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On the Concept of Finitism

Synthese 192 (8):2413-2436 (2015)

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  1. Finitism and Intuitive Knowledge.Charles Parsons - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 249--270.
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  • Partial Realizations of Hilbert's Program.Stephen G. Simpson - 1988 - Journal of Symbolic Logic 53 (2):349-363.
  • Godel's Interpretation of Intuitionism.William Tait - 2006 - Philosophia Mathematica 14 (2):208-228.
    Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...)
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  • Hilbert's Program and the Omega-Rule.Aleksandar Ignjatović - 1994 - Journal of Symbolic Logic 59 (1):322 - 343.
    In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Godel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the ω-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. On (...)
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  • On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the (...)
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  • Absolute Generality.Agustín Rayo & Gabriel Uzquiano (eds.) - 2006 - Oxford University Press.
    The problem of absolute generality has attracted much attention in recent philosophy. Agustin Rayo and Gabriel Uzquiano have assembled a distinguished team of contributors to write new essays on the topic. They investigate the question of whether it is possible to attain absolute generality in thought and language and the ramifications of this question in the philosophy of logic and mathematics.
  • Two (or Three) Notions of Finitism.Mihai Ganea - 2010 - Review of Symbolic Logic 3 (1):119-144.
    Finitism is given an interpretation based on two ideas about strings (sequences of symbols): a replacement principle extracted from Hilberts class 2 can be justified by means of an additional finitistic choice principle, thus obtaining a second equational theory . It is unknown whether is strictly stronger than since 2 may coincide with the class of lower elementary functions.
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  • On Ontology and Realism in Mathematics.Haim Gaifman - 2012 - Review of Symbolic Logic 5 (3):480-512.
  • Existence and Feasibility in Arithmetic.Rohit Parikh - 1971 - Journal of Symbolic Logic 36 (3):494-508.
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  • Benacerraf's Dilemma Revisited.Bob Hale & Crispin Wright - 2002 - European Journal of Philosophy 10 (1):101–129.
  • Hilbert's Epistemology.Philip Kitcher - 1976 - Philosophy of Science 43 (1):99-115.
    Hilbert's program attempts to show that our mathematical knowledge can be certain because we are able to know for certain the truths of elementary arithmetic. I argue that, in the absence of a theory of mathematical truth, Hilbert does not have a complete theory of our arithmetical knowledge. Further, while his deployment of a Kantian notion of intuition seems to promise an answer to scepticism, there is no way to complete Hilbert's epistemology which would answer to his avowed aims.
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  • Collected Papers.Colin McGinn & Gareth Evans - 1988 - Philosophical Review 97 (2):278.
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  • The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940.Paolo Mancosu - 2010 - Oxford University Press.
    At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of .
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  • Reason and Intuition.Charles Parsons - 2000 - Synthese 125 (3):299-315.
  • Seeing Sequences.David Galloway - 1999 - Philosophy and Phenomenological Research 59 (1):93-112.
    This article discusses Charles Parsons’ conception of mathematical intuition. Intuition, for Parsons, involves seeing-as: in seeing the sequences I I I and I I I as the same type, one intuits the type. The type is abstract, but intuiting the type is supposed to be epistemically analogous to ordinary perception of physical objects. And some non-trivial mathematical knowledge is supposed to be intuitable in this way, again in a way analogous to ordinary perceptual knowledge. In particular, the successor axioms are (...)
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  • Parsons on Mathematical Intuition.James Page - 1993 - Mind 102 (406):223-232.
    Charles Parsons has argued that we have the ability to apprehend, or "intuit", certain kinds of abstract objects; that among the objects we can intuit are some which form a model for arithmetic; and that our knowledge that the axioms of arithmetic are true in this model involves our intuition of these objects. I find a problem with Parson's claim that we know this model is infinite through intuition. Unless this problem can be resolved. I question whether our knowledge that (...)
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  • On Some Difficulties Concerning Intuition and Intuitive Knowledge.Charles Parsons - 1993 - Mind 102 (406):233-246.
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  • A Defence of Arbitrary Objects.Kit Fine & Neil Tennant - 1983 - Aristotelian Society Supplementary Volume 57 (1):55 - 89.
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  • Grundlagen der Mathematik. Band I. [REVIEW]Rudolf Carnap - 1939 - Journal of Unified Science (Erkenntnis) 8 (1):184-187.
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  • Mathematical Thought and its Objects.Charles Parsons - 2007 - Cambridge University Press.
    Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition (...)
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  • Philosophy of Mathematics: Selected Readings.Paul Benacerraf & Hilary Putnam (eds.) - 1964 - Cambridge University Press.
    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, and Gödel himself, (...)
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  • The Provenance of Pure Reason: Essays in the Philosophy of Mathematics and its History.William Tait - 2005 - Oup Usa.
    William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gdel.
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  • Die Widerspruchsfreiheit der Allgemeinen Mengenlehre.Wilhelm Ackerman - 1937 - Journal of Symbolic Logic 2 (4):167-167.
  • Grundlagen der Mathematik.David Hilbert & Paul Bernays - 1934 - Springer.
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  • Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.
  • From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s.Paolo Mancosu (ed.) - 1998 - 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. (...)
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  • Remarks on Finitism.William Tait - manuscript
    The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.
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  • Function Theory in an Axiom-Free Equation Calculus.R. L. Goodstein - 1946 - Journal of Symbolic Logic 11 (1):24-26.
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  • Hilbert's Program: An Essay on Mathematical Instrumentalism.Michael Detlefsen - 1986 - Reidel.
    An Essay on Mathematical Instrumentalism M. Detlefsen. THE PHILOSOPHICAL FUNDAMENTALS OF HILBERT'S PROGRAM 1. INTRODUCTION In this chapter I shall attempt to set out Hilbert's Program in a way that is more revealing than ...
  • Die Grundlagen der Mathematik.David Hilbert, Hermann Weyl & Paul Bernays - 1928 - Teubner.
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  • Subsystems of Second-Order Arithmetic.Stephen G. Simpson - 1999 - Springer Verlag.
    Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...
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  • Mathematische Existenz.Oskar Becker - 1928 - Annalen der Philosophie Und Philosophischen Kritik 7:154-155.
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  • Die Widerspruchsfreiheit der Reinen Zahlentheorie.Gerhard Gentzen - 1936 - Journal of Symbolic Logic 1 (2):75-75.
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  • Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes.K. GÖdel - 1958 - Dialectica 12 (3):280.
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  • Parsons on Mathematical Intuition and Obviousness.Michael D. Resnik - 2000 - In Gila Sher & Richard L. Tieszen (eds.), Between Logic and Intuition: Essays in Honor of Charles Parsons. Cambridge University Press. pp. 219--231.
     
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  • The Philosophy of Mathematics.W. D. Hart (ed.) - 1996 - Oxford University Press.
    This volume offers a selection of the most interesting and important work from recent years in the philosophy of mathematics, which has always been closely linked to, and has exerted a significant influence upon, the main stream of analytical philosophy. The issues discussed are of interest throughout philosophy, and no mathematical expertise is required of the reader. Contributors include W.V. Quine, W.D. Hart, Michael Dummett, Charles Parsons, Paul Benacerraf, Penelope Maddy, W.W. Tait, Hilary Putnam, George Boolos, Daniel Isaacson, Stewart Shapiro, (...)
     
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