Oxford: Butterworth Heinemann, 2003. — 289 p. — ISBN: 9780340676530, 0340676531. Numbers and Proofs presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational,...
Mathematical Association of America, 2010. — 320 p. — (Dolciani Mathematical Expositions №42). — ISBN: 9780883853481. Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy'....
Cambridge, Massachusetts, London, England: The MIT Press, 2017. — 1223 p. — ISBN: 0262035537. A textbook that teaches students to read and write proofs using Athena. Proof is the primary vehicle for knowledge generation in mathematics. In computer science, proof has found an additional use: verifying that a particular system (or component, or algorithm) has certain desirable...
2nd. ed. — Stanford: CSLI Publications, 2011. — 618 p. — ISBN 9781575866321. The textbook/software package covers first-order language in a method appropriate for first and second courses in logic . An on-line grading services instantly grades solutions to hundred of computer exercises. It is designed to be used by philosophy instructors teaching a logic course to...
CSLI Publications, 1994. — 266 p. Hyperproof is a system for learning the principles of analytical reasoning and proof construction, consisting of a text and a Macintosh software program. Unlike traditional treatments of first-order logic, Hyperproof combines graphical and sentential information, presenting a set of logical rules for integrating these different forms of...
Seven Bridges Press, 1999. — 597 pp. What do the fields of astronomy, economics, finance, law, mathematics, medicine, physics, and sociology have in common? Not much in the way of subject matter, that's for sure. And not all that much in the way of methodology. What they do have in common, with each other and with many other fields, is their dependence on a certain standard of...
Springer, 2010. — 205 p. — ISBN: 978-1-4419-7022-0. The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The...
Springer, 2015. — 386 p. Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations...
Amsterdam: Elsevier, 1998. — vii + 812 p. — (Studies in Logic and the Foundations of Mathematics, Vol. 137). — ISBN: 0-444-89840-9. Proof theory is the study of proofs as formal objects and is concerned with a broad range of related topics. It is one of the central topics of mathematical logic and has applications in many areas of mathematics, philosophy, and computer science....
Netherlands. Amsterdam.: Elsevier, 1998. — (vii + 812) p. — (Studies in Logic and the Foundations of Mathematics. V. 137 ). — ISBN: 0-444-89840-9, eBook, English. Proof theory is the study of proofs as formal objects and is concerned with a broad range of related topics. It is one of the central topics of mathematical logic and has applications in many areas of mathematics,...
Pearson, 2013. — 489 p. — 3rd ed. — ISBN: 0321797094, 9780321797094
Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their...
4th edition. — New York: Pearson, 2018. — 617 p. — ISBN: 978-0-13-474675-3. As the teaching of calculus in many colleges and universities has become more problemoriented with added emphasis on the use of calculators and computers, the theoretical gap between the material presented in calculus and the mathematical background expected (or at least hoped for) in advanced calculus...
London: Springer-Verlag, 2002. — xvi, 380 p. — (Distinguished Dissertations). — ISBN: 978-1-4471-1113-9, 978-1-4471-0147-5. In recent years, Artificial Intelligence researchers have largely focused their efforts on solving specific problems, with less emphasis on 'the big picture' - automating large scale tasks which require human-level intelligence to undertake. The subject of...
Springer, 2021. — 139 p. — ISBN 978-3-030-68374-0. A compact and easily accessible book, it guides the reader in unravelling the apparent mysteries found in doing mathematical proofs. Simply written, it introduces the art and science of proving mathematical theorems and propositions and equips students with the skill required to tackle the task of proving mathematical...
Springer, 2021. — 139 p. — ISBN 978-3-030-68374-0. A compact and easily accessible book, it guides the reader in unravelling the apparent mysteries found in doing mathematical proofs. Simply written, it introduces the art and science of proving mathematical theorems and propositions and equips students with the skill required to tackle the task of proving mathematical...
Independently published, 2021. — 330 p. This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a...
4th. ed. — Academic Press, 2012. — 296 p. — ISBN: 0123822173. The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs provides basic logic of mathematical proofs and shows how mathematical proofs work . It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive...
CRC Press, 2025. - 169 p. - ISBN 1032686197. A Beginner’s Guide to Mathematical Proof prepares mathematics majors for the transition to abstract mathematics, as well as introducing a wider readership of quantitative science students, such as engineers, to the mathematical structures underlying more applied topics. The text is designed to be easily utilized by both instructor...
College Publications, 2004. — 387 p. This book in categorial proof theory formulates in terms of category theory a generalization close to linear algebra of the notions of distributive lattice and Boolean algebra. These notions of distributive lattice category and Boolean category codify a plausible nontrivial notion of identity of proofs in classical propositional logic, which...
Providence: American Mathematical Society, 1988. — 228 p. In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. Nonclassical logics are used in the theory of computations, in information theory, and for the description of systems of heuristic programming. Intuitionistic logic is a particularly important nonclassical...
Boca Raton: CRC Press, 2024. — 296 p. — (Textbooks in Mathematics). — ISBN 1032595981. This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high-school concepts, such as...
New York: Springer, 2022. — 498 p. Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well...
Cambridge University Press, 2014. — xxviii, 436 p. — ISBN 978-1-107-03650-5. Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential...
Ed. 3.2, corr. — Richard Hammack, 2020. — 380 p. — ISBN: 0989472132. Note: New and corrected edition 3.2 is published on 24.7.2020 by author (in this edition, all known typos, observed in official 3rd. edition (2018), are removed !) This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as...
Ed. 3.4, corrected - Independently published, 2025. - 380 p. - ISBN 0989472132. This book is an introduction to the language and standard proof methods of mathematics. It is a "bridge" from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for...
Virginia Commonwealth University, Department of Mathematics & Applied Mathematics, 2013. — 311 pages.
This is a book about how to prove theorems. Until this point in your education, mathematics has probably been presented as a primarily computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and...
3rd ed. — Richard Hammack, 2018. — 380 p. In this third edition, Chapter 3 (on counting) has been expanded, and a new chapter on calculus proofs has been added. New examples and exercises have been added throughout. My decisions regarding revisions have been guided by both the Amazon reviews and emails from readers, and I am grateful for all comments. I have taken pains to ensure...
Unpublished, 2013. — 313 p. This text is an expansion and refinement of lecture notes developed while teaching proofs courses over the past fourteen years at Virginia Commonwealth University (a large state university) and Randolph-Macon College (a small liberal arts college). I found the needs of these two audiences to be nearly identical, and I wrote this book for them. But I am...
Kluwer, 2000. — 244 p.
Proof theory was developed as part of Hilbert's Programme. According to Hilbert's Programme one could provide mathematics with a firm and secure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be...
Basel: Birkhäuser, 2016. — 430 p. The aim of this volume is to collect original contributions by the best specialists from the area of proof theory, constructivity, and computation and discuss recent trends and results in these areas. Some emphasis will be put on ordinal analysis, reductive proof theory, explicit mathematics and type-theoretic formalisms, and abstract...
Springer International Publishing, Switzerland, 2016. — 364 p. — ISBN: 9783319309651. This is a textbook on proof writing in the area of analysis, balancing a survey of the core concepts of mathematical proof with a tight, rigorous examination of the specific tools needed for an understanding of analysis. Instead of the standard "transition" approach to teaching proofs, wherein...
Springer, 2019. — 123 p. — (Trends in Logic 51). — ISBN: 978-3-030-28920-1. Adamowicz Z., Bigorajska T., Zdanowski K. (Eds.) This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the...
Philadelphia: American Mathematical Society, 2016. — 221 p. The idea for this textbook was conceived as a direct result of my experience teaching the “introduction to proofs” course at Allegheny College. When I first started teaching this course, there were only a handful of appropriate textbooks on the market. My experience teaching from various textbooks clarified in my mind...
CRC Press, 2020. — xvi, 396 p. — (Textbooks in Mathematics). — ISBN: 978-0367338237. An Introduction to Mathematical Proofs presents fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The text uses a methodical, detailed, and highly structured approach to proof techniques and related topics. No...
New York: Springer, 1994. — 202 p. In the last fiffteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose...
Oxford University Press, 2021. — 431 p. — ISBN 978–0–19–289593–6. Введение в теорию доказательств An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard...
Lulu Press, 2019. — 247 p. Metamath is a computer language and an associated computer program for archiving, verifying, and studying mathematical proofs at a very detailed level. The Metamath language incorporates no mathematics per se but treats all mathematical statements as mere sequences of symbols.You provide Metamath with certain special sequences (axioms) that tellit...
Arcler Press, 2023. — 424 p. “The Notion Of Mathematical Proof: Key Rules And Considerations” is an edited book consisting of 16 contemporaneous open-access articles that aim to cover the different aspects of learning and teaching mathematical proof. The first part of this book aims at summing up factors that influence the cognitive development required to successfully...
Unpublished, 2016. — 195 p. Unlike in earlier courses, success in advanced undergraduate mathematics classes (and beyond) does not depend nearly so much on being able to nd the right answer to a question as it does on being able to provide a convincing explanation that the answer is correct. (Mathematicians call this explanation a proof.) This textbook is designed to help students...
Cambridge: Cambridge University Press, 2011. — 280 p. This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included...
Saarbruecken: Lambert Academic Publishing, 2016. — ISBN: 978-3-330-01661-3. The book contains a description of the proof theory of Chrysippus' logic, in terms of an appropriate typed lambda-calculus. [With the author's corrections as of December 27, 2017.]
New York: Chapman and Hall/CRC, 2015. — 406 p. — (Textbooks in Mathematics). — ISBN: 978-1-4822-4688-9. Introduction to Mathematical Proofs helps students develop the necessary skills to write clear, correct, and concise proofs. Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text...
World Scientific Publishing Company, 2023. — 376 p. — eBook ISBN: 978-981-127-210-3. This textbook is aimed at transitioning high-school students who have already developed proficiency in mathematical problem solving from numerical-answer problems to proof-based mathematics. It serves to guide students on how to write and understand mathematical proofs. It covers proof...
Ottawa: University of Ottawa, 2017. - 93 p. These are notes for the course Mathematical Reasoning & Proofs at the University of Ottawa. This is an introduction to rigorous mathematical reasoning and the concept of proof in mathematics. It is a course designed to prepare students for upper-level proof-based mathematics courses. We will discuss the axiomatic method and various...
Cambridge University Press, 2012. — 482 p. — (Perspectives in Logic). — ISBN: 978-0-521-51769-0. This book is about the deep connections between proof theory and recursive function theory. Their interplay has continuously underpinned and motivated the more constructively orientated developments in mathematical logic ever since the pioneering days of Hilbert, Gӧdel, Church,...
MAA Press, AMS, 2022. — 350 p. — (AMS/MAA Textbooks 68). — ISBN 9781470465148ю Доказательства и идеи: прелюдия к высшей математике Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics...
Oxford: Oxford University Press, 1992. — 139 p. — ISBN: 0-19-504672-2. This introduction to Gödel's incompleteness theorems is written for the general mathematician, philosopher, computer scientist and any other curious reader who has at least a nodding acquaintance with the symbolism of first-order logic (the logical connectives and quantifiers) and who can recognize the...
John Wiley & Sons, 1990. — 246 p. This straightforward guide describes the main methods used to prove mathematical theorems. Shows how and when to use each technique such as the contrapositive, induction and proof by contradiction. Each method is illustrated by step-by-step examples. The Second Edition features new chapters on nested quantifiers and proof by cases, and the number...
Version 2.1. — Grand Valley State University, 2017. — 607 p. Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. Another important goal of this text is to provide students with...
USA.: Department of Mathematics. Grand Valley State University (GVSU), Pearson Education, Inc., 2020. — 612 p. — (Series: Open Education Materials). – ASIN B0863V6HJG. Mathematical Reasoning: Writing and Proof is a text for the ?rst college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of...
USA.: Department of Mathematics. Grand Valley State University (GVSU), 2014.- 591 p. - ISBN: 9781492103851, (Series: OPEN EDUCATION MATERIALS ). eBook. English. (This work may be copied, distributed, and/or modified under the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License ). Description . Mathematical Reasoning: Writing and Proof is...
3rd Edition — Cambridge University Press, 2019. — 569 p. — ISBN: 978-1-108-42418-9. Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by...
Third Edition — Cambridge University Press, 2019. — 569 p. — ISBN: 978-1-108-42418-9, 978-1-108-43953-4. Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to...
Перевод с немецкого. — Монография. — М.: Наука, 1979. — 557 с.
Двухтомная монография Д. Гильберта и П. Бернайса "Основания математики" занимает уникальное место в мировой математической литературе. Ее первое немецкое издание, вышедшее в тридцатых годах, подвело итог процессу становления математической логики как самостоятельной математической дисциплины со своей проблематикой и...
Монография. — Перевод с немецкого Н.М. Нагорного. — Под ред. С.И. Адяна. — М.: Наука, Главная редакция физико-математической литературы, 1979. — 557 с. — (Математическая логика и основания математики). Двухтомная монография Д. Гильберта и П. Бернайса "Основания математики" занимает уникальное место в мировой математической литературе. Ее первое немецкое издание, вышедшее в...
М.: Едиториал УРСС, 2003. - 544 с.
А. Г. Драгалин (1941-1998) — выдающийся отечественный логик и математик, оказавший глубокое воздействие на стиль и направление мировых исследований по логике и философии математики.
В настоящее издание включены труды А. Г. Драгалина по интуиционистской теории доказательств, нестандартному анализу, философии математики и автоматическому...
М.: Наука, 1979. — 256 с. — (Математическая логика и основания математики).
Цель этой небольшой книги — изложить важнейшие из методов теории доказательств в интуиционистской логике. Эта теория сейчас не менее богата методами и результатами, чем, например, пользующаяся заслуженной известностью классическая теория моделей. Автор стремился познакомить читателя с основными...
М.: Наука, 1967. — 351 с. — (Математическая логика и основания математики).
Эта книга представляет собой сборник переводов статей по теории логического вывода. Возросший за последнее время интерес к этой области математической логики вызван бурным развитием «машинной логики», в частности, появлением многочисленных работ, посвященных машинному доказательству теорем.
В...
М.: Наука, 1967. — 351 с. — (Математическая логика и основания математики). Эта книга представляет собой сборник переводов статей по теории логического вывода. Возросший за последнее время интерес к этой области математической логики вызван бурным развитием «машинной логики», в частности, появлением многочисленных работ, посвященных машинному доказательству теорем. В сборнике...
М.: Мир, 1981. — 288 с. — (Новое в зарубежной науке. Математика. Выпуск 23).
Сборник работ крупного американского специалиста по математической логике и основаниям математики. В нем дается обзор основных результатов математической теории доказательств и ее методов. Уделяется место происхождению методов теории доказательств и обоснованию интереса к рассматриваемой проблематике....
М.: Мир, 1978. — 412 с.
Перевод с английского Соболева С. К., под ред. Адяна С. И.
Книга посвящена одному из основных разделов математической логики — теории доказательств. Кроме традиционных результатов по системам первого порядка, таких, как устранимость сечении и полнота интуиционистского и классического исчислений
предикатов, неполнота и непротиворечивость арифметики в...
М.: Мир, 1978. — 208 с. Книга посвящена одному из основных разделов математической логики — теории доказательств. Кроме традиционных результатов по системам первого порядка, таких, как устранимость сечении и полнота интуиционистского и классического исчислений предикатов, неполнота и непротиворечивость арифметики в книге приводятся недавние достижения в этой области, включая...
Уважаемые: администратор, модераторы и доверенные пользователи.Друзья, я искренне благодарен Вам за создание подраздела Теория доказательств. Сегодня, когда литература по этой теме увеличивается, особенно сильно на Западе, очень важно иметь отдельный раздел для этого, что Вы и сделали. Слава Богу. Простому пользователю, а также студентам и преподавателям, если они не являются специалистами в области Математической логики, очень трудно разобраться в тонкостях этих вопросов, да ещё и огромной массы литературы основного раздела. Поэтому, как пишет Апостол Иаков: "если угодно будет Господу и живы будем, то... " (Иак. 4:15), - то и в будущем будем поддерживать порядок здесь на благо всем людям. Да благословит Вас Господь, друзья дорогие, и да дарует мир Божий в сердца ваши. С уважением, благодарностью и благословением,
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Простому пользователю, а также студентам и преподавателям, если они не являются специалистами в области Математической логики, очень трудно разобраться в тонкостях этих вопросов, да ещё и огромной массы литературы основного раздела.
Поэтому, как пишет Апостол Иаков: "если угодно будет Господу и живы будем, то... " (Иак. 4:15), - то и в будущем будем поддерживать порядок здесь на благо всем людям. Да благословит Вас Господь, друзья дорогие, и да дарует мир Божий в сердца ваши.
С уважением, благодарностью и благословением,