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"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic 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 ..."
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic 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.
A formal system for Euclid’s elements
 The Review of Symbolic Logic
, 2009
"... Abstract. We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. §1. Introduction. For more than two millennia, Euclid’s Elements was viewed by mathematicians and philosophers alike as a paradigm of rigorous arg ..."
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Abstract. We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. §1. Introduction. For more than two millennia, Euclid’s Elements was viewed by mathematicians and philosophers alike as a paradigm of rigorous argumentation. But the work lost some of its lofty status in the nineteenth century, amidst concerns related to the use of diagrams in its proofs. Recognizing the correctness of Euclid’s inferences was thought to require an “intuitive ” use of these diagrams, whereas, in a proper mathematical
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Understanding proofs
 The Philosophy of Mathematical Practice
, 2008
"... “Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a springcarriage 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 ..."
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“Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a springcarriage 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
Paradoxes in Göttingen
"... In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s ..."
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In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s
Lagrange’s theory of analytical functions and his ideal of purity of method. Archive for History of Exact Sciences. Forthcoming
"... ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on powerseries expansions, on the algorithm ..."
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ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on powerseries expansions, on the algorithm for derivative functions, and the remainder theorem—especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. hal00614606, version 1 12 Aug 2011 1. PRELIMINARIES AND PROPOSALS Foundation of mathematics was a crucial topic for 18thcentury mathematicians. A pivotal aspect of it was the interpretation of the algoritihms of the calculus. This was often referred to as the question of the “metaphysics of the calculus ” 1 (see Carnot 1797, as an example). Around 1800 Lagrange devoted two large treatises to the matter, both of which went through two editions in Lagrange’s
The Pragmatism of Hilbert’s Programme ∗
"... of the GaussWeber monument in memory of the great mathematical and physical tradition of the University of Göttingen. On the occasion of this ..."
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of the GaussWeber monument in memory of the great mathematical and physical tradition of the University of Göttingen. On the occasion of this
Centralized vs. Marketbased and Decentralized DecisionMaking: A Review of the Evidence in Computer Science and Economics
, 2008
"... Within both economics and computer science, many authors have claimed that decentralized or marketbased approaches to decisionmaking are superior in general to centralized approaches. The contrary claim has also been made. Unfortunately, these claims are often supported only by informal or anecdot ..."
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Within both economics and computer science, many authors have claimed that decentralized or marketbased approaches to decisionmaking are superior in general to centralized approaches. The contrary claim has also been made. Unfortunately, these claims are often supported only by informal or anecdotal evidence. In order to assess these competing claims, we present a review of the literatures in economics and in computer science bearing on these issues. Specifically, we report research findings based on empirical evidence and on simulation studies, and we outline the evidence based on formal deductive proofs or on informal and anecdotal evidence. Our main findings from this literature survey are: (i) for efficiency assessments, that there is wider variance in performance of organizations using MarketBased Control (MBC) than in organizations using Centralized Control (CC); (ii) that MBC and CC have the same efficiency on average; which may explain the observation (iii) that human and computer organizations tend to cycle between CC and Decentralized Control (DC) structures. 1