Switch to: References

Citations of:

From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s

Oxford, England: Oxford University Press USA (1997)

Add citations

You must login to add citations.
  1. The development of mathematical logic from Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2009 - In Leila Haaparanta (ed.), The development of modern logic. New York: Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
    Download  
     
    Export citation  
     
    Bookmark   28 citations  
  • (1 other version)Abstract objects.Gideon Rosen - 2008 - Stanford Encyclopedia of Philosophy.
    Download  
     
    Export citation  
     
    Bookmark   93 citations  
  • The pragmatism of Hilbert's programme.Volker Peckhaus - 2003 - Synthese 137 (1-2):141 - 156.
    It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style (...)
    Download  
     
    Export citation  
     
    Bookmark   7 citations  
  • Shaping the Enemy: Foundational Labelling by L.E.J. Brouwer and A. Heyting.Miriam Franchella - 2018 - History and Philosophy of Logic 40 (2):152-181.
    The use of the three labels to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor the...
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • The philosophical background of Weyl's mathematical constructivism.Richard Tieszen - 2000 - Philosophia Mathematica 8 (3):274-301.
    Weyl's inclination toward constructivism in the foundations of mathematics runs through his entire career, starting with Das Kontinuum. Why was Weyl inclined toward constructivism? I argue that Weyl's general views on foundations were shaped by a type of transcendental idealism in which it is held that mathematical knowledge must be founded on intuition. Kant and Fichte had an impact on Weyl but HusserFs transcendental idealism was even more influential. I discuss Weyl's views on vicious circularity, existence claims, meaning, the continuum (...)
    Download  
     
    Export citation  
     
    Bookmark   17 citations  
  • Intuition between the analytic-continental divide: Hermann Weyl's philosophy of the continuum.Janet Folina - 2008 - Philosophia Mathematica 16 (1):25-55.
    Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but his (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Information, possible worlds and the cooptation of scepticism.Luciano Floridi - 2010 - Synthese 175 (1):63 - 88.
    The article investigates the sceptical challenge from an informationtheoretic perspective. Its main goal is to articulate and defend the view that either informational scepticism is radical, but then it is epistemologically innocuous because redundant; or it is moderate, but then epistemologically beneficial because useful. In order to pursue this cooptation strategy, the article is divided into seven sections. Section 1 sets up the problem. Section 2 introduces Borei numbers as a convenient way to refer uniformly to (the data that individuate) (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • (1 other version)Hermann Weyl on intuition and the continuum.John L. Bell - 2000 - Philosophia Mathematica 8 (3):259-273.
    Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.
    Download  
     
    Export citation  
     
    Bookmark   13 citations  
  • (1 other version)Weyl’s Appropriation of Husserl’s and Poincar“s Thought.Richard Feist - 2002 - Synthese 132 (3):273 - 301.
    This article locates Weyl''s philosophy of mathematics and its relationship to his philosophy of science within the epistemological and ontological framework of Husserl''s phenomenology as expressed in the Logical Investigations and Ideas. This interpretation permits a unified reading of Weyl''s scattered philosophical comments in The Continuum and Space-Time-Matter. But the article also indicates that Weyl employed Poincaré''s predicativist concerns to modify Husserl''s semantics and trim Husserl''s ontology. Using Poincaré''s razor to shave Husserl''s beard leads to limitations on the least upper (...)
    Download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Inquisitive Logic.Ivano Ciardelli & Floris Roelofsen - 2011 - Journal of Philosophical Logic 40 (1):55-94.
    This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).
    Download  
     
    Export citation  
     
    Bookmark   62 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   37 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  
  • Brouwer, as never read by Husserl.Mark van Atten - 2003 - Synthese 137 (1-2):3-19.
    Even though Husserl and Brouwer have never discussed each other's work, ideas from Husserl have been used to justify Brouwer's intuitionistic logic. I claim that a Husserlian reading of Brouwer can also serve to justify the existence of choice sequences as objects of pure mathematics. An outline of such a reading is given, and some objections are discussed.
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • How are Concepts of Infinity Acquired?Kazimierz Trzęsicki - 2015 - Studies in Logic, Grammar and Rhetoric 40 (1):179-217.
    Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.
    Download  
     
    Export citation  
     
    Bookmark  
  • Structuring Co-constructive Logic for Proofs and Refutations.James Trafford - 2016 - Logica Universalis 10 (1):67-97.
    This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Co-constructive logic for proofs and refutations.James Trafford - 2014 - Studia Humana 3 (4):22-40.
    This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought.Sören Stenlund - 2015 - Nordic Wittgenstein Review 4 (1):7-92.
    The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...)
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • 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  
  • 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  
  • Carnap’s Early Semantics.Georg Schiemer - 2013 - Erkenntnis 78 (3):487-522.
    This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...)
    Download  
     
    Export citation  
     
    Bookmark   14 citations  
  • Carnap’s early metatheory: scope and limits.Georg Schiemer, Richard Zach & Erich Reck - 2017 - Synthese 194 (1):33-65.
    In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • Intuitionistic logic versus paraconsistent logic. Categorical approach.Mariusz Kajetan Stopa - 2023 - Dissertation, Jagiellonian University
    The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the context (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Análisis de la relación entre el continuo intuitivo y el matemático en "Das Kontinuum".Victor Gonzalez Rojo - 2021 - Revista de Filosofía 46 (2):255-270.
    En este artículo pretendo discutir la conclusión a la que llega Weyl en su libro _El continuo_ sobre la relación entre el continuo intuitivo y el matemático. Esto me sirve a su vez para analizar más profundamente estas ideas, y postular la propiedad de ausencia de espacios vacíos [_Lückenlosigkeit_] como fundamento del continuo intuitivo y, en consecuencia, del matemático. Proponiendo además una alternativa idealista para el tratamiento del problema del continuo.
    Download  
     
    Export citation  
     
    Bookmark  
  • Carnap's Tolerance and Friedman's Revenge.Noah Friedman-Biglin - 2015 - In Pavel Arazim & Michal Dancak (eds.), Logica Yearbook 2014. College Publications. pp. 109 -- 125.
    In this paper, I defend Rudolf Carnap's Principle of Tolerance from an accusation, due to Michael Friedman, that it is self-defeating by prejudicing any debate towards the logically stronger theory. In particular, Friedman attempts to show that Carnap's reconstruction of the debate between classicists and intuitionists over the foundations of mathematics in his book The Logical Syntax of Language, is biased towards the classical standpoint since the metalanguage he constructs to adjudicate between the rival positions is fully classical. I argue (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Hilbert's 'Verunglückter Beweis', the first epsilon theorem, and consistency proofs.Richard Zach - 2004 - History and Philosophy of Logic 25 (2):79-94.
    In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain "general consistency result" due to Bernays. An analysis of the form of this so-called "failed proof" (...)
    Download  
     
    Export citation  
     
    Bookmark   10 citations  
  • Labyrinth of Continua.Patrick Reeder - 2018 - Philosophia Mathematica 26 (1):1-39.
    This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Naturalism in mathematics and the authority of philosophy.Alexander Paseau - 2005 - British Journal for the Philosophy of Science 56 (2):377-396.
    Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the (...)
    Download  
     
    Export citation  
     
    Bookmark   16 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  
  • Wittgenstein and Brouwer.Mathieu Marion - 2003 - Synthese 137 (1-2):103 - 127.
    In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwer's lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgenstein's position, in theTractactus, was already quite close to (...)
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • The Russellian influence on Hilbert and his school.Paolo Mancosu - 2003 - Synthese 137 (1-2):59 - 101.
    The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources (...)
    Download  
     
    Export citation  
     
    Bookmark   12 citations  
  • Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl.Paolo Mancosu & T. A. Ryckman - 2002 - Philosophia Mathematica 10 (2):130-202.
    Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...)
    Download  
     
    Export citation  
     
    Bookmark   24 citations  
  • Replies.Øystein Linnebo - 2023 - Theoria 89 (3):393-406.
    Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • 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  
  • (1 other version)On The Epistemological Justification of Hilbert’s Metamathematics.Javier Legris - 2005 - Philosophia Scientiae 9 (2):225-238.
    The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Continuum, name and paradox.Vojtěch Kolman - 2010 - Synthese 175 (3):351 - 367.
    The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Undetachability of Propositional Content and Its Process of Construction: Another Aspect of Brouwer's Intuitionism.Hiroshi Kaneko - 2006 - Annals of the Japan Association for Philosophy of Science 14 (2):101-112.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Brouwer's conception of language, mind and mathematics'.Hiroshi Kaneko - 2002 - Annals of the Japan Association for Philosophy of Science 11 (1):35-49.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • International Handbook of Research in History, Philosophy and Science Teaching.Michael R. Matthews (ed.) - 2014 - Springer.
    This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...)
    Download  
     
    Export citation  
     
    Bookmark   14 citations  
  • Truth, reflection, and hierarchies.Michael Glanzberg - 2005 - Synthese 142 (3):289 - 315.
    A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unprob-lematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of (...)
    Download  
     
    Export citation  
     
    Bookmark   15 citations  
  • Conceptions of the continuum.Solomon Feferman - unknown
    Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.
    Download  
     
    Export citation  
     
    Bookmark   26 citations  
  • Some Logical Notations for Pragmatic Assertions.Massimiliano Carrara, Daniele Chiffi & Ahti-Veikko Pietarinen - 2020 - Logique Et Analyse 251:297 - 315.
    The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of these (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Gödel's Incompleteness Theorems.Panu Raatikainen - 2013 - The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (Ed.).
    Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot (...)
    Download  
     
    Export citation  
     
    Bookmark   22 citations  
  • Syntax meets semantics during brain logical computations.Arturo Tozzi, James F. Peters, Andrew And Alexander Fingelkurts & Leonid Perlovsky - 2018 - Progress in Biophysics and Molecular Biology 140:133-141.
    The discrepancy between syntax and semantics is a painstaking issue that hinders a better comprehension of the underlying neuronal processes in the human brain. In order to tackle the issue, we at first describe a striking correlation between Wittgenstein's Tractatus, that assesses the syntactic relationships between language and world, and Perlovsky's joint language-cognitive computational model, that assesses the semantic relationships between emotions and “knowledge instinct”. Once established a correlation between a purely logical approach to the language and computable psychological activities, (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • The Theory of Relations, Complex Terms, and a Connection Between λ and ε Calculi.Edward N. Zalta - manuscript
    This paper introduces a new method of interpreting complex relation terms in a second-order quantified modal language. We develop a completely general second-order modal language with two kinds of complex terms: one kind for denoting individuals and one kind for denoting n-place relations. Several issues arise in connection with previous, algebraic methods for interpreting the relation terms. The new method of interpreting these terms described here addresses those issues while establishing an interesting connection between λ and ε calculi. The resulting (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • (1 other version)Predicativity.Solomon Feferman - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford and New York: Oxford University Press. pp. 590-624.
    What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...)
    Download  
     
    Export citation  
     
    Bookmark   30 citations  
  • The proper explanation of intuitionistic logic: on Brouwer's demonstration of the Bar Theorem.Mark van Atten & Göran Sundholm - 2008 - In Mark van Atten, Pascal Boldini, Michel Bourdeau & Gerhard Heinzmann (eds.), One Hundred Years of Intuitionism : The Cerisy Conference. Birkhäuser Basel. pp. 60-77.
    Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.
    Download  
     
    Export citation  
     
    Bookmark   7 citations  
  • .[author unknown] - unknown
    Download  
     
    Export citation  
     
    Bookmark