Search results for 'Jeremy D. Avigad' (try it on Scholar)

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  1.  9
    Jeremy D. Avigad (2002). Review: Sergei N. Artemov, Explicit Provability and Constructive Semantics. [REVIEW] Bulletin of Symbolic Logic 8 (3):432-433.
  2. Jeremy D. Avigad (2002). Artemov Sergei N.. Explicit Provability and Constructive Semantics. The Bulletin of Symbolic Logic, Vol. 7 , Pp. 1–36. Bulletin of Symbolic Logic 8 (3):432-433.
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  3.  1
    Lev D. Beklemishev, Stephen Cook, Olivier Lessmann, Simon Thomas, Jeremy Avigad, Arnold Beckmann, Tim Carlson, Robert L. Constable & Kosta Došen (2003). 2002 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium'02. Bulletin of Symbolic Logic 9 (1):71.
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  4. Jeremy Avigad, Notes on II-Conservativity, W-Submodels, and the Collection Schema.
    Jeremy Avigad. Notes on II-conservativity, w-submodels, and the Collection Schema.
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  5.  29
    Jeremy Avigad, Computability and Convergence.
    For most of its history, mathematics was fairly constructive: • Euclidean geometry was based on geometric construction. • Algebra sought explicit solutions to equations. Analysis, probability, etc. were focused on calculations. Nineteenth century developments in analysis challenged this view. A sequence (an) in a metric space is said Cauchy if for every ε > 0, there is an m such that for every n, n ≥ m, d (a n , a n ) < ε.
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  6.  27
    Jesse Hughes, Steve Awodey, Dana Scott, Jeremy Avigad & Lawrence Moss, A Study of Categorres of Algebras and Coalgebras.
    This thesis is intended t0 help develop the theory 0f coalgebras by, Hrst, taking classic theorems in the theory 0f universal algebras amd dualizing them and, second, developing an interna] 10gic for categories 0f coalgebras. We begin with an introduction t0 the categorical approach t0 algebras and the dual 110tion 0f coalgebras. Following this, we discuss (c0)a,lg€bra.s for 2. (c0)monad and develop 2. theory 0f regular subcoalgebras which will be used in the interna] logic. We also prove that categories 0f (...)
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  7. E. Dean J. Avigad & J. Mumma (2009). Sur Quelques Points d'Algebre Homologique. Review of Symbolic Logic 2 (4):700-768.
     
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  8.  48
    Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris Van Doorn & Jakob von Raumer, The Lean Theorem Prover.
    Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API (...)
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  9. Jeremy Avigad (2000). Interpreting Classical Theories in Constructive Ones. Journal of Symbolic Logic 65 (4):1785-1812.
    A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.
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  10. Jeremy Avigad, Edward Dean & John Mumma (2009). A Formal System for Euclid's Elements. Review of Symbolic Logic 2 (4):700--768.
    We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
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  11. Jeremy Avigad (2004). Forcing in Proof Theory. Bulletin of Symbolic Logic 10 (3):305-333.
    Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...)
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  12.  8
    Jeremy Avigad, Mathematics and Language.
    This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes that we view mathematics as a system of conventions and norms that is designed to help us make sense of the world and reason efficiently. Like any designed system, it can perform well or poorly, and the philosophy of mathematics has a role to play in helping (...)
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  13. Jeremy Avigad & Harvey Friedman, Combining Decision Procedures for the Reals.
    We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local'’ decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let $Tadd[QQ]$ be the first-order theory of the real numbers in the language with symbols $0, 1, +, -.
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  14.  22
    Jeremy Avigad, Local Stability of Ergodic Averages.
    We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, (...)
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  15. Jeremy Avigad, Steven Kieffer & Harvey Friedman, A Language for Mathematical Knowledge Management.
    We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of that language that provides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these definitions, under various complexity (...)
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  16.  50
    Jeremy Avigad (2006). Mathematical Method and Proof. Synthese 153 (1):105 - 159.
    On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...)
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  17.  82
    Jeremy Avigad (2010). Understanding, Formal Verification, and the Philosophy of Mathematics. Journal of the Indian Council of Philosophical Research 27:161-197.
    The philosophy of mathematics has long been concerned with deter- mining the means that are appropriate for justifying claims of mathemat- ical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary math- ematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The as- sociated values are often loosely classied as (...)
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  18.  60
    Jeremy Avigad (2003). Number Theory and Elementary Arithmetic. Philosophia Mathematica 11 (3):257-284.
    is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
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  19.  32
    Jeremy Avigad & Richard Zach, The Epsilon Calculus. Stanford Encyclopedia of Philosophy.
    The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term..
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  20.  20
    Jeremy Avigad (1996). Formalizing Forcing Arguments in Subsystems of Second-Order Arithmetic. Annals of Pure and Applied Logic 82 (2):165-191.
    We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.
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  21.  58
    Jeremy Avigad & Jeffrey Helzner (2002). Transfer Principles in Nonstandard Intuitionistic Arithmetic. Archive for Mathematical Logic 41 (6):581-602.
    Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...)
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  22. Jeremy Avigad, Philosophy of Mathematics.
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
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  23. Jeremy Avigad, Eliminating Definitions and Skolem Functions.
    two elements, one can eliminate definitions with a polynomial bound on the increase in proof length. In any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions with a polynomial bound on the increase in proof length.
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  24.  19
    Jeremy Avigad, Weak Theories of Nonstandard Arithmetic and Analysis.
    A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.
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  25.  4
    Jeremy Avigad, Leonardo de Moura & Soonho Kong, Theorem Proving in Lean.
    Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point (...)
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  26.  4
    Rebecca Morris & Jeremy Avigad, Character and Object.
    In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation. In this essay, we (...)
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  27.  33
    Jeremy Avigad (2006). Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals. In Jose Ferreiros Jeremy Gray (ed.), The architecture of modern mathematics.
    Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...)
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  28.  23
    Jeremy Avigad (2006). Fundamental Notions of Analysis in Subsystems of Second-Order Arithmetic. Annals of Pure and Applied Logic 139 (1):138-184.
    We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic. In particular, we explore issues having to do with distances, closed subsets and subspaces, closures, bases, norms, and projections. We pay close attention to variations that arise when formalizing definitions and theorems, and study the relationships between them.
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  29.  37
    Jeremy Avigad, Eliminating Definitions and Skolem Functions in First-Order Logic.
    From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.
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  30.  24
    Jeremy Avigad (2002). Update Procedures and the 1-Consistency of Arithmetic. Mathematical Logic Quarterly 48 (1):3-13.
    The 1-consistency of arithmetic is shown to be equivalent to the existence of fixed points of a certain type of update procedure, which is implicit in the epsilon-substitution method.
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  31.  31
    Jeremy Avigad (2009). The Metamathematics of Ergodic Theory. Annals of Pure and Applied Logic 157 (2):64-76.
    The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns (...)
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  32.  66
    Jeremy Avigad (2007). Philosophy of Mathematics: 5 Questions. In V. F. Hendricks & Hannes Leitgeb (eds.), Philosophy of Mathematics: Five Questions. Automatic Press/VIP
    In 1977, when I was nine years old, Doubleday released Asimov on Numbers, a collection of essays that had first appeared in Isaac Asimov’s Science Fiction and Fantasy column. My mother, recognizing my penchant for science fiction and mathematics, bought me a copy as soon as it hit the bookstores. The essays covered topics such as number systems, combinatorial curiosities, imaginary numbers, and π. I was especially taken, however, by an essay titled “Varieties of the infinite,” which included a photograph (...)
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  33.  34
    Jeremy Avigad (1996). On the Relationship Between ATR 0 And. Journal of Symbolic Logic 61 (3):768-779.
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  34.  31
    Jeremy Avigad, Algebraic Proofs of Cut Elimination.
    Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as special (...)
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  35.  24
    Jeremy Avigad, Kevin Donnelly, David Gray & Paul Raff, A Formally Verified Proof of the Prime Number Theorem.
    The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.
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  36.  3
    Leonardo de Moura, Jeremy Avigad, Soonho Kong & Cody Roux, Elaboration in Dependent Type Theory.
    To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary mathematical text, and resolving ambiguities in mathematical expressions. We refer to the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one as elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean’s (...)
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  37.  23
    Jeremy Avigad (2002). Saturated Models of Universal Theories. Annals of Pure and Applied Logic 118 (3):219-234.
    A notion called Herbrand saturation is shown to provide the model-theoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the model-theoretic version.
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  38.  39
    Jeremy Avigad (2009). Marcus Giaquinto. Visual Thinking in Mathematics: An Epistemological Study. [REVIEW] Philosophia Mathematica 17 (1):95-108.
    Published in 1891, Edmund Husserl's first book, Philosophie der Arithmetik, aimed to ‘prepare the scientific foundations for a future construction of that discipline’. His goals should seem reasonable to contemporary philosophers of mathematics: "…through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. 1"But the ensuing strategy for grounding mathematical knowledge (...)
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  39.  22
    Jeremy Avigad (1998). An Effective Proof That Open Sets Are Ramsey. Archive for Mathematical Logic 37 (4):235-240.
    Solovay has shown that if $\cal{O}$ is an open subset of $P(\omega)$ with code $S$ and no infinite set avoids $\cal{O}$ , then there is an infinite set hyperarithmetic in $S$ that lands in $\cal{O}$ . We provide a direct proof of this theorem that is easily formalizable in $ATR_0$.
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  40.  58
    Jeremy Avigad, Computers in Mathematical Inquiry.
    In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: some (...)
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  41.  16
    Jeremy Avigad, Proof Mining.
    Hilbert’s program: • Formalize abstract, infinitary, nonconstructive mathematics. • Prove consistency using only finitary methods.
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  42.  57
    Jeremy Avigad, “Clarifying the Nature of the Infinite”: The Development of Metamathematics and Proof Theory.
    We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
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  43.  25
    Jeremy Avigad & Richard Sommer (1999). The Model-Theoretic Ordinal Analysis of Theories of Predicative Strength. Journal of Symbolic Logic 64 (1):327-349.
    We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.
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  44.  15
    Jeremy Avigad (1998). Predicative Functionals and an Interpretation of ⌢ID<Ω. Annals of Pure and Applied Logic 92 (1):1-34.
    In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifier-free theory T axiomatizing the primitive recursive functionals of finite type.
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  45.  39
    Jeremy Avigad, Formalizing O Notation in Isabelle/Hol.
    We describe a formalization of asymptotic O notation using the Isabelle/HOL proof assistant.
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  46.  5
    Jeremy Avigad (1998). Predicative Functionals and an Interpretation of ⌢ID<Ω. Annals of Pure and Applied Logic 92 (1):1-34.
    In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifier-free theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel's T to theories Pn of “predicative” functionals, which are defined using Martin-Löf's universes of transfinite types. We then extend Gödel's interpretation to the theories of arithmetic inductive definitions IDn, so that each IDn is interpreted in the corresponding Pn. Since the strengths of the theories IDn are cofinal in the ordinal Γ0, as (...)
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  47.  19
    Jeremy Avigad, Plausibly Hard Combinatorial Tautologies.
    We present a simple propositional proof system which consists of a single axiom schema and a single rule, and use this system to construct a sequence of combinatorial tautologies that, when added to any Frege system, p-simulates extended-Frege systems.
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  48.  38
    Jeremy Avigad, Proof Theory.
    At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...)
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  49.  26
    Jeremy Avigad & Richard Sommer (1997). A Model-Theoretic Approach to Ordinal Analysis. Bulletin of Symbolic Logic 3 (1):17-52.
    We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.
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  50.  41
    Jeremy Avigad, Understanding Proofs.
    “Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).
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