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  1. Yehoshua Bar-Hillel (ed.) (1970). Mathematical Logic and Foundations of Set Theory. Amsterdam,North-Holland Pub. Co..
    LN , so f lies in the elementary submodel M'. Clearly co 9 M' . It follows that 6 = {f(n): n em} is included in M'. Hence the ordinals of M' form an initial ...
  2. Alessandro Andretta, Keith Kearnes & Domenico Zambella (eds.) (2008). Logic Colloquium 2004: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Torino, Italy, July 25-31, 2004. [REVIEW] Cambridge University Press.
    Highlights of this volume from the 2004 Annual European Meeting of the Association for Symbolic Logic (ASL) include a tutorial survey of the recent highpoints of universal algebra, written by a leading expert; explorations of foundational questions; a quartet of model theory papers giving an excellent reflection of current work in model theory, from the most abstract aspect "abstract elementary classes" to issues around p-adic integration.
  3. Mary Tiles (1991). Mathematics and the Image of Reason. Routledge.
    A thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic.
  4. John Pottage (1983). Geometrical Investigations: Illustrating the Art of Discovery in the Mathematical Field. Addison-Wesley.
  5. Dale Gottlieb (1980). Ontological Economy: Substitutional Quantification and Mathematics. Oxford University Press.
  6. Nancy Rodgers (2000). Learning to Reason: An Introduction to Logic, Sets and Relations. Wiley.
    Learn how to develop your reasoning skills and how to write well-reasoned proofs Learning to Reason shows you how to use the basic elements of mathematical language to develop highly sophisticated, logical reasoning skills. You'll get clear, concise, easy-to-follow instructions on the process of writing proofs, including the necessary reasoning techniques and syntax for constructing well-written arguments. Through in-depth coverage of logic, sets, and relations, Learning to Reason offers a meaningful, integrated view of modern mathematics, cuts through confusing terms and (...)
  7. George R. Exner (1997). An Accompaniment to Higher Mathematics. Springer.
    This text prepares undergraduate mathematics students to meet two challenges in the study of mathematics, namely, to read mathematics independently and to understand and write proofs. The book begins by teaching how to read mathematics actively, constructing examples, extreme cases, and non-examples to aid in understanding an unfamiliar theorem or definition (a technique famililar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. The book then turns to proofs, (...)
  8. Alfred North Whitehead & Bertrand Russell (1962/1997). Principia Mathematica, to *56. Cambridge University Press.
    The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will wish (...)
  9. H. Kirchner & Christophe Ringeissen (eds.) (2000). Frontiers of Combining Systems: Third International Workshop, Frocos 2000, Nancy, France, March 22-24, 2000: Proceedings. [REVIEW] Springer.
    This book constitutes the refereed proceedings of the Third International Workshop on Frontiers of Combining Systems, FroCoS 2000, held in Nancy, France, in March 2000.The 14 revised full papers presented together with four invited papers were carefully reviewed and selected from a total of 31 submissions. Among the topics covered are constraint processing, interval narrowing, rewriting systems, proof planning, sequent calculus, type systems, model checking, theorem proving, declarative programming, logic programming, and equational theories.
  10. Thomas J. Jech (1971). Lectures in Set Theory. New York,Springer-Verlag.
  11. Chin-Liang Chang (1973/1987). Symbolic Logic and Mechanical Theorem Proving. Academic Press.
    This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4–9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
  12. Donald Gillies (ed.) (1992). Revolutions in Mathematics. Oxford University Press.
    Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...)
  13. Shaughan Lavine (1994). Understanding the Infinite. Harvard University Press.
    An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...
  14. Michael D. Resnik (1980). Frege and the Philosophy of Mathematics. Cornell University Press.
  15. Stephen David Ross (1994). Locality and Practical Judgment: Charity and Sacrifice. Fordham University Press.
    This work completes Ross's trilogy examining the inexhaustible complexity of the world and our relation to our surroundings.
  16. Charles Castonguay (1972). Meaning and Existence in Mathematics. New York,Springer-Verlag.
  17. Stewart Shapiro (ed.) (1985). Intentional Mathematics. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..
    Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
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  18. Paolo Mancosu (ed.) (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press.
    From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...)
  19. Hilary Putnam (1979). Mathematics, Matter, and Method. Cambridge University Press.
    Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an (...)
  20. Imre Lakatos (1978). Mathematics, Science, and Epistemology. Cambridge University Press.
    Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.
  21. Jody Azzouni (2010). Talking About Nothing: Numbers, Hallucinations, and Fictions. Oxford University Press.
    Numbers -- Hallucinations -- Fictions -- Scientific languages, ontology, and truth -- Truth conditions and semantics.
  22. George Boolos, John Burgess, Richard P. & C. Jeffrey (2007). Computability and Logic. Cambridge University Press.
    Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel’s incompleteness theorems, but also a large number of optional topics, from Turing’s theory of computability to Ramsey’s theorem. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, a (...)
  23. Jody Azzouni (2004). Deflating Existential Commitment: A Case for Nominalism. OUP USA.
    If we must take mathematical statements to be true, must we also believe in the existence of abstract invisible mathematical objects accessible only by the power of pure thought? Jody Azzouni says no, and he claims that the way to escape such commitments is to accept (as an essential part of scientific doctrine) true statements which are about objects that don't exist in any sense at all. Azzouni illustrates what the metaphysical landscape looks like once we avoid a militant Realism (...)
  24. Philip Kitcher (1983). The Nature of Mathematical Knowledge. Oxford University Press.
    This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...
  25. Michał Heller & W. H. Woodin (eds.) (2011). Infinity: New Research Frontiers. Cambridge University Press.
    Machine generated contents note: Introduction Rudy Rucker; Part I. Perspectives on Infinity from History: 1. Infinity as a transformative concept in science and theology Wolfgang Achtner; Part II. Perspectives on Infinity from Mathematics: 2. The mathematical infinity Enrico Bombieri; 3. Warning signs of a possible collapse of contemporary mathematics Edward Nelson; Part III. Technical Perspectives on Infinity from Advanced Mathematics: 4. The realm of the infinite W. Hugh Woodin; 5. A potential subtlety concerning the distinction between determinism and nondeterminism W. (...)
  26. Charles S. Chihara (1973). Ontology and the Vicious-Circle Principle. Ithaca [N.Y.]Cornell University Press.
  27. J. Sicha (1974). A Metaphysics of Elementary Mathematics. Amherst, University of Massachusetts Press.
  28. Ludwig Wittgenstein (1978). Remarks on the Foundations of Mathematics. B. Blackwell.
  29. Matthias Baaz (ed.) (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press.
    Machine generated contents note: Part I. Historical Context - Gödel's Contributions and Accomplishments: 1. The impact of Gödel's incompleteness theorems on mathematics Angus Macintyre; 2. Logical hygiene, foundations, and abstractions: diversity among aspects and options Georg Kreisel; 3. The reception of Gödel's 1931 incompletabilty theorems by mathematicians, and some logicians, to the early 1960s Ivor Grattan-Guinness; 4. 'Dozent Gödel will not lecture' Karl Sigmund; 5. Gödel's thesis: an appreciation Juliette C. Kennedy; 6. Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on (...)
  30. Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. Routledge.
    Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen has brought together (...)
  31. Eli Maor (1987/1991). To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press.
    Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes (...)
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  32. Ian Stewart & David Tall (1977). The Foundations of Mathematics. Oxford University Press.
    The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books.
  33. Philip J. Davis (1986/2005). Descartes' Dream: The World According to Mathematics. Dover Publications.
    Philosopher Rene Descartes visualized a world unified by mathematics, in which all intellectual issues could be resolved rationally by local computation. This series of provocative essays takes a modern look at the seventeenth-century thinker’s dream, examining the physical and intellectual influences of mathematics on society, particularly in light of technological advances. They survey the conditions that elicit the application of mathematic principles; the effectiveness of these applications; and how applied mathematics constrain lives and transform perceptions of reality. Highly suitable for (...)
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  34. Gila Sher & Richard L. Tieszen (eds.) (2000). Between Logic and Intuition: Essays in Honor of Charles Parsons. Cambridge University Press.
    This collection of new essays offers a 'state-of-the-art' conspectus of major trends in the philosophy of logic and philosophy of mathematics. A distinguished group of philosophers addresses issues at the centre of contemporary debate: semantic and set-theoretic paradoxes, the set/class distinction, foundations of set theory, mathematical intuition and many others. The volume includes Hilary Putnam's 1995 Alfred Tarski lectures, published here for the first time.
  35. Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.
    Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...)
  36. Amir D. Aczel (2000). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. Four Walls Eight Windows.
    From the end of the 19th century until his death, one of history's most brilliant mathematicians languished in an asylum. The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for his (...)
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  37. W. S. Anglin (1996). Mathematics, a Concise History and Philosophy. Springer.
    This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble coping with vast amounts of reading. Furthermore, there are many detailed explanations of the important mathematical procedures (...)
  38. Bryan H. Bunch (1982/1997). Mathematical Fallacies and Paradoxes. Dover Publications.
    Stimulating, thought-provoking analysis of a number of the most interesting intellectual inconsistencies in mathematics, physics and language. Delightful elucidations of methods for misunderstanding the real world of experiment (Aristotle’s Circle paradox), being led astray by algebra (De Morgan’s paradox) and other mind-benders. Some high school algebra and geometry is assumed; any other math needed is developed in text. Reprint of 1982 ed.
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  39. George Polya (1990). Mathematics and Plausible Reasoning. Princeton University Press.
    Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.
  40. Thomas J. McKay (2006). Plural Predication. Oxford University Press.
    Plural predication is a pervasive part of ordinary language. We can say that some people are fifty in number, are surrounding a building, come from many countries, and are classmates. These predicates can be true of some people without being true of any one of them; they are non-distributive predications. However, the apparatus of modern logic does not allow a place for them. Thomas McKay here explores the enrichment of logic with non-distributive plural predication and quantification. His book will be (...)
  41. Alonzo Church (1956). Introduction to Mathematical Logic. Princeton, Princeton University Press.
    This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.
  42. Philippe Smets (ed.) (1988). Non-Standard Logics for Automated Reasoning. Academic Press.
  43. Philip J. Davis (1995/1982). The Mathematical Experience. Birkhäuser.
    Presents general information about meteorology, weather, and climate and includes more than thirty activities to help study these topics, including making a ...
  44. Donald C. Benson (1999). The Moment of Proof: Mathematical Epiphanies. Oxford University Press.
    When Archimedes, while bathing, suddenly hit upon the principle of buoyancy, he ran wildly through the streets of Syracuse, stark naked, crying "eureka!" In The Moment of Proof, Donald Benson attempts to convey to general readers the feeling of eureka--the joy of discovery--that mathematicians feel when they first encounter an elegant proof. This is not an introduction to mathematics so much as an introduction to the pleasures of mathematical thinking. And indeed the delights of this book are many and varied. (...)
  45. C. S. Jenkins (2008). Grounding Concepts: An Empirical Basis for Arithmetical Knowledge. OUP Oxford.
    Grounding Concepts tackles the issue of arithmetical knowledge, developing a new position which respects three intuitions which have appeared impossible to satisfy simultaneously: a priorism, mind-independence realism, and empiricism. -/- Drawing on a wide range of philosophical influences, but avoiding unnecessary technicality, a view is developed whereby arithmetic can be known through the examination of empirically grounded concepts. These are concepts which, owing to their relationship to sensory input, are non-accidentally accurate representations of the mind-independent world. Examination of such concepts (...)
  46. Charles Chihara (2004). A Structural Account of Mathematics. Clarendon Press.
    A Structural Account of Mathematics will be required reading for anyone working in this field.
  47. D. S. Bridges (1987). Varieties of Constructive Mathematics. Cambridge University Press.
    This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.
  48. Clare Boucher (2000). The Six Blind Men and the Elephant: A Traditional Indian Story. Candlewick Press.
  49. Robert S. Tragesser (1984). Husserl and Realism in Logic and Mathematics. Cambridge University Press.
    In this book Robert Tragesser sets out to determine the conditions under which a realist ontology of mathematics and logic might be justified, taking as his starting point Husserl's treatment of these metaphysical problems. He does not aim primarily at an exposition of Husserl's phenomenology, although many of the central claims of phenomenology are clarified here. Rather he exploits its ideas and methods to show how they can contribute to answering Michael Dummet's question 'Realism or Anti-Realism?'. In doing so he (...)
  50. Joseph Mazur (2005). Euclid in the Rainforest: Discovering Universal Truth in Logic and Math. Pi Press.
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