Results for 'Finitism'

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  1. Strict Finitism and the Happy Sorites.Ofra Magidor - 2012 - Journal of Philosophical Logic 41 (2):471-491.
    Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at (...)
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  2. On the Concept of Finitism.Luca Incurvati - 2015 - Synthese 192 (8):2413-2436.
    At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as (...)
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  3.  45
    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 (...)
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  4.  87
    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 (...)
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  5. How Discrete Patterns Emerge From Algorithmic Fine-Tuning: A Visual Plea for Kroneckerian Finitism.Ivahn Smadja - 2010 - Topoi 29 (1):61-75.
    This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding (...)
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  6. Aristotelian Finitism.Tamer Nawar - 2015 - Synthese 192 (8):2345-2360.
    It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle (...)
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  7.  47
    What Finitism Could Not Be.Matthias Schirn & Karl-Georg Niebergall - 2003 - Critica 35 (103):43-68.
    In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by (...)
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  8.  40
    Wittgenstein, Finitism, and the Foundations of Mathematics. [REVIEW]David Stern - 2001 - Dialogue 40 (3):624-625.
    More than half of Wittgenstein’s writings from the years between his return to philosophy in 1929 and the completion of Part I of the Philosophical Investigations in 1945 are about issues in the philosophy of mathematics. In 1929 he wrote that “There is no religious denomination in which so much sin has been committed through the misuse of metaphorical expressions as in mathematics”. But what sins, and which misuses, was he criticizing in his writings on the philosophy of mathematics? Wittgenstein, (...)
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  9. Finitism and the Beginning of the Universe.Stephen Puryear - 2014 - Australasian Journal of Philosophy 92 (4):619-629.
    Many philosophers have argued that the past must be finite in duration because otherwise reaching the present moment would have involved something impossible, namely, the sequential occurrence of an actual infinity of events. In reply, some philosophers have objected that there can be nothing amiss in such an occurrence, since actually infinite sequences are ‘traversed’ all the time in nature, for example, whenever an object moves from one location in space to another. This essay focuses on one of the two (...)
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  10.  20
    The Unfolding of Non-Finitist Arithmetic.Solomon Feferman & Thomas Strahm - 2000 - Annals of Pure and Applied Logic 104 (1-3):75-96.
    The unfolding of schematic formal systems is a novel concept which was initiated in Feferman , Gödel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic . In particular, we examine two restricted unfoldings and , as well as a full unfolding, . The principal results then state: is equivalent to ; is equivalent to ; is equivalent to . Thus is proof-theoretically equivalent (...)
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  11.  53
    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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  12.  79
    Strict Finitism.Crispin Wright - 1982 - Synthese 51 (2):203 - 282.
    Dummett's objections to the coherence of the strict finitist philosophy of mathematics are thus, at the present time at least, ill-taken. We have so far no definitive treatment of Sorites paradoxes; so no conclusive ground for dismissing Dummett's response — the response of simply writing off a large class of familiar, confidently handled expressions as semantically incoherent. I believe that cannot be the right response, if only because it threatens to open an unacceptable gulf between the insight into his own (...)
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  13. 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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  14.  88
    Gödel on Intuition and on Hilbert's Finitism.W. W. Tait - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
    There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” (...)
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  15.  53
    A Defense of Strict Finitism.J. P. Van Bendegem - 2012 - Constructivist Foundations 7 (2):141-149.
    Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an (...)
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  16. Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program.Solomon Feferman - 2008 - Dialectica 62 (2):179-203.
    This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.
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  17. Strict Finitism Refuted?Ofra Magidor - 2007 - Proceedings of the Aristotelian Society 107 (1pt3):403-411.
    In his paper ‘Wang’s Paradox’, Michael Dummett provides an argument for why strict finitism in mathematics is internally inconsistent and therefore an untenable position. Dummett’s argument proceeds by making two claims: (1) Strict finitism is committed to the claim that there are sets of natural numbers which are closed under the successor operation but nonetheless have an upper bound; (2) Such a commitment is inconsistent, even by finitistic standards. -/- In this paper I claim that Dummett’s argument fails. (...)
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  18. Wittgenstein and Finitism.Mathieu Marion - 1995 - Synthese 105 (2):141 - 176.
    In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of (...)
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  19.  58
    Unfolding Finitist Arithmetic.Solomon Feferman & Thomas Strahm - 2010 - Review of Symbolic Logic 3 (4):665-689.
    The concept of the (full) unfolding of a schematic system is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted ? The program to determine for various systems of foundational significance was previously carried out for a system of nonfinitist arithmetic, ; it was shown that is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system (...)
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  20. Finitism, Divisibilty, and the Beginning of the Universe: Replies to Loke and Dumsday.Stephen Puryear - 2016 - Australasian Journal of Philosophy 94 (4):808-813.
    Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies.
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  21.  74
    Do the Right Thing! Rule Finitism, Rule Scepticism and Rule Following.Wes Sharrock & Graham Button - 1999 - Human Studies 22 (2-4):193-210.
    Rule following is often made an unnecessary mystery in the philosophy of social science. One form of mystification is the issue of 'rule finitism', which raises the puzzle as to how a learner can possibly extend the rule to applications beyond those examples which have been given as instruction in the rule. Despite the claim that this problem originated in the work of Wittgenstein, it is clear that his philosophical method is designed to evaporate, not perpetuate, such problems. The (...)
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  22.  38
    Extensions of the Finitist Point of View.Matthias Schirn & Karl-Georg Niebergall - 2001 - History and Philosophy of Logic 22 (3):135-161.
    Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...)
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  23.  30
    Kant and Finitism.W. W. Tait - 2016 - Journal of Philosophy 113 (5/6):261-273.
    An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism”, namely the idea of constructions f from ‘arbitrary’ or ‘generic’ objects of various types. The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number and arithmetic, he was not referring to (...)
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  24.  27
    On Tait on Kant and Finitism.W. Sieg - 2016 - Journal of Philosophy 113 (5/6):274-285.
    In his “Kant and Finitism” Tait attempts to connect his analysis of finitist arithmetic with Kant’s perspective on arithmetic. The examination of this attempt is the basis for a distinctive view on the dramatic methodological shift from Kant to Dedekind and Hilbert. Dedekind’s 1888 essay “Was sind und was sollen die Zahlen?” gives a logical analysis of arithmetic, whereas Hilbert’s 1899 book “Grundlagen der Geometrie” presents such an analysis of geometry or, as Hilbert puts it, of our spatial intuition. (...)
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  25. 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 (...)
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  26.  70
    A Defense of Strict Finitism.J. P. Bendegem - 2012 - Constructivist Foundations 7 (2):141-149.
    Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for (...)
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  27.  53
    The Philosophy of Strict Finitism.Ernest J. Welti - 1987 - Theoria 2 (2):575-582.
    The philosolphy of strict finitism is a research programme containing developmental theory and mathematics as its main branches. The first branch is concerned with the ontogenetic and historicaldevelopment of various concepts of infinity. The frame work is Jean Piaget’s genetic epistemology. Based upon these develop mental studies, the mathematical branch introduces a new concept of infinity into mathematics. Cantor propagated the actual infinite, Brouwer and the constructivists the potential infinite. Still more radical is strict finitism, favoring the natural (...)
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  28.  7
    Finitist Set Theory in Ontological Modeling.Avril Styrman & Aapo Halko - 2018 - Applied Ontology 13 (2):107-133.
    This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as (...)
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  29.  70
    A Middle Position Between Meaning Finitism and Meaning Platonism.Jussi Haukioja - 2005 - International Journal of Philosophical Studies 13 (1):35 – 51.
    David Bloor and Crispin Wright have argued, independently, that the proper lesson to draw from Wittgenstein's so-called rule-following considerations is the rejection of meaning Platonism. According to Platonism, the meaningfulness of a general term is constituted by its connection with an abstract entity, the (possibly) infinite extension of which is determined independently of our classificatory practices. Having rejected Platonism, both Bloor and Wright are driven to meaning finitism, the view that the question of whether a meaningful term correctly applies (...)
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  30. Finitism = PRA? On a Thesis of W.W. Tait.Matthias Schirn & Karl-Georg Niebergall - 2005 - Reports on Mathematical Logic:3-24.
    In his paper `Finitism' , W.W.~Tait maintained that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argued that all finitist reasoning is essentially primitive recursive. In our paper, we attempt to show that his thesis ``The finitist functions are precisely the primitive recursive functions'' is disputable and that another, likewise defended by (...)
     
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  31.  39
    Dummett's Intuitionism is Not Strict Finitism.Samuel William Mitchell - 1992 - Synthese 90 (3):437 - 458.
    Michael Dummett's anti-realism is founded on the semantics of natural language which, he argues, can only be satisfactorily given in mathematics by intuitionism. It has been objected that an analog of Dummett's argument will collapse intuitionism into strict finitism. My purpose in this paper is to refute this objection, which I argue Dummett does not successfully do. I link the coherence of strict finitism to a view of confirmation — that our actual practical abilities cannot confirm we know (...)
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  32. Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.
  33.  82
    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 (...)
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  34.  24
    Wittgenstein, Finitism, and the Foundations of Mathematics.Paolo Mancosu & Mathieu Marion - 2001 - Philosophical Review 110 (2):286.
    It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. In the (...)
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  35.  14
    Mechanism, Mentalism and Metamathematics: An Essay on Finitism.Judson Webb - 1980 - Kluwer Academic Publishers.
  36.  52
    Wittgenstein, Finitism, and the Foundations of Mathematics.Mathieu Marion - 1998 - Oxford University Press.
    This pioneering book demonstrates the crucial importance of Wittgenstein's philosophy of mathematics to his philosophy as a whole. Marion traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. In particular, he illuminates the work of the neglected 'transitional period' between the Tractatus and the Investigations.
  37.  21
    Quine’s “Predilection” for Finitism.Gary Ebbs - 2016 - Metascience 25 (1):31-36.
  38. 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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  39.  20
    Mechanism, Mentalism and Metamathematics: An Essay on Finitism.Stewart Shapiro & Judson Chambers Webb - 1980 - Journal of Symbolic Logic 51 (2):472.
  40.  42
    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 (...)
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  41.  32
    Finitism and Divisibility: A Reply to Puryear.Travis Dumsday - 2016 - Australasian Journal of Philosophy 94 (3):596-601.
    Puryear develops an objection against a prominent attempt to show that the universe must have a temporal beginning. Here I formulate a reply.
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  42.  26
    More Infinity for a Better Finitism.Sam Sanders - 2010 - Annals of Pure and Applied Logic 161 (12):1525-1540.
    Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed in around 1995 by Patrick Suppes and Richard Sommer. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes. Here, we discuss the inherent problems and limitations of the classical nonstandard framework and propose a much-needed refinement of ERNA, called , in the spirit of Karel Hrbacek’s stratified set theory. We study the metamathematics of and its (...)
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  43.  35
    On the Interpretation of Non-Finitist Proofs–Part II.G. Kreisel - 1952 - Journal of Symbolic Logic 17 (1):43-58.
  44.  21
    Finitism in Geometry.Jean-Paul Van Bendegem - 2008 - Stanford Encyclopedia of Philosophy.
  45.  17
    On Finitism and the Beginning of the Universe: A Reply to Stephen Puryear.Andrew Ter Ern Loke - 2016 - Australasian Journal of Philosophy 94 (3):591-595.
    ABSTRACTStephen Puryear argues that William Lane Craig's view, that time as duration is logically prior to the potentially infinite divisions that we make of it, involves the idea that time is prior to any parts we conceive within it. He objects that PWT entails the Priority of the Whole with respect to Events, and that it subverts the argument, used by proponents of the Kalam Cosmological Argument such as Craig, against an eternal past based on the impossibility of traversing an (...)
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  46.  9
    Nominalistic Ordinals, Recursion on Higher Types, and Finitism.Maria Hämeen-Anttila - 2019 - Bulletin of Symbolic Logic 25 (1):101-124.
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  47.  25
    Strict Finitism, Feasibility, and the Sorites.Walter Dean - 2018 - Review of Symbolic Logic 11 (2):295-346.
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  48. Hilbert's Finitism and the Notion of Infinity.Karl-Georg Niebergall & Matthias Schirn - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
  49. Counting Steps: A Finitist Interpretation of Objective Probability in Physics.Amit Hagar & Giuseppe Sergioli - 2015 - Epistemologia 37 (2):262-275.
    We propose a new interpretation of objective deterministic chances in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set--up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the 'distance' (in terms of physical resources) from a physical state to another, and (3) the size of the set of time--complexity functions that are compatible with the physical (...)
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  50.  37
    On the Interpretation of Non-Finitist Proofs—Part I.G. Kreisel - 1951 - Journal of Symbolic Logic 16 (4):241-267.
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