Search results for 'mathematics' (try it on Scholar)

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  1. William Bragg Ewald (ed.) (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press.score: 21.0
    This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here (...)
     
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  2. Simon B. Duffy (ed.) (2006). Virtual Mathematics: The Logic of Difference. Clinamen.score: 19.0
    Of all twentieth century philosophers, it is Gilles Deleuze whose work agitates most forcefully for a worldview privileging becoming over being, difference over sameness; the world as a complex, open set of multiplicities. Nevertheless, Deleuze remains singular in enlisting mathematical resources to underpin and inform such a position, refusing the hackneyed opposition between ‘static’ mathematical logic versus ‘dynamic’ physical world. This is an international collection of work commissioned from foremost philosophers, mathematicians and philosophers of science, to address the wide range (...)
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  3. Jeffry L. Hirst (2006). Reverse Mathematics of Separably Closed Sets. Archive for Mathematical Logic 45 (1):1-2.score: 19.0
    This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.
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  4. Sam Sanders & Keita Yokoyama (2012). The Dirac Delta Function in Two Settings of Reverse Mathematics. Archive for Mathematical Logic 51 (1-2):99-121.score: 19.0
    The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property ${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$ of the Dirac delta function. (...)
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  5. Jeffry L. Hirst (1999). Reverse Mathematics of Prime Factorization of Ordinals. Archive for Mathematical Logic 38 (3):195-201.score: 19.0
    One of the earliest applications of Cantor's Normal Form Theorem is Jacobstahl's proof of the existence of prime factorizations of ordinals. Applying the techniques of reverse mathematics, we show that the full strength of the Normal Form Theorem is used in this proof.
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  6. Justin Clarke-Doane (2012). Morality and Mathematics: The Evolutionary Challenge. Ethics 122 (2):313-340.score: 18.0
    It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for (...)
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  7. Frank Plumpton Ramsey (1960). The Foundations of Mathematics and Other Logical Essays. Paterson, N.J.,Littlefield, Adams.score: 18.0
    THE FOUNDATIONS OF MATHEMATICS () PREFACE The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with ...
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  8. Ian Hacking (2011). Why is There Philosophy of Mathematics AT ALL? South African Journal of Philosophy 30 (1):1-15.score: 18.0
    Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination (...)
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  9. Stewart Shapiro (2011). Epistemology of Mathematics: What Are the Questions? What Count as Answers? Philosophical Quarterly 61 (242):130-150.score: 18.0
    A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of (...)
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  10. Antony Eagle (2008). Mathematics and Conceptual Analysis. Synthese 161 (1):67–88.score: 18.0
    Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number (...)
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  11. John P. Burgess & Gideon A. Rosen (1997). A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press.score: 18.0
    Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured (...)
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  12. John Bigelow (1988). The Reality of Numbers: A Physicalist's Philosophy of Mathematics. Oxford University Press.score: 18.0
    Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.
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  13. Paola Cantù, Bolzano Versus Kant: Mathematics as a Scientia Universalis. Philosophical Papers Dedicated to Kevin Mulligan.score: 18.0
    The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome (...)
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  14. Mikhail G. Katz & Thomas Mormann, Infinitesimals and Other Idealizing Completions in Neo-Kantian Philosophy of Mathematics.score: 18.0
    We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a sophisticated (...)
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  15. Mark Colyvan, Indispensability Arguments in the Philosophy of Mathematics. Stanford Encyclopedia of Philosophy.score: 18.0
    One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these (...)
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  16. Paul Benacerraf & Hilary Putnam (eds.) (1983). Philosophy of Mathematics: Selected Readings. Cambridge University Press.score: 18.0
    The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, (...)
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  17. Richard L. Tieszen (2005). Phenomenology, Logic, and the Philosophy of Mathematics. Cambridge University Press.score: 18.0
    Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this book is divided into three parts. Part I, Reason, Science, and Mathematics contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay oN phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some (...)
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  18. Carlo Cellucci (1996). Mathematical Logic: What has It Done for the Philosophy of Mathematics? In Piergiorgio Odifreddi (ed.), Kreiseliana. About and Around Georg Kreisel, pp. 365-388. A K Peters.score: 18.0
    onl y to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics.
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  19. Paola Cantù (2010). Aristotle's Prohibition Rule on Kind-Crossing and the Definition of Mathematics as a Science of Quantities. Synthese 174 (2):225 - 235.score: 18.0
    The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in (...)
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  20. Nicholas Maxwell (2010). Wisdom Mathematics. Friends of Wisdom Newsletter (6):1-6.score: 18.0
    For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could (...)
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  21. Hilary Putnam (1979). Mathematics, Matter, and Method. Cambridge University Press.score: 18.0
    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 (...)
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  22. Michael D. Resnik (1997). Mathematics as a Science of Patterns. New York ;Oxford University Press.score: 18.0
    This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of (...)--the view that mathematics is about things that really exist. (shrink)
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  23. James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge.score: 18.0
    Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
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  24. Penelope Maddy (1997). Naturalism in Mathematics. Oxford University Press.score: 18.0
    Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both (...)
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  25. M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.score: 18.0
    Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual (...)
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  26. Simon Friederich (2010). Structuralism and Meta-Mathematics. Erkenntnis 73 (1):67 - 81.score: 18.0
    The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro (2005), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one (...)
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  27. Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. Routledge.score: 18.0
    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 (...)
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  28. Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.score: 18.0
    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 (...)
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  29. Feng Ye (2010). What Anti-Realism in Philosophy of Mathematics Must Offer. Synthese 175 (1):13 - 31.score: 18.0
    This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. (...)
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  30. Guillermo E. Rosado Haddock (2006). Husserl's Philosophy of Mathematics: Its Origin and Relevance. [REVIEW] Husserl Studies 22 (3):193-222.score: 18.0
    This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.
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  31. Pasquale Frascolla (1994). Wittgenstein's Philosophy of Mathematics. Routledge.score: 18.0
    Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.
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  32. Bob Hale (ed.) (2001). The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press.score: 18.0
    Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as (...)
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  33. Paolo Mancosu (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford University Press.score: 18.0
    The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting (...)
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  34. James Robert Brown (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge.score: 18.0
    1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
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  35. Alison Pease & Andrew Aberdein (2011). Five Theories of Reasoning: Interconnections and Applications to Mathematics. Logic and Logical Philosophy 20 (1-2):7-57.score: 18.0
    The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [39], Toulmin’s argumentation layout [52], Lakatos’s theory of reasoning in mathematics [23], Pollock’s notions of counterexample [44], and (...)
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  36. Nathan U. Salmon (2005). Metaphysics, Mathematics, and Meaning. Oxford University Press.score: 18.0
    Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.
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  37. Stewart Shapiro (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.score: 18.0
    Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable (...)
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  38. Javier Echeverría, Andoni Ibarra & Thomas Mormann (eds.) (1992). The Space of Mathematics: Philosophical, Epistemological, and Historical Explorations. W. De Gruyter.score: 18.0
    The Protean Character of Mathematics SAUNDERS MAC LANE (Chicago) 1. Introduction The thesis of this paper is that mathematics is protean. ...
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  39. Helen De Cruz (2006). Towards a Darwinian Approach to Mathematics. Foundations of Science 11 (1-2):157-196.score: 18.0
    In the past decades, recent paradigm shifts in ethology, psychology, and the social sciences have given rise to various new disciplines like cognitive ethology and evolutionary psychology. These disciplines use concepts and theories of evolutionary biology to understand and explain the design, function and origin of the brain. I shall argue that there are several good reasons why this approach could also apply to human mathematical abilities. I will review evidence from various disciplines (cognitive ethology, cognitive psychology, cognitive archaeology and (...)
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  40. Jean-Pierre Marquis (2013). Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics. Synthese 190 (12):2141-2164.score: 18.0
    In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, (...)
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  41. Matthias Schirn (ed.) (1998). The Philosophy of Mathematics Today. Clarendon Press.score: 18.0
    This comprehensive volume gives a panorama of the best current work in this lively field, through twenty specially written essays by the leading figures in the field. All essays deal with foundational issues, from the nature of mathematical knowledge and mathematical existence to logical consequence, abstraction, and the notions of set and natural number. The contributors also represent and criticize a variety of prominent approaches to the philosophy of mathematics, including platonism, realism, nomalism, constructivism, and formalism.
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  42. Stewart Shapiro (ed.) (2005). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press.score: 18.0
    Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on (...)
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  43. Mark Steiner (1998). The Applicability of Mathematics as a Philosophical Problem. Harvard University Press.score: 18.0
    This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...
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  44. Mirja Hartimo (ed.) (2010). Phenomenology and Mathematics. Springer.score: 18.0
    This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.
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  45. Mark McEvoy (2013). Experimental Mathematics, Computers and the a Priori. Synthese 190 (3):397-412.score: 18.0
    In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There (...)
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  46. G. T. Kneebone (1963/2001). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover.score: 18.0
    Graduate-level historical study is ideal for students intending to specialize in the topic, as well as those who only need a general treatment. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics, emphasizing Hilbert’s metamathematics. Part III focuses on the philosophy of mathematics. Each chapter has extensive supplementary notes; a detailed appendix charts modern developments.
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  47. Charles Parsons (1983). Mathematics in Philosophy: Selected Essays. Cornell University Press.score: 18.0
    This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.
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  48. Imre Lakatos (1978). Mathematics, Science, and Epistemology. Cambridge University Press.score: 18.0
    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.
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  49. M. Redhead (2004). Mathematics and the Mind. British Journal for the Philosophy of Science 55 (4):731-737.score: 18.0
    Granted that truth is valuable we must recognize that certifiable truth is hard to come by, for example in the natural and social sciences. This paper examines the case of mathematics. As a result of the work of Gödel and Tarski we know that truth does not equate with proof. This has been used by Lucas and Penrose to argue that human minds can do things which digital computers can't, viz to know the truth of unprovable arithmetical statements. The (...)
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  50. Stewart Shapiro (2000). Thinking About Mathematics: The Philosophy of Mathematics. Oxford University Press.score: 18.0
    This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that (...) is logic (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy. (shrink)
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