62 found
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  1. Wilfried Sieg (1999). Hilbert's Programs: 1917-1922. 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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  2.  8
    Wilfried Sieg (1985). Fragments of Arithmetic. 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.  18
    Wilfried Sieg & Dirk Schlimm (2014). Dedekind's Abstract Concepts: Models and Mappings. 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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  4. Wilfried Sieg (2005). Only Two Letters: The Correspondence Between Herbrand and Gödel. 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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  5.  76
    Wilfried Sieg & Dirk Schlimm (2005). Dedekind's Analysis of Number: Systems and Axioms. Synthese 147 (1):121 - 170.
    Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.
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  6.  27
    Wilfried Sieg (1994). Mechanical Procedures and Mathematical Experience. In Alexander George (ed.), Mathematics and Mind. Oxford University Press 71--117.
    Wilfred Sieg. Mechanical Procedures and Mathematical Experience.
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  7.  7
    Wilfried Sieg (1991). Herbrand Analyses. 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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  8.  24
    Wilfried Sieg & John Byrnes (1999). An Abstract Model For Parallel Computations: Gandy’s Thesis. The Monist 82 (1):150-164.
    Wilfried Sieg and John Byrnes. AnModel for Parallel Computation: Gandy's Thesis.
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  9.  23
    Wilfried Sieg, Church Without Dogma: Axioms for Computability.
    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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  10.  12
    Wilfried Sieg, Calculations by Man and Machine: Conceptual Analysis.
    Wilfried Sieg. Calculations by Man and Machine: Conceptual Analysis.
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  11.  27
    Wilfried Sieg (1997). Step by Recursive Step: Church's Analysis of Effective Calculability. 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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  12.  38
    Wilfried Sieg (1988). Hilbert's Program Sixty Years Later. Journal of Symbolic Logic 53 (2):338-348.
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  13.  2
    Wilfried Sieg & Dirk Schlimm (2005). Dedekind’s Analysis of Number: Systems and Axioms. Synthese 147 (1):121-170.
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  14.  47
    Wilfried Sieg & John Byrnes (1998). Normal Natural Deduction Proofs (in Classical Logic). 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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  15.  27
    Wilfried Sieg (1990). Relative Consistency and Accessible Domains. Synthese 84 (2):259 - 297.
    Wilfred Sieg. Relative Consistency and Accesible Domains.
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  16. Solomon Feferman, John W. Dawson, Warren Goldfarb, Charles Parsons & Wilfried Sieg (2004). Kurt Gödel Collected Works IV-V: Correspondence. Bulletin of Symbolic Logic 10 (4):558-563.
     
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  17.  19
    Wilfried Sieg (2014). The Ways of Hilbert's Axiomatics: Structural and Formal. 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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  18.  26
    Wilfried Sieg (1984). Foundations for Analysis and Proof Theory. Synthese 60 (2):159 - 200.
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  19.  10
    Wilfried Sieg & Frank Pfenning (1998). Note by the Guest Editors. Studia Logica 60 (1):1-1.
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  20.  1
    Wilfried Sieg (2007). The AProS Project: Strategic Thinking & Computational Logic. 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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  21.  39
    Wilfried Sieg & Mark Ravaglia, David Hilbert and Paul Bernays, Grundlagen der Mathematik I and II: A Landmark.
    Wilfred Sieg and Mark Ravaglia. David Hilbert and Paul Bernays, Grundlagen der Mathematik I and II: A Landmark.
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  22.  18
    Wilfried Sieg & John Byrnes, K-Graph Machines: Generalizing Turing's Machines and Arguments.
    Wilfred Sieg and John Byrnes. K-Graph Machines: Generalizing Turing's Machines and Arguments.
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  23.  19
    Wilfried Sieg, Unification For Quantified Formulae.
    — 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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  24.  29
    Wilfried Sieg, Toward Finitist Proof Theory.
    This is a summary of developments analysed in my (1997A). A first version of that paper was presented at the workshop Modern Mathematical Thought in Pittsburgh (September 21-24, 1995).
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  25.  15
    Wilfried Sieg & Stanley S. Wainer, Program Transformation and Proof Transformation.
    Wilfred Sieg and Stanley S. Wainer. Program Transformation and Proof Transformation.
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  26.  9
    Wilfried Sieg, Calculations by Man and Machine: Mathematical Presentation.
    Wilfried Sieg. Calculations by Man and Machine: Mathematical Presentation.
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  27.  25
    Wilfried Sieg, Formal Systems, Church Turing Thesis, and Gödel's Theorems: Three Contributions to The MIT Encyclopedias of Cognitive Science.
    Wilfried Sieg. Formal Systems, Church Turing Thesis, and Gödel's Theorems: Three Contributions to The MIT Encyclopedias of Cognitive Science.
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  28.  23
    Daniele Mundici & Wilfried Sieg, Mathematics Studies Machines.
    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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  29.  9
    Sergei Artemov, Peter Koellner, Michael Rabin, Jeremy Avigad, Wilfried Sieg, William Tait & Haim Gaifman (2006). Of the Association for Symbolic Logic. Bulletin of Symbolic Logic 12 (3-4):503.
  30. Wilfried Sieg & Rossella Lupacchini, Computing Machines.
    Any thorough discussion of computing machines requires the examination of rigorous concepts of computation and is facilitated by the distinction between mathematical, symbolic and physical computations. The delicate connection between the three kinds of computations and the underlying questions, "What are machines?" and "When are they computing?", motivate an extensive theoretical and historical discussion. The relevant outcome of this..
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  31.  10
    Wilfried Sieg (1983). Review: Neil Tennant, Natural Logic. [REVIEW] Journal of Symbolic Logic 48 (1):215-217.
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  32.  2
    Wilfried Sieg, Mechanisms and Search: Aspects of Proof Theory.
    Wilfred Sieg. Mechanisms and Search: Aspects of Proof Theory.
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  33.  17
    Wilfrid Hodges & Wilfried Sieg (1988). A Symposium on Hilbert's Program. Journal of Symbolic Logic 53 (2):337.
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  34.  19
    Wilfried Sieg, Proof Theory.
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  35.  15
    Wilfried Sieg & Richard Scheines, Searching for Proofs.
    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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  36.  4
    Wilfried Sieg (1990). Review: Stephen G. Simpson, Friedman's Research on Subsystems of Second Order Arithmetic. [REVIEW] Journal of Symbolic Logic 55 (2):870-874.
  37.  16
    Daniele Mundici & Wilfried Sieg, Computability Theory.
    Daniele Mundici and Wilfred Sieg. Computability Theory.
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  38.  14
    Richard Scheines & Wilfried Sieg, Computer Environments for Proof Construction.
    Richard Scheines and Wilfred Sieg. Computer Environments for Proof Construction.
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  39. Wilfried Sieg (1977). Trees in Metamathematics. Dissertation, Stanford University
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  40.  5
    Sergei Artemov, Peter Koellner, Michael Rabin, Jeremy Avigad, Wilfried Sieg, William Tait & Haim Gaifman (2006). The Hilton New York Hotel New York, NY December 27–29, 2005. Bulletin of Symbolic Logic 12 (3).
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  41.  10
    Wilfried Sieg & John Byrnes, Generalizing Turing's Machine and Arguments.
    Wilfred Sieg and John Byrnes. Generalizing Turing's Machine and Arguments.
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  42.  11
    Wilfried Sieg & Clinton Field (2005). Automated Search for Gödel’s Proofs. Annals of Pure and Applied Logic 133 (1):319-338.
    Wilfred Sieg and Clinton Field. Automated Search for Gödel's Proofs.
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  43.  4
    Wilfried Sieg & Saverio Cittadini, Normal Natural Deduction Proof (In Non-Classical Logics).
    Wilfred Sieg and Saverio Cittadini. Normal Natural Deduction Proof (In Non-Classical Logics.
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  44.  9
    Wilfried Sieg & John Byrnes, Gödel, Turing, and K-Graph Machines.
    Wilfried Sieg and John Byrnes. Gödel, Turing, and K-Graph Machines.
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  45.  3
    Wilfried Sieg, Church's Thesis, "Consistency", "Formalization", "Proof Theory" : Dictionary Entries.
    Wilfred Sieg. “Church's Thesis”, “Consistency”, “Formalization”, “Proof Theory”: Dictionary Entries.
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  46.  1
    Wilfried Sieg (1950). Formal Systems, Properties Of. Bulletin of Symbolic Logic 3:154-180.
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  47.  1
    Wilfried Sieg (2009). Hilbert's Proof Theory. In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier 5--321.
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  48.  2
    Wilfried Sieg, Aspects of Mathematical Experience.
    Wilfred Sieg. Aspects of Mathematical Experience.
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  49.  5
    Wilfried Sieg & Rosella Lupiccini, Computing Machines: Entry for the Second Edition of the Encyclopedia of Philsophy.
    Wilfred Sieg and Rosella Lupiccini. Computing Machines: Entry for the Second Edition of the Encyclopedia of Philsophy.
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  50.  5
    Wilfried Sieg, Intercalculation Calculi for Classical Logic.
    Wilfred Sieg. Intercalculation Calculi for Classical Logic.
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