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The Foundations of Mathematics

In ¸ Itevanheijenoort1967. Harvard University Press (1927)

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  1. On what Hilbert aimed at in the foundations.Besim Karakadılar - manuscript
    Hilbert's axiomatic approach was an optimistic take over on the side of the logical foundations. It was also a response to various restrictive views of mathematics supposedly bounded by the reaches of epistemic elements in mathematics. A complete axiomatization should be able to exclude epistemic or ontic elements from mathematical theorizing, according to Hilbert. This exclusion is not necessarily a logicism in similar form to Frege's or Dedekind's projects. That is, intuition can still have a role in mathematical reasoning. Nevertheless, (...)
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  • (12 other versions)Отвъд машината на Тюринг: квантовият компютър.Vasil Penchev - 2014 - Sofia: BAS: ISSK (IPS).
    Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer might (...)
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  • Wittgenstein’s Critique of the Additive Conception of Language.James F. Conant - 2020 - Nordic Wittgenstein Review 9.
    This paper argues that Wittgenstein, both early and late, rejects the idea that the logically simpler and more fundamental case is that of "the mere sign" and that what a meaningful symbol is can be explained through the elaboration of an appropriately supplemented conception of the sign: the sign plus something. Rather the sign, in the logically fundamental case of its mode of occurrence, is an internal aspect of the symbol. The Tractatus puts this point as follows: “The sign is (...)
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  • Diagrams in Mathematics.Carlo Cellucci - 2019 - Foundations of Science 24 (3):583-604.
    In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...)
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  • The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas' Goedelian Thesis.Bhupinder Singh Anand - 2016 - Cognitive Systems Research 40:35-45.
    We consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth (...)
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  • Is Mathematics Problem Solving or Theorem Proving?Carlo Cellucci - 2017 - Foundations of Science 22 (1):183-199.
    The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...)
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  • 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 (...)
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  • Hilbert's Objectivity.Lydia Patton - 2014 - Historia Mathematica 41 (2):188-203.
    Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly "meaningless" signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl (1949); Kitcher (1976)). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates (...)
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  • Category theory and the foundations of mathematics: Philosophical excavations.Jean-Pierre Marquis - 1995 - Synthese 103 (3):421 - 447.
    The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 (...)
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  • Arguing on the Toulmin Model: New Essays in Argument Analysis and Evaluation.David Hitchcock & Bart Verheij (eds.) - 2006 - Dordrecht, Netherland: Springer.
    In The Uses of Argument, Stephen Toulmin proposed a model for the layout of arguments: claim, data, warrant, qualifier, rebuttal, backing. Since then, Toulmin’s model has been appropriated, adapted and extended by researchers in speech communications, philosophy and artificial intelligence. This book assembles the best contemporary reflection in these fields, extending or challenging Toulmin’s ideas in ways that make fresh contributions to the theory of analysing and evaluating arguments.
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  • Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method.Carlo Cellucci - 2013 - Dordrecht, Netherland: Springer.
    This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...)
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  • 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 (...)
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  • Does Homotopy Type Theory Provide a Foundation for Mathematics?James Ladyman & Stuart Presnell - 2016 - British Journal for the Philosophy of Science:axw006.
    Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be (...)
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  • Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...)
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  • Knowledge of Mathematics without Proof.Alexander Paseau - 2015 - British Journal for the Philosophy of Science 66 (4):775-799.
    Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...)
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  • On the imaginative constructivist nature of design: a theoretical approach.Akin Osman Kazakci - unknown
    Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in the current design research. By contrast, in other fields such as Mathematics or Psychology, the notion of constructivism is seen as a foundational issue. The present paper defends the point of view that forms of constructivism in design need to be identified and integrated as (...)
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  • On How Logic Became First-Order.Matti Eklund - 1996 - Nordic Journal of Philosophical Logic 1 (2):147-67.
    Added by a category editor--not an official abstract. -/- Discusses the history (and reasons for the history) implicit in the title, as well as the author's view on same.
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  • Formalizing Medieval Logical Theories: Suppositio, Consequentiae and Obligationes.Catarina Dutilh Novaes - 2007 - Dordrecht, Netherland: Springer.
    This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...)
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  • On an alleged refutation of Hilbert's program using gödel's first incompleteness theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  • Advances in Natural Deduction: A Celebration of Dag Prawitz's Work.Luiz Carlos Pereira & Edward Hermann Haeusler (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...)
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  • Walter Dubislav’s Philosophy of Science and Mathematics.Nikolay Milkov - 2016 - Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (1):96-116.
    Walter Dubislav (1895–1937) was a leading member of the Berlin Group for scientific philosophy. This “sister group” of the more famous Vienna Circle emerged around Hans Reichenbach’s seminars at the University of Berlin in 1927 and 1928. Dubislav was to collaborate with Reichenbach, an association that eventuated in their conjointly conducting university colloquia. Dubislav produced original work in philosophy of mathematics, logic, and science, consequently following David Hilbert’s axiomatic method. This brought him to defend formalism in these disciplines as well (...)
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  • Epistemic truth and excluded middle.Cesare Cozzo - 1998 - Theoria 64 (2-3):243-282.
    Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution.
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  • On the unreasonable reliability of mathematical inference.Brendan Philip Larvor - 2022 - Synthese 200 (4):1-16.
    In, Jeremy Avigad makes a novel and insightful argument, which he presents as part of a defence of the ‘Standard View’ about the relationship between informal mathematical proofs and their corresponding formal derivations. His argument considers the various strategies by means of which mathematicians can write informal proofs that meet mathematical standards of rigour, in spite of the prodigious length, complexity and conceptual difficulty that some proofs exhibit. He takes it that showing that and how such strategies work is a (...)
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  • Pluralism in Mathematics: A New Position in Philosophy of Mathematics.Michèle Friend - 2013 - Dordrecht, Netherland: Springer.
    The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...)
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  • On the Relationship between Parts and Wholes in Husserl's Phenomenology.Ettore Casari - 2007 - In Luciano Boi, Pierre Kerszberg & Frédéric Patras (eds.), Rediscovering Phenomenology. Phenomenological Essays on Mathematical Beings, Physical Reality, Perception and Consciousness. Hal Ccsd. pp. 67-102.
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  • Why history matters to philosophy of physics.Thomas Ryckman - 2015 - Studies in History and Philosophy of Science Part A 50:4-12.
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  • Reflections on gödel's and Gandy's reflections on Turing's thesis.David Israel - 2002 - Minds and Machines 12 (2):181-201.
    We sketch the historical and conceptual context of Turing's analysis of algorithmic or mechanical computation. We then discuss two responses to that analysis, by Gödel and by Gandy, both of which raise, though in very different ways. The possibility of computation procedures that cannot be reduced to the basic procedures into which Turing decomposed computation. Along the way, we touch on some of Cleland's views.
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  • On computable numbers with an application to the AlanTuringproblem.C. F. Huws & J. C. Finnis - 2017 - Artificial Intelligence and Law 25 (2):181-203.
    This paper explores the question of whether or not the law is a computable number in the sense described by Alan Turing in his 1937 paper ‘On computable numbers with an application to the Entscheidungsproblem.’ Drawing upon the legal, social, and political context of Alan Turing’s own involvement with the law following his arrest in 1952 for the criminal offence of gross indecency, the article explores the parameters of computability within the law and analyses the applicability of Turing’s computability thesis (...)
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  • Confronting Ideals of Proof with the Ways of Proving of the Research Mathematician.Norma B. Goethe & Michèle Friend - 2010 - Studia Logica 96 (2):273-288.
    In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.
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  • Why should we abandon the mental logic hypothesis?Luca Bonatti - 1994 - Cognition 50 (1-3):17-39.
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  • On Bourbaki’s axiomatic system for set theory.Maribel Anacona, Luis Carlos Arboleda & F. Javier Pérez-Fernández - 2014 - Synthese 191 (17):4069-4098.
    In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom (...)
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  • 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.
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  • Formalizations après la lettre: Studies in Medieval Logic and Semantics.Catarina Dutilh Novaes - 2006 - Dissertation, Leiden University
    This thesis is on the history and philosophy of logic and semantics. Logic can be described as the ‘science of reasoning’, as it deals primarily with correct patterns of reasoning. However, logic as a discipline has undergone dramatic changes in the last two centuries: while for ancient and medieval philosophers it belonged essentially to the realm of language studies, it has currently become a sub-branch of mathematics. This thesis attempts to establish a dialogue between the modern and the medieval traditions (...)
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  • La méthode axiomatique durant la crise des fondements.Mathieu Bélanger - 2013 - In . Les Cahiers D'Ithaque.
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  • logicism, intuitionism, and formalism - What has become of them?Sten Lindstr©œm, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.) - 2008 - Berlin, Germany: Springer.
    The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...)
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