70 found
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  1. Dedekind’s Analysis of Number: Systems and Axioms.Wilfried Sieg & Dirk Schlimm - 2005 - Synthese 147 (1):121-170.
    Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.
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  2.  40
    Fragments of Arithmetic.Wilfried Sieg - 1985 - Annals of Pure and Applied Logic 28 (1):33-71.
    We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...)
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  3. Hilbert's Programs: 1917–1922.Wilfried Sieg - 1999 - Bulletin of Symbolic Logic 5 (1):1-44.
    Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...)
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  4.  82
    Mechanical Procedures and Mathematical Experience.Wilfried Sieg - 1994 - In Alexander George (ed.), Mathematics and Mind. Oxford University Press. pp. 71--117.
    Wilfred Sieg. Mechanical Procedures and Mathematical Experience.
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  5.  26
    Hilbert's Programs and Beyond.Wilfried Sieg - 2013 - Oup Usa.
    David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.
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  6.  42
    The Ways of Hilbert's Axiomatics: Structural and Formal.Wilfried Sieg - 2014 - Perspectives on Science 22 (1):133-157.
    Hilbert gave lectures on the foundations of mathematics throughout his career. Notes for many of them have been preserved and are treasures of information; they allow us to reconstruct the path from Hilbert's logicist position, deeply influenced by Dedekind and presented in lectures starting around 1890, to the program of finitist proof theory in the early 1920s. The development toward proof theory begins, in some sense, in 1917 when Hilbert gave his talk Axiomatisches Denken in Zürich. This talk is rooted (...)
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  7.  10
    Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf.Wilfried Sieg & Patrick Walsh - forthcoming - Review of Symbolic Logic:1-44.
    Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame (...)
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  8.  31
    Dedekind’s Structuralism: Creating Concepts and Deriving Theorems.Wilfried Sieg & Rebecca Morris - 2018 - In Erich Reck (ed.), Logic, Philosophy of Mathematics, and their History: Essays in Honor W.W. Tait. College Publications.
    Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...)
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  9.  49
    Step by Recursive Step: Church's Analysis of Effective Calculability.Wilfried Sieg - 1997 - Bulletin of Symbolic Logic 3 (2):154-180.
    Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...)
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  10. An Abstract Model For Parallel Computations: Gandy’s Thesis.Wilfried Sieg & John Byrnes - 1999 - The Monist 82 (1):150-164.
    Wilfried Sieg and John Byrnes. AnModel for Parallel Computation: Gandy's Thesis.
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  11.  22
    Herbrand Analyses.Wilfried Sieg - 1991 - Archive for Mathematical Logic 30 (5-6):409-441.
    Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is to show (...)
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  12.  40
    Church Without Dogma: Axioms for Computability.Wilfried Sieg - unknown
    Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work (...)
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  13.  6
    Gödel’s Philosophical Challenge.Wilfried Sieg - 2020 - Studia Semiotyczne 34 (1):57-80.
    The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that (...)
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  14. Relative Consistency and Accessible Domains.Wilfried Sieg - 1990 - Synthese 84 (2):259 - 297.
    Wilfred Sieg. Relative Consistency and Accesible Domains.
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  15.  25
    Calculations by Man and Machine: Conceptual Analysis.Wilfried Sieg - unknown
    Wilfried Sieg. Calculations by Man and Machine: Conceptual Analysis.
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  16.  66
    Hilbert's Program Sixty Years Later.Wilfried Sieg - 1988 - Journal of Symbolic Logic 53 (2):338-348.
  17.  67
    Dedekind's Abstract Concepts: Models and Mappings.Wilfried Sieg & Dirk Schlimm - 2014 - Philosophia Mathematica:nku021.
    Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts, models, and mappings.
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  18. Only Two Letters: The Correspondence Between Herbrand and Gödel.Wilfried Sieg - 2005 - Bulletin of Symbolic Logic 11 (2):172-184.
    Two young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Gödel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Bérarde (Isère). Herbrand's letter played a significant role in the development of computability theory. Gödel asserted in his 1934 Princeton Lectures and on later occasions that it suggested to him a crucial part of (...)
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  19.  64
    K-Graph Machines: Generalizing Turing's Machines and Arguments.Wilfried Sieg & John Byrnes - unknown
    Wilfred Sieg and John Byrnes. K-Graph Machines: Generalizing Turing's Machines and Arguments.
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  20.  11
    Natural Logic.Wilfried Sieg - 1983 - Journal of Symbolic Logic 48 (1):215-217.
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  21.  37
    Searching for Proofs.Wilfried Sieg & Richard Scheines - unknown
    The Carnegie Mellon Proof Tutor project was motivated by pedagogical concerns: we wanted to use a "mechanical" (i.e. computerized) tutor for teaching students..
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  22.  59
    Foundations for Analysis and Proof Theory.Wilfried Sieg - 1984 - Synthese 60 (2):159 - 200.
  23.  69
    Normal Natural Deduction Proofs (in Classical Logic).Wilfried Sieg & John Byrnes - 1998 - Studia Logica 60 (1):67-106.
    Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? This informal (...)
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  24.  11
    Fragments of Arithmetic.Wilfried Sieg - 1987 - Journal of Symbolic Logic 52 (4):1054-1055.
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  25.  8
    The AProS Project: Strategic Thinking & Computational Logic.Wilfried Sieg - 2007 - Logic Journal of the IGPL 15 (4):359-368.
    The paper discusses tools for teaching logic used in Logic & Proofs, a web-based introduction to modern logic that has been taken by more than 1,300 students since the fall of 2003. The tools include a wide array of interactive learning environments or cognitive mini-tutors; most important among them is the Carnegie Proof Lab. The Proof Lab is a sophisticated interface for constructing natural deduction proofs and is central, as strategically guided discovery of proofs is the distinctive focus of the (...)
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  26.  33
    Note by the Guest Editors.Wilfried Sieg & Frank Pfenning - 1998 - Studia Logica 60 (1):1-1.
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  27.  19
    Calculations by Man and Machine: Mathematical Presentation.Wilfried Sieg - unknown
    Wilfried Sieg. Calculations by Man and Machine: Mathematical Presentation.
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  28.  36
    Proof Theory.Wilfried Sieg - unknown
  29.  7
    Hilbert's Proof Theory.Wilfried Sieg - 2009 - In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier. pp. 5--321.
  30.  40
    Unification For Quantified Formulae.Wilfried Sieg - unknown
    — via appropriate substitutions — syntactically identical. The method can be applied directly to quantifierfree formulae and, in this paper, will b e extended in a natural and strai ghlforward way to quantified formulae.
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  31.  41
    Review: Stephen G. Simpson, Friedman's Research on Subsystems of Second Order Arithmetic. [REVIEW]Wilfried Sieg - 1990 - Journal of Symbolic Logic 55 (2):870-874.
  32.  23
    Normal Natural Deduction Proof (In Non-Classical Logics).Wilfried Sieg & Saverio Cittadini - unknown
    Wilfred Sieg and Saverio Cittadini. Normal Natural Deduction Proof (In Non-Classical Logics.
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  33.  23
    Generalizing Turing's Machine and Arguments.Wilfried Sieg & John Byrnes - unknown
    Wilfred Sieg and John Byrnes. Generalizing Turing's Machine and Arguments.
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  34.  24
    Computability Theory.Daniele Mundici & Wilfried Sieg - unknown
    Daniele Mundici and Wilfred Sieg. Computability Theory.
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  35.  22
    Program Transformation and Proof Transformation.Wilfried Sieg & Stanley S. Wainer - unknown
    Wilfred Sieg and Stanley S. Wainer. Program Transformation and Proof Transformation.
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  36.  10
    Mechanisms and Search: Aspects of Proof Theory.Wilfried Sieg - unknown
    Wilfred Sieg. Mechanisms and Search: Aspects of Proof Theory.
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  37.  7
    Reductions of Theories for Analysis.Wilfried Sieg, Georg Dorn & P. Weingartner - 1990 - Journal of Symbolic Logic 55 (1):354-354.
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  38.  18
    Of the Association for Symbolic Logic.Sergei Artemov, Peter Koellner, Michael Rabin, Jeremy Avigad, Wilfried Sieg, William Tait & Haim Gaifman - 2006 - Bulletin of Symbolic Logic 12 (3-4):503.
  39.  15
    The Hilton New York Hotel New York, NY December 27–29, 2005.Sergei Artemov, Peter Koellner, Michael Rabin, Jeremy Avigad, Wilfried Sieg, William Tait & Haim Gaifman - 2006 - Bulletin of Symbolic Logic 12 (3).
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  40. A Graphical Presentation of Gandy's Parallel Machines'.John Byrnes & Wilfried Sieg - 1996 - Bulletin of Symbolic Logic 2:452-3.
  41.  4
    Feferman on Foundations.Gerhard Jäger & Wilfried Sieg (eds.) - 2017 - Springer.
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  42. Kurt Gödel Collected Works IV-V: Correspondence.Solomon Feferman, John W. Dawson, Warren Goldfarb, Charles Parsons & Wilfried Sieg - 2004 - Bulletin of Symbolic Logic 10 (4):558-563.
  43. Collected Works. Vol. IV: Correspondence A-G. Vol. V: Correspondence H-Z.Kurt Gödel, Solomon Feferman, John W. Dawson, Warren Goldfarb, Charles Parsons & Wilfried Sieg - 2004 - Tijdschrift Voor Filosofie 66 (1):165-166.
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  44.  37
    A Symposium on Hilbert's Program.Wilfrid Hodges & Wilfried Sieg - 1988 - Journal of Symbolic Logic 53 (2):337.
  45.  33
    Mathematics Studies Machines.Daniele Mundici & Wilfried Sieg - unknown
    Machines were introduced as calculating devices to simulate operations carried out by human computors following fixed algorithms: this is true for the early mechanical calculators devised by Pascal and Leibniz, for the analytical engine built by Babbage, and the theoretical machines introduced by Turing. The distinguishing feature of the latter is their universality: They are claimed to be able to capture any algorithm whatsoever and, conversely, any procedure they can carry out is evidently algorithmic. The study of such "paper machines" (...)
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  46.  23
    In Memoriam: Solomon Feferman.Charles Parsons & Wilfried Sieg - 2017 - Bulletin of Symbolic Logic 23 (3):337-344.
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  47.  5
    An Experimental Comparison of Alternative Proof Construction Environments.Richard Scheines & Wilfried Sieg - unknown
    : "In this paper we compare computerized environments in which students complete proof construction exercises in formal logic. Afterbeing given a pretest for logical aptitude, three matched groups were presented identical course material on logic for approximately five weeks by a computer. During the treatment, all students were required to complete several hundred proof construction exercises. The three groups did the exercises and the midterm in different environments. The group with a more sophisticated interface performed better on the midterm. Nearly (...)
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  48.  28
    Computer Environments for Proof Construction.Richard Scheines & Wilfried Sieg - unknown
    Richard Scheines and Wilfred Sieg. Computer Environments for Proof Construction.
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  49. Acting and Reflecting the Interdisciplinary Turn in Philosophy.Wilfried Sieg - 1990
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  50. A Brief Note on Gödel, Nagel, Minds, and Machines.Wilfried Sieg - 2017 - In G. Jaeger & W. Sieg (eds.), Feferman on Foundations. Springer.
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