Switch to: References

Citations of:

Finitism

Journal of Philosophy 78 (9):524-546 (1981)

Add citations

You must login to add citations.
  1. Hilbert's Metamathematical Problems and Their Solutions.Besim Karakadilar - 2008 - Dissertation, Boston University
    This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Strict finitism, feasibility, and the sorites.Walter Dean - 2018 - Review of Symbolic Logic 11 (2):295-346.
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Takeuti's Well-Ordering Proof: Finitistically Fine?Eamon Darnell & Aaron Thomas-Bolduc - 2018 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics The CSHPM 2017 Annual Meeting in Toronto, Ontario. New York: Birkhäuser.
    If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • The Implicit Commitment of Arithmetical Theories and Its Semantic Core.Carlo Nicolai & Mario Piazza - 2019 - Erkenntnis 84 (4):913-937.
    According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Predicativity through transfinite reflection.Andrés Cordón-Franco, David Fernández-Duque, Joost J. Joosten & Francisco Félix Lara-martín - 2017 - Journal of Symbolic Logic 82 (3):787-808.
    Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.For a set of formulas Γ, define predicative oracle reflection for T over Γ ) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then$$\forall \,\lambda (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • Numbers and functions in Hilbert's finitism.Richard Zach - 1998 - Taiwanese Journal for History and Philosophy of Science 10:33-60.
    David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...)
    Download  
     
    Export citation  
     
    Bookmark   9 citations  
  • Hilbert’s Program.Richard Zach - 2014 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab.
    In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...)
    Download  
     
    Export citation  
     
    Bookmark   28 citations  
  • Hilbert's program relativized: Proof-theoretical and foundational reductions.Solomon Feferman - 1988 - Journal of Symbolic Logic 53 (2):364-384.
    Download  
     
    Export citation  
     
    Bookmark   64 citations  
  • The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory.Michael Rathjen - 2005 - Synthese 147 (1):81-120.
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Recursive Functions and Metamathematics: Problems of Completeness and Decidability, Gödel's Theorems.Rod J. L. Adams & Roman Murawski - 1999 - Dordrecht, Netherland: Springer Verlag.
    Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.
    Download  
     
    Export citation  
     
    Bookmark   11 citations  
  • Hilbert’s Finitism: Historical, Philosophical, and Metamathematical Perspectives.Richard Zach - 2001 - Dissertation, University of California, Berkeley
    In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...)
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • The Paradox of the Knower revisited.Walter Dean & Hidenori Kurokawa - 2014 - Annals of Pure and Applied Logic 165 (1):199-224.
    The Paradox of the Knower was originally presented by Kaplan and Montague [26] as a puzzle about the everyday notion of knowledge in the face of self-reference. The paradox shows that any theory extending Robinson arithmetic with a predicate K satisfying the factivity axiom K → A as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Consistency, Models, and Soundness.Matthias Schirn - 2010 - Axiomathes 20 (2):153-207.
    This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Mathematical instrumentalism, Gödel’s theorem, and inductive evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • Transfer and a Supremum Principle for ERNA.Chris Impens & Sam Sanders - 2008 - Journal of Symbolic Logic 73 (2):689 - 710.
    Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Patrick Suppes and Richard Sommer, who also proved its consistency inside PRA. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes, of which Michal Rössler and Emil Jeřábek have recently proposed a weakened version. We add a Π₁-transfer principle to ERNA and prove the consistency of the extended theory inside PRA. In this extension of ERNA a σ₁-supremum (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • The Crisis in the Foundations of Mathematics.J. Ferreiros - 2008 - In T. Gowers (ed.), Princeton Companion to Mathematics. Princeton University Press.
    A general introduction to the celebrated foundational crisis, discussing how the characteristic traits of modern mathematics (acceptance of the notion of an “arbitrary” function proposed by Dirichlet; wholehearted acceptance of infinite sets and the higher infinite; a preference “to put thoughts in the place of calculations” and to concentrate on “structures” characterized axiomatically; a reliance on “purely existential” methods of proof) provoked extensive polemics and alternative approaches. Going beyond exclusive concentration on the paradoxes, it also discusses the role of the (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • Proof theory in philosophy of mathematics.Andrew Arana - 2010 - Philosophy Compass 5 (4):336-347.
    A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • How to be a structuralist all the way down.Elaine Landry - 2011 - Synthese 179 (3):435 - 454.
    This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", (...)
    Download  
     
    Export citation  
     
    Bookmark   16 citations  
  • Reconstructing Hilbert to construct category theoretic structuralism.Elaine Landry - unknown
    This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Varieties of Finitism.Manuel Bremer - 2007 - Metaphysica 8 (2):131-148.
    I consider here several versions of finitism or conceptions that try to work around postulating sets of infinite size. Restricting oneself to the so-called potential infinite seems to rest either on temporal readings of infinity (or infinite series) or on anti-realistic background assumptions. Both these motivations may be considered problematic. Quine’s virtual set theory points out where strong assumptions of infinity enter into number theory, but is implicitly committed to infinity anyway. The approaches centring on the indefinitely large and the (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • The Price of Mathematical Scepticism.Paul Blain Levy - 2022 - Philosophia Mathematica 30 (3):283-305.
    This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.
    Download  
     
    Export citation  
     
    Bookmark  
  • 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 (...)
    Download  
     
    Export citation  
     
    Bookmark   13 citations  
  • 2000 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium 2000.Carol Wood - 2001 - Bulletin of Symbolic Logic 7 (1):82-163.
    Download  
     
    Export citation  
     
    Bookmark  
  • Philosophy of mathematics.Leon Horsten - 2008 - Stanford Encyclopedia of Philosophy.
    If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...)
    Download  
     
    Export citation  
     
    Bookmark   25 citations  
  • The practice of finitism: Epsilon calculus and consistency proofs in Hilbert's program.Richard Zach - 2003 - Synthese 137 (1-2):211 - 259.
    After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...)
    Download  
     
    Export citation  
     
    Bookmark   36 citations  
  • 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 (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • 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 (...)
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • Partial realizations of Hilbert's program.Stephen G. Simpson - 1988 - Journal of Symbolic Logic 53 (2):349-363.
    Download  
     
    Export citation  
     
    Bookmark   52 citations  
  • Fragment of nonstandard analysis with a finitary consistency proof.Michal Rössler & Emil Jeřábek - 2007 - Bulletin of Symbolic Logic 13 (1):54-70.
    We introduce a nonstandard arithmetic $NQA^-$ based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by $NQA^+$ ), with a weakened external open minimization schema. A finitary consistency proof for $NQA^-$ formalizable in PRA is presented. We also show interesting facts about the strength of the theories $NQA^-$ and $NQA^+$ ; $NQA^-$ is mutually interpretable with $I\Delta_0 + EXP$ , and on the other hand, $NQA^+$ interprets the theories IΣ1 and $WKL_0$.
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • Elementary realizability.Zlatan Damnjanovic - 1997 - Journal of Philosophical Logic 26 (3):311-339.
    A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π₁⁰-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π₁⁰ formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π₁⁰-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Finitist Axiomatic Truth.Sato Kentaro & Jan Walker - 2023 - Journal of Symbolic Logic 88 (1):22-73.
    Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $, of Kripke–Feferman truth (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • A theory of implicit commitment.Mateusz Łełyk & Carlo Nicolai - 2022 - Synthese 200 (4):1-26.
    The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • The collapse of the Hilbert program: A variation on the gödelian theme.Saul A. Kripke - 2022 - Bulletin of Symbolic Logic 28 (3):413-426.
    The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • Strict Finitism's Unrequited Love for Computational Complexity.Noel Arteche - manuscript
    As a philosophy of mathematics, strict finitism has been traditionally concerned with the notion of feasibility, defended mostly by appealing to the physicality of mathematical practice. This has led the strict finitists to influence and be influenced by the field of computational complexity theory, under the widely held belief that this branch of mathematics is concerned with the study of what is “feasible in practice”. In this paper, I survey these ideas and contend that, contrary to popular belief, complexity theory (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Strict Finitism and the Logic of Mathematical Applications.Feng Ye - 2011 - Dordrecht, Netherland: Springer.
    This book intends to show that radical naturalism, nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry. The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to the (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Hilbert izlencesinin izinde adcılık adına yeni bulgular.Besim Karakadılar - manuscript
    Hilbert izlencesinin kanıt kuramsal amacı tarihsel gelişimi içinde özetlendikten sonra arka plandaki model-kuramsal motivasyonu belirtilmektedir. Hilbert'in nihai hedefinin matematiğin temellerine ilişkin tüm epistemolojik ve ontolojik varsayımlardan arındırılmış bir matematik kuramı geliştirmek olduğu savunulmaktadır. Yakın geçmişte mantıktaki bazı gelişmelerin Hilbert izlencesinin yalnızca adcı varsayımlar temelinde sürdürülebileceğine ilişkin yeni bir bakış açısı sağladığı öne sürülmektedir.
    Download  
     
    Export citation  
     
    Bookmark  
  • Computational Complexity Theory and the Philosophy of Mathematics†.Walter Dean - 2019 - Philosophia Mathematica 27 (3):381-439.
    Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Hypatia's silence.Martin Fischer, Leon Horsten & Carlo Nicolai - 2021 - Noûs 55 (1):62-85.
    Hartry Field distinguished two concepts of type‐free truth: scientific truth and disquotational truth. We argue that scientific type‐free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non‐classical logical treatment.
    Download  
     
    Export citation  
     
    Bookmark   11 citations  
  • Hilbert's Program Revisited.Panu Raatikainen - 2003 - Synthese 137 (1-2):157-177.
    After sketching the main lines of Hilbert's program, certain well-known andinfluential interpretations of the program are critically evaluated, and analternative interpretation is presented. Finally, some recent developments inlogic related to Hilbert's program are reviewed.
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • The Finitistic Consistency of Heck’s Predicative Fregean System.Luís Cruz-Filipe & Fernando Ferreira - 2015 - Notre Dame Journal of Formal Logic 56 (1):61-79.
    Frege’s theory is inconsistent. However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak. We also prove the finitistic consistency of the extension of Heck’s theory to $\Delta^{1}_{1}$-comprehension and of Heck’s ramified predicative second-order system.
    Download  
     
    Export citation  
     
    Bookmark  
  • Things that can and things that cannot be done in PRA.Ulrich Kohlenbach - 2000 - Annals of Pure and Applied Logic 102 (3):223-245.
    It is well known by now that large parts of mathematical reasoning can be carried out in systems which are conservative over primitive recursive arithmetic PRA . On the other hand there are principles S of elementary analysis which are known to be equivalent to arithmetical comprehension and therefore go far beyond the strength of PRA . In this paper we determine precisely the arithmetical and computational strength of weaker function parameter-free schematic versions S− of S, thereby exhibiting different levels (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Mathematical Intuition and Natural Numbers: A Critical Discussion. [REVIEW]Felix Mühlhölzer - 2010 - Erkenntnis 73 (2):265-292.
    Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • A hierarchy of hereditarily finite sets.Laurence Kirby - 2008 - Archive for Mathematical Logic 47 (2):143-157.
    This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy.
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • L'infinité des nombres premiers : une étude de cas de la pureté des méthodes.Andrew Arana - 2011 - Les Etudes Philosophiques 97 (2):193.
    Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Kilka uwag o tezie Churcha i aksjomacie Hilberta.Adam Olszewski - 2006 - Zagadnienia Filozoficzne W Nauce 38.
    Download  
     
    Export citation  
     
    Bookmark  
  • Is unsaying polite?Berislav Žarnić - 2011 - In Majda Trobok, Nenad Miščević & Berislav Žarnić (eds.), Between Logic and Reality: Modeling Inference, Action and Understanding. Dordrecht and New York: Springer. pp. 201--224.
    This paper is divided in five sections. Section 11.1 sketches the history of the distinction between speech act with negative content and negated speech act, and gives a general dynamic interpretation for negated speech act. “Downdate semantics” for AGM contraction is introduced in Section 11.2. Relying on semantically interpreted contraction, Section 11.3 develops the dynamic semantics for constative and directive speech acts, and their external negations. The expressive completeness for the formal variants of natural language utterances, none of which is (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Different senses of finitude: An inquiry into Hilbert’s finitism.Sören Stenlund - 2012 - Synthese 185 (3):335-363.
    This article develops a critical investigation of the epistemological core of Hilbert's foundational project, the so-called the finitary attitude. The investigation proceeds by distinguishing different senses of 'number' and 'finitude' that have been used in the philosophical arguments. The usual notion of modern pure mathematics, i.e. the sense of number which is implicit in the notion of an arbitrary finite sequence and iteration is one sense of number and finitude. Another sense, of older origin, is connected with practices of counting (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Finitary Set Theory.Laurence Kirby - 2009 - Notre Dame Journal of Formal Logic 50 (3):227-244.
    I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.
    Download  
     
    Export citation  
     
    Bookmark   11 citations  
  • Critical study of Michael Potter’s Reason’s Nearest Kin. [REVIEW]Richard Zach - 2005 - Notre Dame Journal of Formal Logic 46 (4):503-513.
    Critical study of Michael Potter, Reason's Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. x + 305 pages.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • 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.
    Download  
     
    Export citation  
     
    Bookmark   5 citations