The Nature of Sets

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
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  1. Mathematical Logic, the Theory of Algorithms, and the Theory of Sets.S. I. Adi͡an (ed.) - 1977 - American Mathematical Society.
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  2. Review of Some Iterative Root-Finding Methods From a Dynamical Point of View. [REVIEW]Sergio Amat, Sonia Busquier & Sergio Plaza - 2004 - Scientia 10:3-35.
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  3. V = L and Intuitive Plausibility in Set Theory. A Case Study.Tatiana Arrigoni - 2011 - Bulletin of Symbolic Logic 17 (3):337-360.
    What counts as an intuitively plausible set theoretic content (notion, axiom or theorem) has been a matter of much debate in contemporary philosophy of mathematics. In this paper I develop a critical appraisal of the issue. I analyze first R. B. Jensen's positions on the epistemic status of the axiom of constructibility. I then formulate and discuss a view of intuitiveness in set theory that assumes it to hinge basically on mathematical success. At the same time, I present accounts of (...)
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  4. The Hyperuniverse Program.Tatiana Arrigoni & Sy-David Friedman - 2013 - Bulletin of Symbolic Logic 19 (1):77-96.
    The Hyperuniverse Program is a new approach to set-theoretic truth which is based on justifiable principles and leads to the resolution of many questions independent from ZFC. The purpose of this paper is to present this program, to illustrate its mathematical content and implications, and to discuss its philosophical assumptions.
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  5. Philosophy of Mathematics.Jeremy Avigad - manuscript
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
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  6. Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences.Jody Azzouni - 1994 - Cambridge University Press.
    Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...)
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  7. Sets Whose Members Might Not Exist + Essentialism Possible Worlds.T. Baldwin - unknown
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  8. Mathematical Logic and Foundations of Set Theory.Yehoshua Bar-Hillel (ed.) - 1970 - 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 ...
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  9. Hypersets.J. Barwise & L. Moss - 1991 - The Mathematical Intelligencer 13:31-41.
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  10. Retrieving the Mathematical Mission of the Continuum Concept From the Transfinitely Reductionist Debris of Cantor’s Paradise. Extended Abstract.Edward G. Belaga - forthcoming - International Journal of Pure and Applied Mathematics.
    What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...)
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  11. The Elusiveness of Sets.Max Black - 1971 - Review of Metaphysics 24 (4):614-636.
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  12. A Certain Conception of the Calculus of Rough Sets.Zbigniew Bonikowski - 1992 - Notre Dame Journal of Formal Logic 33 (3):412-421.
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  13. The Simple Consistency of Naive Set Theory Using Metavaluations.Ross T. Brady - 2014 - Journal of Philosophical Logic 43 (2-3):261-281.
    The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the (...)
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  14. E Pluribus Unum: Plural Logic and Set Theory.John P. Burgess - 2004 - Philosophia Mathematica 12 (3):193-221.
    A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.
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  15. Is the Continuum Hypothesis True, False, or Neither?David J. Chalmers - manuscript
    Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non professionals.
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  16. Does Everyone Love Everyone? The Psychology of Iterative Reasoning.Paolo Cherubini & PN Johnson-Laird - 2004 - Thinking and Reasoning 10 (1):31 – 53.
    When a quantified premise such as: Everyone loves anyone who loves someone, occurs with a premise such as: Anne loves Beth, it follows immediately that everyone loves Anne. It also follows that Carol loves Diane, where these two individuals are in the domain of discourse. According to the theory of mental models, this inference requires the quantified premise to be used again to update a model of specific individuals. The paper reports four experiments examining such iterative inferences. Experiment 1 confirmed (...)
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  17. Impure Sets Are Not Located: A Fregean Argument.Roy T. Cook - 2012 - Thought: A Journal of Philosophy 1 (3):219-229.
    It is sometimes suggested that impure sets are spatially co-located with their members (and hence are located in space). Sets, however, are in important respects like numbers. In particular, sets are connected to concepts in much the same manner as numbers are connected to concepts—in both cases, they are fundamentally abstracts of (or corresponding to) concepts. This parallel between the structure of sets and the structure of numbers suggests that the metaphysics of sets and the metaphysics of numbers should parallel (...)
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  18. Understanding the Infinite I: Niceness, Robustness, and Realism.D. Corfield - 2010 - Philosophia Mathematica 18 (3):253-275.
    This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed ‘niceness’ by the mathematician Michiel Hazewinkel, a concept we compare to the ‘robustness’ described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but (...)
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  19. Set Theory: Constructive and Intuitionistic Zf.Laura Crosilla - 2010 - Stanford Encyclopedia of Philosophy.
    Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic. They were introduced in the 1970's and they represent a formal context within which to codify mathematics based on intuitionistic logic. They are formulated on the basis of the standard first order language of Zermelo-Fraenkel set theory and make no direct use of inherently constructive ideas. In working in constructive and intuitionistic ZF we can thus (...)
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  20. Sobre los orígenes de la Matemática abstracta.Domínguez José Ferreiros - 1992 - Theoria 7 (1/2/3):473-498.
    Dedekind used to refer to Riemann as his main model concerning mathematical methodology, particularly regarding the use of abstract notions as a basis for mathematical theories. So, in passages written in 1876 and 1895 he compared his approach to ideal theory with Riemann’s theory of complex functions. In this paper, I try to make sense of those declarations, showing the role of abstract notions in Riemann’s function theory, its influence on Dedekind, and the importance of the methodological principle of avoiding (...)
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  21. Zermelo: Boundary Numbers and Domains of Sets Continued.Heinz-Dieter Ebbinghaus - 2006 - History and Philosophy of Logic 27 (4):285-306.
    Towards the end of his 1930 paper on boundary numbers and domains of sets Zermelo briefly discusses the questions of consistency and of the existence of an unbounded sequence of strongly inaccessible cardinals, deferring a detailed discussion to a later paper which never appeared. In a report to the Emergency Community of German Science from December 1930 about investigations in progress he mentions that some of the intended extensions of these topics had been worked out and were nearly ready for (...)
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  22. An Introduction to Mathematical Reasoning: Lectures on Numbers, Sets, and Functions.Peter J. Eccles - 1997 - Cambridge University Press.
    The purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic (...)
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  23. Is the Continuum Hypothesis a Definite Mathematical Problem?Solomon Feferman - manuscript
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  24. On Arbitrary Sets and ZFC.José Ferreirós - 2011 - Bulletin of Symbolic Logic 17 (3):361-393.
    Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what (...)
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  25. Nonstandard Set Theory.Peter Fletcher - 1989 - Journal of Symbolic Logic 54 (3):1000-1008.
    Nonstandard set theory is an attempt to generalise nonstandard analysis to cover the whole of classical mathematics. Existing versions (Nelson, Hrbáček, Kawai) are unsatisfactory in that the unlimited idealisation principle conflicts with the wish to have a full theory of external sets. I re-analyse the underlying requirements of nonstandard set theory and give a new formal system, stratified nonstandard set theory, which seems to meet them better than the other versions.
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  26. Sets As Mereological Tropes.Peter Forrest - 2002 - Metaphysica 3 (1).
    Either from concrete examples such as tomatoes on a plate, an egg carton full of eggs and so on, or simply because of the braces notation, we come to have some intuitions about the sorts of things sets might be. (See Maddy 1990.) First we tend to think of a set of particulars as itself a particular thing.. Second, even after the distinction between settheory and mereology has been carefully explained we tend to think of the members of a set (...)
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  27. Implementing Mathematical Objects in Set Theory.Thomas Forster - 2007 - Logique Et Analyse 50 (197):79-86.
    In general little thought is given to the general question of how to implement mathematical objects in set theory. It is clear that—at various times in the past—people have gone to considerable lengths to devise implementations with nice properties. There is a litera- ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that makes every set an ordered pair. The implementation of ordinals as Von Neumann ordinals is (...)
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  28. Relevant First-Order Logic LP# and Curry’s Paradox Resolution.Jaykov Foukzon - 2015 - Pure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015 DOI: 10.11648/J.Pamj.S.2015040101.12.
    In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.
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  29. Foundations of Set Theory.A. A. Fraenkel, Y. Bar-Hillel & A. Levy - 1973 - North Holland.
    HISTORICAL INTRODUCTION In Abstract Set Theory) the elements of the theory of sets were presented in a chiefly generic way: the fundamental concepts were ...
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  30. The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number.Gottlob Frege - 1950 - Northwestern University Press.
    § i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...
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  31. The Mathematical Meaning of Mathematical Logic.Harvey Friedman - manuscript
    Each of these theorems and concepts arose from very specific considerations of great general interest in the foundations of mathematics (f.o.m.). They each serve well defined purposes in f.o.m. Naturally, the preferred way to formulate them for mathe-matical logicians is in terms that are close to their roots in f.o.m.
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  32. The Interpretation of Set Theory in Mathematical Predication Theory.Harvey M. Friedman - unknown
    This paper was referred to in the Introduction in our paper [Fr97a], “The Axiomatization of Set Theory by Separation, Reducibility, and Comprehension.” In [Fr97a], all systems considered used the axiom of Extensionality. This is appropriate in a set theoretic context.
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  33. Bemerkungen Zu den Paradoxien von Russell Und Burali-Forti.K. Grelling & L. Nelson - 1907 - Abhandlungen Der Fries'schen Schule (Neue Serie) 2:300-334.
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  34. Cantor's Absolute in Metaphysics and Mathematics.Kai Hauser - 2013 - International Philosophical Quarterly 53 (2):161-188.
    This paper explores the metaphysical roots of Cantor’s conception of absolute infinity in order to shed some light on two basic issues that also affect the mathematical theory of sets: the viability of Cantor’s distinction between sets and inconsistent multiplicities, and the intrinsic justification of strong axioms of infinity that are studied in contemporary set theory.
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  35. Cantor's Concept of Set in the Light of Plato's Philebus.Kai Hauser - 2010 - Review of Metaphysics 63 (4):783-805.
    In explaining his concept of set Cantor intimates a connection with the metaphysical scheme put forward in Plato’s Philebus to determine the place of pleasure. We argue that these determinations capture key ideas of Cantorian set theory and, moreover, extend to intuitions which continue to play a central role in the modern mathematics of infinity.
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  36. Analysis of Bilateral Iterative Networks.Frederick C. Hennie - 1959 - Journal of Symbolic Logic 24 (3):259-260.
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  37. Discovery of Clusters From Proximity Data: An Approach Using Iterative Adjustment of Binary Classifications.Shoji Hirano & Shusaku Tsumoto - 2008 - In S. Iwata, Y. Oshawa, S. Tsumoto, N. Zhong, Y. Shi & L. Magnani (eds.), Communications and Discoveries From Multidisciplinary Data. Springer. pp. 251--268.
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  38. Universal Sets for Pointsets Properly on the N Th Level of the Projective Hierarchy.Greg Hjorth, Leigh Humphries & Arnold W. Miller - 2013 - Journal of Symbolic Logic 78 (1):237-244.
    The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.
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  39. Airport ‘86 Revisited: Toward a Unified Indefinite Any.Larry Horn - manuscript
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  40. The Mathematical Development of Set Theory From Cantor to Cohen.Akihiro Kanamori - 1996 - Bulletin of Symbolic Logic 2 (1):1-71.
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  41. A Hierarchy of Languages, Logics, and Mathematical Theories.Charles W. Kastner - manuscript
    We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. (...)
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  42. Gödel's Modernism: On Set Theoretic Incompleteness, Revisited.Juliette Kennedy - 2009 - In Sten Lindström, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.), Logicism, Intuitionism and Formalism: What has become of them? Springer.
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  43. The Nature of Mathematical Knowledge.Philip Kitcher - 1983 - 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 ...
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  44. Groundedness - Its Logic and Metaphysics.Jönne Kriener - 2014 - Dissertation, Birkbeck College, University of London
    In philosophical logic, a certain family of model constructions has received particular attention. Prominent examples are the cumulative hierarchy of well-founded sets, and Kripke's least fixed point models of grounded truth. I develop a general formal theory of groundedness and explain how the well-founded sets, Cantor's extended number-sequence and Kripke's concepts of semantic groundedness are all instances of the general concept, and how the general framework illuminates these cases. Then, I develop a new approach to a grounded theory of proper (...)
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  45. Encoded Pilots for Iterative Receiver Improvement.Hyuck M. Kwon, Khurram Hassan, Ashutosh Goyal, Mi-Kyung Oh, Dong-Jo Park & Yong Hoon Lee - 2005 - Complexity 4:5.
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  46. Zermelo and Russell's Paradox: Is There a Universal Set?G. Landini - 2013 - Philosophia Mathematica 21 (2):180-199.
    Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied the criteria and (...)
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  47. Michael Potter Tom Ricketts, Eds. The Cambridge Companion to Frege. Cambridge: Cambridge University Press, 2010. Isbn 978-0-521-62479-4. Pp. XVII+639. [REVIEW]G. Landini - 2012 - Philosophia Mathematica 20 (3):372-387.
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  48. Finite Mathematics.Shaughan Lavine - 1995 - Synthese 103 (3):389 - 420.
    A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form (...)
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  49. Iterativity Vs. Habituality: On the Iterative Interpretation of Perfective Sentences.Alessandro Lenci & Pier Marco Bertinetto - 2000 - In Achille Varzi, James Higginbotham & Fabio Pianesi (eds.), Speaking of Events. Oxford University Press.
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  50. Mathematical Knowledge.Mary Leng, Alexander Paseau & Michael D. Potter (eds.) - 2007 - Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. (...)
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