA, before he passed away on May 10, 2017, at the age of 91. Takeuti was one of the founders of Proof Theory, a branch of mathematical logic that originated from Hilbert's program about the consistency of mathematics. Based on Gentzen's pioneering works of proof theory in the 1930s, he proposed a conjecture in 1953 concerning the essential nature of formal proofs of higher-order logic now known as Takeuti's fundamental conjecture and of which he gave a partial positive solution. His arguments on the conjecture and proof theory in general have had great influence on the later developments of mathematical logic, philosophy of mathematics, and applications of mathematical logic to theoretical computer science. Takeuti's work ranged over the whole spectrum of mathematical logic, including set theory, computability theory, Boolean valued analysis, fuzzy logic, bounded arithmetic, and theoretical computer science. He wrote many monographs and textbooks both in English and in Japanese, and his monumental monograph Proof Theory, published in 1975, has long been a standard reference of proof theory. He had a wide range of interests covering virtually all areas of mathematics and extending to physics. His publications include many Japanese books for students and general readers about mathematical logic, mathematics in general, and connections between mathematics and physics, as well as many essays for Japanese science magazines. This volume is a collection of papers based on the Symposium on Advances in Mathematical Logic 2018. The symposium was held September 18–20, 2018, at Kobe University, Japan, and was dedicated to the memory of Professor Gaisi Takeuti.

In: LICS '07: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, Washington, DC, USA, 2007, ... 114(10), 882–894 (2007) Hales, T.C.: Flyspeck: A blueprint for the formal proof of the Kepler conjecture (2008).

Author: Jeffrey C. Lagarias

Publisher: Springer Science & Business Media

ISBN: 9781461411291

Category: Mathematics

Page: 456

View: 513

The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the "cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.

LOGIC The scope of the identity conjecture is to some extent left open by Prawitz . ... Minimal implicational logic is , however , the natural technical starting point for structural proof theory , since the ideas needed for typical ...

Author: Filip Wideback

Publisher:

ISBN: 9122019405

Category: Philosophy

Page: 85

View: 460

Hodel, R. E., An Introduction to Mathematical Logic, PWS-Kent, Boston, MA, 1995. ... Schwartz, D., Conjecture and Proof: An Introduction to Mathematical Thinking, Saunders College Publishing, Philadelphia, PA, 1997.

Author: Antonella Cupillari

Publisher: Academic Press

ISBN: 9780080537900

Category: Mathematics

Page: 192

View: 289

The Nuts and Bolts of Proof instructs students on the basic logic of mathematical proofs, showing how and why proofs of mathematical statements work. It provides them with techniques they can use to gain an inside view of the subject, reach other results, remember results more easily, or rederive them if the results are forgotten.A flow chart graphically demonstrates the basic steps in the construction of any proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems. * The "List of Symbols" has been extended. * Set Theory section has been strengthened with more examples and exercises. * Addition of "A Collection of Proofs"

(ii) MOD(q)[W]sh Reg(A) = MOD(q)[Reg) Proof. The proofs of (i) => (ii) in both parts are identical to the argument in the preceding theorem. ... I. The foregoing theorems lead us to our central conjecture: IX.3.4 Conjecture.

Author: Howard Straubing

Publisher: Springer Science & Business Media

ISBN: 9781461202899

Category: Computers

Page: 227

View: 796

The study of the connections between mathematical automata and for mal logic is as old as theoretical computer science itself. In the founding paper of the subject, published in 1936, Turing showed how to describe the behavior of a universal computing machine with a formula of first order predicate logic, and thereby concluded that there is no algorithm for deciding the validity of sentences in this logic. Research on the log ical aspects of the theory of finite-state automata, which is the subject of this book, began in the early 1960's with the work of J. Richard Biichi on monadic second-order logic. Biichi's investigations were extended in several directions. One of these, explored by McNaughton and Papert in their 1971 monograph Counter-free Automata, was the characterization of automata that admit first-order behavioral descriptions, in terms of the semigroup theoretic approach to automata that had recently been developed in the work of Krohn and Rhodes and of Schiitzenberger. In the more than twenty years that have passed since the appearance of McNaughton and Papert's book, the underlying semigroup theory has grown enor mously, permitting a considerable extension of their results. During the same period, however, fundamental investigations in the theory of finite automata by and large fell out of fashion in the theoretical com puter science community, which moved to other concerns.

Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory.This book is designed to be a general overview of the topic and provide you with the structured knowledge to familiarize yourself ...

Author: Edited by: Kisak

Publisher: CreateSpace

ISBN: 1515055248

Category:

Page: 228

View: 652

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments. A proof must demonstrate that a statement is always true rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture. Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory.

Logic And Mathematic Proofs Pdf

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