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The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field.

The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly suited to independent study, and provide an unparalleled opportunity to learn from one of the leading authorities in the field.

Collecting, curating, and illuminating over 75 years of progress since Tarski's seminal work in 1941, this textbook in two volumes offers a landmark, unified treatment of the increasingly relevant field of relation algebras. Clear and insightful prose guides the reader through material previously only available in scattered, highly-technical journal articles. Students and experts alike will appreciate the work as both a textbook and invaluable reference for the community.

The second volume of a pair that charts relation algebras from novice to expert level, this text brings the well-grounded reader to the frontiers of research. Building on the foundations established in the preceding Introduction to Relation Algebras, this volume advances the reader into the deeper mathematical results of the past few decades. Such material offers an ideal preparation for research in relation algebras and Boolean algebras with operators.Arranged in a modular fashion, this text offers the opportunity to explore any of several areas in detail; topics include canonical extensions, completions, representations, varieties, and atom structures. Each chapter offers a complete account of one such avenue of development, including a historical section and substantial number of exercises. The clarity of exposition and comprehensive nature of each module make this an ideal text for the independent reader entering the field, while researchers will value it as a reference for years to come.

Collecting, curating, and illuminating over 75 years of progress since Tarski's seminal work in 1941, this textbook in two volumes offers a landmark, unified treatment of the increasingly relevant field of relation algebras. Clear and insightful prose guides the reader through material previously only available in scattered, highly-technical journal articles. Students and experts alike will appreciate the work as both a textbook and invaluable reference for the community. Note that this volume contains numerous, essential references to the previous volume, Introduction to Relation Algebras. The reader is strongly encouraged to secure at least electronic access to the first book in order to make use of the second.

What is algebra? For some, it is an abstract language of x's and y’s. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. Taming the Unknown considers how these two seemingly different types of algebra evolved and how they relate. Victor Katz and Karen Parshall explore the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century.

Defining algebra originally as a collection of techniques for determining unknowns, the authors trace the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. They show how similar problems were tackled in Alexandrian Greece, in China, and in India, then look at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era.

Taming the Unknown follows algebra’s remarkable growth through different epochs around the globe.

IN 1959 I lectured on Boolean algebras at the University of Chicago. A mimeographed version of the notes on which the lectures were based circulated for about two years; this volume contains those notes, corrected and revised. Most of the corrections were suggested by Peter Crawley. To judge by his detailed and precise suggestions, he must have read every word, checked every reference, and weighed every argument, and I am lIery grateful to hirn for his help. This is not to say that he is to be held responsible for the imperfec tions that remain, and, in particular, I alone am responsible for all expressions of personal opinion and irreverent view point. P. R. H. Ann Arbor, Michigan ] anuary, 1963 Contents Section Page 1 1 Boolean rings ............................ . 2 Boolean algebras ......................... . 3 9 3 Fields of sets ............................ . 4 Regular open sets . . . . . . . . . . . . . . . . . . . 12 . . . . . . 5 Elementary relations. . . . . . . . . . . . . . . . . . 17 . . . . . 6 Order. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . 7 Infinite operations. . .. . . . . . . . . . . . . . . . . 25 . . . . . 8 Subalgebras . . . . . . . . . . . . . . . . . . . . .. . . . 31 . . . . . . 9 Homomorphisms . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . 10 Free algebras . . . . . . . . . . . . . . . . . . . . . . 40 . . . . . . . 11 Ideals and filters. . . . . . . . . . . . . . . . . . . . 47 . . . . . . 12 The homomorphism theorem. . . . . . . . . . . . .. . . 52 . . 13 Boolean a-algebras . . . . . . . . . . . . . . . . . . 55 . . . . . . 14 The countable chain condition . . . . . . . . . . . . 61 . . . 15 Measure algebras . . . . . . . . . . . . . . . . . . . 64 . . . . . . . 16 Atoms.. . . . .. . . . . .. .. . . . ... . . . . .. . . ... . . .. 69 17 Boolean spaces . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . 18 The representation theorem. . . . . . . . . . . . . . 77 . . . 19 Duali ty for ideals . . . . . . . . . . . . . . . . . .. . . 81 . . . . . 20 Duality for homomorphisms . . . . . . . . . . . . . . 84 . . . . 21 Completion . . . . . . . . . . . . . . . . . . . . . . . 90 . . . . . . . . 22 Boolean a-spaces . . . . . . . . . . . . . . . . . .. . . 97 . . . . . 23 The representation of a-algebras . . . . . . . . .. . . 100 . 24 Boolean measure spaces . . . . . . . . . . . . . .. . . 104 . . . 25 Incomplete algebras . . . . . . . . . . . . . . . .. . . 109 . . . . . 26 Products of algebras . . . . . . . . . . . . . . . .. . . 115 . . . . 27 Sums of algebras . . . . . . . . . . . . . . . . . .. . . 119 . . . . . 28 Isomorphisms of factors . . . . . . . . . . . . . .. . . 122 . . .
without a properly developed inconsistent calculus based on infinitesimals, then in consistent claims from the history of the calculus might well simply be symptoms of confusion. This is addressed in Chapter 5. It is further argued that mathematics has a certain primacy over logic, in that paraconsistent or relevant logics have to be based on inconsistent mathematics. If the latter turns out to be reasonably rich then paraconsistentism is vindicated; while if inconsistent mathematics has seri ous restriytions then the case for being interested in inconsistency-tolerant logics is weakened. (On such restrictions, see this chapter, section 3. ) It must be conceded that fault-tolerant computer programming (e. g. Chapter 8) finds a substantial and important use for paraconsistent logics, albeit with an epistemological motivation (see this chapter, section 3). But even here it should be noted that if inconsistent mathematics turned out to be functionally impoverished then so would inconsistent databases. 2. Summary In Chapter 2, Meyer's results on relevant arithmetic are set out, and his view that they have a bearing on G8del's incompleteness theorems is discussed. Model theory for nonclassical logics is also set out so as to be able to show that the inconsistency of inconsistent theories can be controlled or limited, but in this book model theory is kept in the background as much as possible. This is then used to study the functional properties of various equational number theories.
For more than 30 years, the author has studied the model-theoretic aspects of the theory of valued fields and multi-valued fields. Many of the key results included in this book were obtained by the author whilst preparing the manuscript. Thus the unique overview of the theory, as developed in the book, has been previously unavailable.
The book deals with the theory of valued fields and mutli-valued fields. The theory of Prüfer rings is discussed from the `geometric' point of view. The author shows that by introducing the Zariski topology on families of valuation rings, it is possible to distinguish two important subfamilies of Prüfer rings that correspond to Boolean and near Boolean families of valuation rings. Also, algebraic and model-theoretic properties of multi-valued fields with near Boolean families of valuation rings satisfying the local-global principle are studied. It is important that this principle is elementary, i.e., it can be expressed in the language of predicate calculus. The most important results obtained in the book include a criterion for the elementarity of an embedding of a multi-valued field and a criterion for the elementary equivalence for multi-valued fields from the class defined by the additional natural elementary conditions (absolute unramification, maximality and almost continuity of local elementary properties). The book concludes with a brief chapter discussing the bibliographic references available on the material presented, and a short history of the major developments within the field.
This textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems. Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Gödel’s Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics. Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.
An entertaining look at the origins of mathematical symbols

While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In Enlightening Symbols, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted.

Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics.

From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM). It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets). In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set). It is a theory of `invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports.

After explaining the motivation for using invariant sets in the experimental sciences as well as the connections with the nominal approach, admissible sets and Gandy machines (Chapter 1), the authors present in Chapter 2 the basics of invariant sets and show that the principles of constructing FSM have historical roots both in the definition of Tarski `logical notions' and in the Erlangen Program of Klein for the classification of various geometries according to invariants under suitable groups of transformations. Furthermore, the consistency of various choice principles is analyzed in FSM. Chapter 3 examines whether it is possible to obtain valid results by replacing the notion of infinite sets with the notion of invariant sets in the classical ZF results. The authors present techniques for reformulating ZF properties of algebraic structures in FSM. In Chapter 4 they generalize FM set theory by providing a new set of axioms inspired by the theory of amorphous sets, and so defining the extended Fraenkel-Mostowski (EFM) set theory. In Chapter 5 they define FSM semantics for certain process calculi (e.g., fusion calculus), and emphasize the links to the nominal techniques used in computer science. They demonstrate a complete equivalence between the new FSM semantics (defined by using binding operators instead of side conditions for presenting the transition rules) and the known semantics of these process calculi.

The book is useful for researchers and graduate students in computer science and mathematics, particularly those engaged with logic and set theory.

Heyting'88 Summer School and Conference on Mathematical Logic, held September 13 - 23, 1988 in Chaika, Bulgaria, was honourably dedicated to Arend Heyting's 90th anniversary. It was organized by Sofia University "Kliment Ohridski" on the occasion of its centenary and by the Bulgarian Academy of Sciences, with sponsorship of the Association for Symbolic Logic. The Meeting gathered some 115 participants from 19 countries. The present volume consists of invited and selected papers. Included are all the invited lectures submitted for publication and the 14 selected contributions, chosen out of 56 submissions by the Selection Committee. The selection was made on the basis of reports of PC members, an average of 4 per sLlbmission. All the papers are concentrated on the topics of the Meeting: Recursion Theory, Modal and Non-classical Logics, Intuitionism and Constructivism, Related Applications to Computer and Other Sciences, Life and Work of Arend Heyting. I am pleased to thank all persons and institutions that contributed to the success of the Meeting: sponsors, Programme Committee members and additional referees, the members of the Organizing Committee, our secretaries K. Lozanova and L. Nikolova, as well as K. Angelov, V. Bozhichkova, A. Ditchev, D. Dobrev, N. Dimitrov, R. Draganova, G. Gargov, N. Georgieva, M. Janchev, P. Marinov, S. Nikolova, S. Radev, I. Soskov, A. Soskova and v. Sotirov, who helped in the organization, Plenum Press and at last but not least all participants in the Meeting and contributors to this volume.
A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R. Dedekind, a. Holder, D. Gilbert, B. Neumann, A. I. Mal'cev, P. Hall, G. Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam ple, partially ordered groups with interpolation property were intro duced in F. Riesz's fundamental paper [1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G. Birkhoff, H. Nakano and P.
Imagine algebra class meets The Hitchhiker’s Guide to the Galaxy...

Meet JJ, an unusual character with a unique vantage position from which he can measure and monitor humanity’s progress. Armed with a device that compels all around it to tell the truth, JJ offers a satirical evaluation of our attitudes to numeracy and logic, touching upon several aspects of life on Earth along the way, from the criminal justice system and people’s use of language to highway driving and modern art.

A collection of mathematically-flavored stories and jokes, interlaced with puzzles, paradoxes and problems, fuse together in an entertaining, free-flowing narrative that will engage and amuse anyone with an interest in the issues confronting society today. JJ demonstrates how a lack of elementary mathematical knowledge can taint our work and general thinking and reflects upon the importance of what is arguably our most valuable weapon against ignorance: a sound mathematical education.

What is JJ’s prognosis for our future? There’s only one way to find out...

"Numbers, logic, human behavior and aliens: this unique book blends them all into a captivating narrative of serious talk and satire, where wit and scholarly details are counterpointed by instructive puzzles and mathematical fun. A ‘must read’ for anybody who appreciates humor and culture."

Stanislav Potapenko, Department of Civil and Environmental Engineering, University of Waterloo, Canada

"... a real delight. Constanda has managed to intertwine stories, puzzles, logic and some very rich mathematics concepts into a very readable, enjoyable novel... I believe this book should be in the personal library of every high school mathematics teacher."

Tom Becvar, St Louis University High School, USA

"...a highly readable, unique and fascinating combination of humor, mathematics and social commentary that is factual, educational and, more importantly, understandable. Given what is taking place in today’s society, J.J. Moon brings mathematics back to earth! I’ll never go through the car buying process again without thinking about SCAM 16!"

Jerry Hoopert, VP / Chief Administrative Officer, Tulsa Teachers Credit Union, USA

“Proofs and Fundamentals: A First Course in Abstract Mathematics” 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo– Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition.
The evaluation of a logical formula can be viewed as a game played by two opponents, one trying to show that the formula is true and the other trying to prove it is false. This correspondence has been known for a very long time and has inspired numerous research directions. In this book, the author extends this connection between logic and games to the class of automatic structures, where relations are recognized by synchronous finite automata.

In model-checking games for automatic structures, two coalitions play against each other with a particular kind of hierarchical imperfect information. The investigation of such games leads to the introduction of a game quantifier on automatic structures, which connects alternating automata with the classical model-theoretic notion of a game quantifier. This study is then extended, determining the memory needed for strategies in infinitary games on the one hand, and characterizing regularity-preserving Lindström quantifiers on the other. Counting quantifiers are investigated in depth: it is shown that all countable omega-automatic structures are in fact finite-word automatic and that the infinity and uncountability set quantifiers are definable in MSO over countable linear orders and over labeled binary trees.

This book is based on the PhD thesis of Lukasz Kaiser, which was awarded with the E.W. Beth award for outstanding dissertations in the fields of logic, language, and information in 2009. The work constitutes an innovative study in the area of algorithmic model theory, demonstrating the deep interplay between logic and computability in automatic structures. It displays very high technical and presentational quality and originality, advances significantly the field of algorithmic model theory and raises interesting new questions, thus emerging as a fruitful and inspiring source for future research.

This book is an example of fruitful interaction between (non-classical) propo sitionallogics and (classical) model theory which was made possible due to categorical logic. Its main aim consists in investigating the existence of model completions for equational theories arising from propositional logics (such as the theory of Heyting algebras and various kinds of theories related to proposi tional modal logic ). The existence of model-completions turns out to be related to proof-theoretic facts concerning interpretability of second order propositional logic into ordinary propositional logic through the so-called 'Pitts' quantifiers' or 'bisimulation quantifiers'. On the other hand, the book develops a large number of topics concerning the categorical structure of finitely presented al gebras, with related applications to propositional logics, both standard (like Beth's theorems) and new (like effectiveness of internal equivalence relations, projectivity and definability of dual connectives such as difference). A special emphasis is put on sheaf representation, showing that much of the nice categor ical structure of finitely presented algebras is in fact only a restriction of natural structure in sheaves. Applications to the theory of classifying toposes are also covered, yielding new examples. The book has to be considered mainly as a research book, reporting recent and often completely new results in the field; we believe it can also be fruitfully used as a complementary book for graduate courses in categorical and algebraic logic, universal algebra, model theory, and non-classical logics. 1.
Focusing on an approach of solving rigorous problems and learning how to prove, this volume is concentrated on two specific content themes, elementary number theory and algebraic polynomials. The benefit to readers who are moving from calculus to more abstract mathematics is to acquire the ability to understand proofs through use of the book and the multitude of proofs and problems that will be covered throughout. This book is meant to be a transitional precursor to more complex topics in analysis, advanced number theory, and abstract algebra. To achieve the goal of conceptual understanding, a large number of problems and examples will be interspersed through every chapter. The problems are always presented in a multi-step and often very challenging, requiring the reader to think about proofs, counter-examples, and conjectures. Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high-achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge. In the past, PNA has been taught in a "problem solving in middle school” course (twice), to a quite advanced high school students course (three semesters), and three times as a secondary resource for a course for future high school teachers. PNA is suitable for secondary math teachers who look for material to encourage and motivate more high achieving students.
The chief purpose of the book is to present, in detail, a compilation of proofs of the Cantor-Bernstein Theorem (CBT) published through the years since the 1870's. Over thirty such proofs are surveyed.

The book comprises five parts. In the first part the discussion covers the role of CBT and related notions in the writings of Cantor and Dedekind. New views are presented, especially regarding the general proof of CBT obtained by Cantor, his proof of the Comparability Theorem, the ruptures in the Cantor-Dedekind correspondence and the origin of Dedekind's proof of CBT.

The second part covers the first CBT proofs published (1896-1901). The works of the following mathematicians is considered in detail: Schröder, Bernstein, Bore, Schoenflies and Zermelo. Here a subtheme of the book is launched; it concerns the research project following Bernstein's Division Theorem (BDT).

In its third part the book covers proofs that emerged during the period when the logicist movement was developed (1902-1912). It covers the works of Russell and Whitehead, Jourdain, Harward, Poincaré, J. König, D. König (his results in graph theory), Peano, Zermelo, Korselt. Also Hausdorff's paradox is discussed linking it to BDT.

In the fourth part of the book are discussed the developments of CBT and BDT (including the inequality-BDT) in the hands of the mathematicians of the Polish School of Logic, including Sierpiński, Banach, Tarski, Lindenbaum, Kuratowski, Sikorski, Knaster, the British Whittaker, and Reichbach.

Finally, in the fifth part, the main discussion concentrates on the attempts to port CBT to intuitionist mathematics (with results by Brouwer, Myhill, van Dalen and Troelstra) and to Category Theory (by Trnková and Koubek).

The second purpose of the book is to develop a methodology for the comparison of proofs. The core idea of this methodology is that a proof can be described by two descriptors, called gestalt and metaphor. It is by comparison of their descriptors that the comparison of proofs is obtained. The process by which proof descriptors are extracted from a proof is named 'proof-processing', and it is conjectured that mathematicians perform proof-processing habitually, in the study of proofs.
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