This category needs an editor. We encourage you to help if you are qualified.
Volunteer, or read more about what this involves.
Related categories
Subcategories:
322 found
Search inside:
(import / add options)   Sort by:
1 — 50 / 322
Material to categorize
  1. Douglas Bridges & Steeve Reeves (1999). Constructive Mathematics in Theory and Programming Practice. Philosophia Mathematica 7 (1):65-104.
    The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). it gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-L6f 's theory of types as a formal system for BISH.
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
Logicism in Mathematics
  1. Alice Ambrose (1933). A Controversy in the Logic of Mathematics. Philosophical Review 42 (6):594-611.
  2. Irving H. Anellis (1987). Russell and Engels: Two Approaches to a Hegelian Philosophy of Mathematics. Philosophia Mathematica (2):151-179.
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  3. Irving H. Anellis (1987). Russell's Earliest Interpretation of Cantorian Set Theory, 1896–1900. Philosophia Mathematica (1):1-31.
  4. G. A. Antonelli (2010). Notions of Invariance for Abstraction Principles. Philosophia Mathematica 18 (3):276-292.
    The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance (...)
    Remove from this list | Direct download (12 more)  
     
    My bibliography  
     
    Export citation  
  5. Alexander Bird (1997). The Logic in Logicism. Dialogue 36 (02):341--60.
    Frege's logicism consists of two theses: (1) the truths of arithmetic are truths of logic; (2) the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G are (...)
    Remove from this list | Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  6. Kenneth Blackwell, Nicholas Griffin & Bernard Linsky (eds.) (2011). Principia Mathematica at 100. Bertrand Russell Research Centre.
  7. George Boolos (1990). The Standard of Equality of Numbers. In , Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge University Press. 261--77.
    Remove from this list |
     
    My bibliography  
     
    Export citation  
  8. George S. Boolos (ed.) (1990). Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge University Press.
    This volume is a report on the state of philosophy in a number of significant areas.
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  9. Andrew Boucher, Who Needs (to Assume) Hume's Principle?
    Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  10. Andrew Boucher, Who Needs (to Assume) Hume's Principle? July 2006.
    In the Foundations of Arithmetic, Frege famously developed a theory which today goes by the name of logicism - that it is possible to prove the truths of arithmetic using only logical principles and definitions. Logicism fell out of favor for various reasons, most spectacular of which was that the system, which Frege thought would definitively prove his thesis, turned out to be inconsistent. In the early 1980s a movement called neo-logicism was begun by Crispin Wright. Neo-logicism holds that Frege (...)
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  11. Otavio Bueno (2001). Logicism Revisited. Principia 5 (1-2):99-124.
    In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, I argue (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  12. Rudolf Carnap (1983). The Logicist Foundations of Mathematics. In Paul Benacerraf & Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings. Cambridge University Press. 41--52.
  13. R. T. Cook (2012). RICHARD G. HECK, Jr. Frege's Theorem. Oxford: Clarendon Press, 2011. ISBN 978-0-19-969564-5. Pp. Xiv + 307. Philosophia Mathematica 20 (3):346-359.
  14. Roy T. Cook (2013). Patricia A. Blanchette. Frege's Conception of Logic. Oxford University Press, 2012. ISBN 978-0-19-926925-9 (Hbk). Pp. Xv + 256. [REVIEW] Philosophia Mathematica (1):nkt029.
  15. Roy T. Cook & Philip A. Ebert (2005). Abstraction and Identity. Dialectica 59 (2):121–139.
    A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.
    Remove from this list | Direct download (11 more)  
     
    My bibliography  
     
    Export citation  
  16. Boudewijn de Bruin (2008). Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number. Philosophia Mathematica 16 (3):354-373.
    Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...)
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  17. William Demopoulos (2013). Logicism and its Philosophical Legacy. Cambridge University Press.
    Frege's analysis of arithmetical knowledge -- Carnap's thesis -- On extending 'empiricism, semantics and ontology' to the realism-instrumentalism controversy -- Carnap's analysis of realism -- Bertrand Russell's The analysis of matter: its historical context and contemporary interest (with Michael Friedman) -- On the rational reconstruction of our theoretical knowledge -- Three views of theoretical knowledge -- Frege and the rigorization of analysis -- The philosophical basis of our knowledge of number -- The 1910 Principia's theory of functions and classes -- (...)
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  18. William Demopoulos (ed.) (1995). Frege's Philosophy of Mathematics. Harvard University Press.
  19. William Demopoulus & William Bell (1993). Frege's Theory of Concepts and Objects and the Interpretation of Second-Order Logict. Philosophia Mathematica 1 (2):139-156.
    This paper casts doubt on a recent criticism of Frege's theory of concepts and extensions by showing that it misses one of Frege's most important contributions: the derivation of the infinity of the natural numbers. We show how this result may be incorporated into the conceptual structure of Zermelo- Fraenkel Set Theory. The paper clarifies the bearing of the development of the notion of a real-valued function on Frege's theory of concepts; it concludes with a brief discussion of the claim (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  20. Michael A. E. Dummett (1991). Frege: Philosophy of Mathematics. Harvard University Press.
    In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  21. P. A. Ebert (2011). Guillermo E. Rosado Haddock. A Critical Introduction to the Philosophy of Gottlob Frege. Aldershot, Hampshire, and Burlington, Vermont: Ashgate Publishing, 2006. Isbn 978-0-7546-5471-1. Pp. X+157. [REVIEW] Philosophia Mathematica 19 (3):363-367.
  22. Philip A. Ebert & Marcus Rossberg (2009). Ed Zalta's Version of Neo-Logicism: A Friendly Letter of Complaint. In Hannes Leitgeb & Alexander Hieke (eds.), Reduction – Abstraction – Analysis. Ontos. 11--305.
    In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  23. Philip A. Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415 - 441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  24. Matti Eklund (2009). Bad Company and Neo-Fregean Philosophy. Synthese 170 (3):393 - 414.
    A central element in neo-Fregean philosophy of mathematics is the focus on abstraction principles, and the use of abstraction principles to ground various areas of mathematics. But as is well known, not all abstraction principles are in good standing. Various proposals for singling out the acceptable abstraction principles have been presented. Here I investigate what philosophical underpinnings can be provided for these proposals; specifically, underpinnings that fit the neo-Fregean's general outlook. Among the philosophical ideas I consider are: general views on (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  25. Fernando Ferreira & Kai F. Wehmeier (2002). On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze. Journal of Philosophical Logic 31 (4):301-311.
    It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...)
    Remove from this list | Direct download (9 more)  
     
    My bibliography  
     
    Export citation  
  26. José Ferreirós (2009). Hilbert, Logicism, and Mathematical Existence. Synthese 170 (1):33 - 70.
    David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  27. Gottlob Frege (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. 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. ...
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  28. Gottlob Frege, Philip A. Ebert & Marcus Rossberg (eds.) (2013). Basic Laws of Arithmetic. Oxford University Press.
  29. S. Gandon & B. Halimi (2013). Introduction: Logicism Today. Philosophia Mathematica 21 (2):129-132.
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  30. Sébastien Gandon (2008). Which Arithmetization for Which Logicism? Russell on Relations and Quantities in The Principles of Mathematics. History and Philosophy of Logic 29 (1):1-30.
    This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible 'by logical principles from logical principles' does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV-V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...)
    Remove from this list | Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  31. Bonnie Gold & Roger Simons (eds.) (2008). Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America.
    This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to ...
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  32. N. Griffin (2013). Bernard Linsky. The Evolution of Principia Mathematica: Bertrand Russell's Manuscripts and Notes for the Second Edition. Cambridge: Cambridge University Press, 2011. ISBN 978-1-107-00327-9. Pp. Vii + 407. [REVIEW] Philosophia Mathematica 21 (3):403-411.
    Remove from this list | Direct download (11 more)  
     
    My bibliography  
     
    Export citation  
  33. Bob Hale (2000). Reals by Abstractiont. Philosophia Mathematica 8 (2):100--123.
    On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...)
    Remove from this list | Direct download (9 more)  
     
    My bibliography  
     
    Export citation  
  34. Bob Hale (1999). Frege's Philosophy of Mathematics. Philosophical Quarterly 49 (194):92–104.
  35. William H. Hanson (1990). Second-Order Logic and Logicism. Mind 99 (393):91-99.
    Some widely accepted arguments in the philosophy of mathematics are fallacious because they rest on results that are provable only by using assumptions that the con- clusions of these arguments seek to undercut. These results take the form of bicon- ditionals linking statements of logic with statements of mathematics. George Boolos has given an argument of this kind in support of the claim that certain facts about second-order logic support logicism, the view that mathematics—or at least part of it—reduces to (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  36. William S. Hatcher (1982). The Logical Foundations of Mathematics. Pergamon Press.
  37. Richard Heck (2011). Ramified Frege Arithmetic. Journal of Philosophical Logic 40 (6):715-735.
    Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.
    Remove from this list | Direct download (8 more)  
     
    My bibliography  
     
    Export citation  
  38. Richard Heck (2011). A Logic for Frege's Theorem. In Frege's Theorem. Oxford University Press.
    It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...)
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  39. Richard Heck (2005). Julius Caesar and Basic Law V. Dialectica 59 (2):161–178.
    This paper dates from about 1994: I rediscovered it on my hard drive in the spring of 2002. It represents an early attempt to explore the connections between the Julius Caesar problem and Frege's attitude towards Basic Law V. Most of the issues discussed here are ones treated rather differently in my more recent papers "The Julius Caesar Objection" and "Grundgesetze der Arithmetik I 10". But the treatment here is more accessible, in many ways, providing more context and a better (...)
    Remove from this list | Direct download (9 more)  
     
    My bibliography  
     
    Export citation  
  40. Richard Heck (2000). Cardinality, Counting, and Equinumerosity. Notre Dame Journal of Formal Logic 41 (3):187-209.
    Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...)
    Remove from this list | Direct download (9 more)  
     
    My bibliography  
     
    Export citation  
  41. Richard Heck (1999). Grundgesetze der Arithmetic I §10. Philosophia Mathematica 7 (3):258-292.
    In section 10 of Grundgesetze, Frege confronts an indeterm inacy left by his stipulations regarding his ‘smooth breathing’, from which names of valueranges are formed. Though there has been much discussion of his arguments, it remains unclear what this indeterminacy is; why it bothers Frege; and how he proposes to respond to it. The present paper attempts to answer these questions by reading section 10 as preparatory for the (fallacious) proof, given in section 31, that every expression of Frege's formal (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  42. Richard Heck (ed.) (1997). Language, Truth, and Logic. Oxford University Press.
    A Festschrift for Michael Dummett. Includes papers by Christopher Peacocke, Alexander George, Sanford Shieh, John McDowell, Jason Stanley, John Campbell, Barry Taylor, Crispin Wright, George Boolos, Charles Parsons, and Richard Heck.
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  43. Richard Heck (1993). Critical Notice of Michael Dummett, Frege: Philosophy of Mathematics. Philosophical Quarterly 43:223-33.
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  44. Claire Hill (2002). W. Demopoulos (Ed.), Frege's Philosophy of Mathematics, and W. W. Tait (Ed.), Early Analytic Philosophy, Frege, Russell, Wittgenstein, Essays in Honor of Leonard Linsky. [REVIEW] Synthese 133 (3):441-452.
  45. Harold T. Hodes (1984). Logicism and the Ontological Commitments of Arithmetic. Journal of Philosophy 81 (3):123-149.
  46. Luca Incurvati (2007). On Some Consequences of the Definitional Unprovability of Hume's Principle. In Pierre Joray (ed.), Contemporary Perspectives on Logicism and the Foundations of Mathematics. CDRS.
    Remove from this list | Direct download  
     
    My bibliography  
     
    Export citation  
  47. Ivan Kasa (2010). A Puzzle About Ontological Commitments: Reply to Ebert. Philosophia Mathematica 18 (1):102-105.
    This note refutes P. Ebert’s argument against Epistemic Rejectionism.
    Remove from this list | Direct download (11 more)  
     
    My bibliography  
     
    Export citation  
  48. L. H. Kauffman (2012). The Russell Operator. Constructivist Foundations 7 (2):112-115.
    Context: The question of how to understand the epistemology of set theory has been a longstanding problem in the foundations of mathematics since Cantor formulated the theory in the 19th century, and particularly since Bertrand Russell articulated his paradox in the early twentieth century. The theory of types pioneered by Russell and Whitehead was simplified by mathematicians to a single distinction between sets and classes. The question of the meaning of this distinction and its necessity still remains open. Problem: I (...)
    Remove from this list | Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  49. Joongol Kim (forthcoming). A Logical Foundation of Arithmetic. Studia Logica:1-32.
    The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework (ALA) that takes seriously ordinary locutions like ‘at least n Fs’, ‘n more Fs than Gs’ and ‘n times as many Fs as Gs’, instead of paraphrasing them away in terms of expressions of the form ‘the number of Fs’. It will be shown that the basic concepts of arithmetic can be intuitively defined in the language of ALA, and (...)
    Remove from this list | Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
1 — 50 / 322