A Combination Of Geometry Theorem Proving And Nonstandard Analysis With Application To Newton S Principia
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A Combination Of Geometry Theorem Proving And Nonstandard Analysis With Application To Newton S Principia
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Author : Jacques Fleuriot
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-09-30
A Combination Of Geometry Theorem Proving And Nonstandard Analysis With Application To Newton S Principia written by Jacques Fleuriot and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-30 with Mathematics categories.
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague. In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.
Automated Deduction In Geometry
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Author : Jürgen Richter-Gebert
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-09-12
Automated Deduction In Geometry written by Jürgen Richter-Gebert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-09-12 with Computers categories.
This book constitutes the thoroughly refereed post-proceedings of the Third International Workshop on Automated Deduction in Geometry, ADG 2000, held in Zurich, Switzerland, in September 2000. The 16 revised full papers and two invited papers presented were carefully selected for publication during two rounds of reviewing and revision from a total of initially 31 submissions. Among the issues addressed are spatial constraint solving, automated proving of geometric inequalities, algebraic proof, semi-algebraic proofs, geometrical reasoning, computational synthetic geometry, incidence geometry, and nonstandard geometric proofs.
A Combination Of Geometry Theorem Proving And Nonstandard Analysis With Application To Newton S Principia
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Author : Jacques D. Fleuriot
language : en
Publisher:
Release Date : 1999
A Combination Of Geometry Theorem Proving And Nonstandard Analysis With Application To Newton S Principia written by Jacques D. Fleuriot and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Automatic theorem proving categories.
Abstract: "Sir Isaac Newton's Philosophiæ Naturalis Principia Mathematica (the Principia) was first published in 1687 and set much of the foundations that led to profound changes in modern science. Despite the influence of the work, the elegance of the geometrical techniques used by Newton is little known since the demonstrations of most of the theorems set out in it are usually done using calculus. Newton's reasoning also goes beyond the traditional boundaries of Euclidean geometry with the presence of both motion and infinitesimals. This thesis describes the mechanization of Lemmas and Propositions from the Principia using formal tools developed in the generic theorem prover Isabelle. We discuss the formalization of a geometry theory based on existing methods from automated geometry theorem proving. The theory contains extra geometric notions, including definitions of the ellipse and its tangent, that enable us to deal with the motion of bodies and other physical aspects. We introduce the formalization of a theory of filters and ultrafilters, and the purely definitional construction of the hyperreal numbers of Nonstandard Analysis (NSA). The hyperreals form a proper field extension of the reals that contains new types of numbers including infinitesimals and infinite numbers. By combining notions from NSA and geometry theorem proving, we propose an 'infinitesimal' geometry in which quantities can be infinitely small. This approach then reveals new properties of the geometry that only hold because infinitesimal elements are allowed. We also mechanize some analytic geometry and use it to verify the geometry theories of Isabelle. We then report on the main application of this framework. We discuss the formalization of several results from the Principia and give a detailed case study of one of its most important propositions: the Propositio Kepleriana. An anomaly is revealed in Newton's reasoning through our rigorous mechanization. Finally, we present the formalization of a portion of mathematical analysis using the nonstandard approach. We mechanize both standard and nonstandard definitions of familiar concepts, prove their equivalence, and use nonstandard arguments to provide intuitive yet rigorous proofs of many of their properties."
Automated Deduction Cade 15
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Author : Claude Kirchner
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-06-24
Automated Deduction Cade 15 written by Claude Kirchner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-06-24 with Computers categories.
This book constitutes the refereed proceedings of the 15th International Conference on Automated Deduction, CADE-15, held in Lindau, Germany, in July 1998. The volume presents three invited contributions together with 25 revised full papers and 10 revised system descriptions; these were selected from a total of 120 submissions. The papers address all current issues in automated deduction and theorem proving based on resolution, superposition, model generation and elimination, or connection tableau calculus, in first-order, higher-order, intuitionistic, or modal logics, and describe applications to geometry, computer algebra, or reactive systems.
Theorem Proving In Higher Order Logics
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Author : Mark Aagaard
language : en
Publisher: Springer
Release Date : 2007-07-23
Theorem Proving In Higher Order Logics written by Mark Aagaard and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-07-23 with Computers categories.
This volume is the proceedings of the 13th International Conference on Theo rem Proving in Higher Order Logics (TPHOLs 2000) held 14-18 August 2000 in Portland, Oregon, USA. Each of the 55 papers submitted in the full rese arch category was refereed by at least three reviewers who were selected by the program committee. Because of the limited space available in the program and proceedings, only 29 papers were accepted for presentation and publication in this volume. In keeping with tradition, TPHOLs 2000 also offered a venue for the presen tation of work in progress, where researchers invite discussion by means of a brief preliminary talk and then discuss their work at a poster session. A supplemen tary proceedings containing associated papers for work in progress was published by the Oregon Graduate Institute (OGI) as technical report CSE-00-009. The organizers are grateful to Bob Colwell, Robin Milner and Larry Wos for agreeing to give invited talks. Bob Colwell was the lead architect on the Intel P6 microarchitecture, which introduced a number of innovative techniques and achieved enormous commercial success. As such, he is ideally placed to offer an industrial perspective on the challenges for formal verification. Robin Milner contributed many key ideas to computer theorem proving, and to functional programming, through his leadership of the influential Edinburgh LCF project.
A Combination Of Nonstandard Analysis And Geometry Theorem Proving With Application To Newton S Principia
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Author : Jacques D. Fleuriot
language : en
Publisher:
Release Date : 1997
A Combination Of Nonstandard Analysis And Geometry Theorem Proving With Application To Newton S Principia written by Jacques D. Fleuriot and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Automatic theorem proving categories.
Abstract: "The theorem prover Isabelle is used to formalize and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains 'infinitesimal' elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton's proofs from being mechanised using only the existing geometry theorem proving (GTP) techniques. Using concepts from Robinson's Nonstandard Analysis (NSA) and a powerful geometric theory, we introduce the concept of an infinitesimal geometry in which quantities can be infinitely small or infinitesimal. We reveal and prove new properties of this geometry that only hold because infinitesimal elements are allowed and use them to prove lemmas and theorems from the Principia."
Automated Deduction In Geometry
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Author : Xiao-Shan Gao
language : en
Publisher: Springer Science & Business Media
Release Date : 1999-10-13
Automated Deduction In Geometry written by Xiao-Shan Gao and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-10-13 with Computers categories.
The Second International Workshop on Automated Deduction in Geometry (ADG ’98) was held in Beijing, China, August 1–3, 1998. An increase of interest in ADG ’98 over the previous workshop ADG ’96 is represented by the notable number of more than 40 participants from ten countries and the strong tech- cal program of 25 presentations, of which two one-hour invited talks were given by Professors Wen-tsun ̈ Wu and Jing-Zhong Zhang. The workshop provided the participants with a well-focused forum for e?ective exchange of new ideas and timely report of research progress. Insight surveys, algorithmic developments, and applications in CAGD/CAD and computer vision presented by active - searchers, together with geometry software demos, shed light on the features of this second workshop. ADG ’98 was hosted by the Mathematics Mechanization Research Center (MMRC) with ?nancial support from the Chinese Academy of Sciences and the French National Center for Scienti?c Research (CNRS), and was organized by the three co-editors of this proceedings volume. The papers contained in the volume were selected, under a strict refereeing procedure, from those presented at ADG ’98 and submitted afterwards. Most of the 14 accepted papers were carefully revised and some of the revised versions were checked again by external reviewers. We hope that these papers cover some of the most recent and signi?cant research results and developments and re?ect the current state-of-the-art of ADG.
Automated Deduction In Geometry
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Author : Xiao-lu Gao
language : en
Publisher: Springer
Release Date : 2003-06-26
Automated Deduction In Geometry written by Xiao-lu Gao and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-06-26 with Computers categories.
The Second International Workshop on Automated Deduction in Geometry (ADG ’98) was held in Beijing, China, August 1–3, 1998. An increase of interest in ADG ’98 over the previous workshop ADG ’96 is represented by the notable number of more than 40 participants from ten countries and the strong tech- cal program of 25 presentations, of which two one-hour invited talks were given by Professors Wen-tsun ̈ Wu and Jing-Zhong Zhang. The workshop provided the participants with a well-focused forum for e?ective exchange of new ideas and timely report of research progress. Insight surveys, algorithmic developments, and applications in CAGD/CAD and computer vision presented by active - searchers, together with geometry software demos, shed light on the features of this second workshop. ADG ’98 was hosted by the Mathematics Mechanization Research Center (MMRC) with ?nancial support from the Chinese Academy of Sciences and the French National Center for Scienti?c Research (CNRS), and was organized by the three co-editors of this proceedings volume. The papers contained in the volume were selected, under a strict refereeing procedure, from those presented at ADG ’98 and submitted afterwards. Most of the 14 accepted papers were carefully revised and some of the revised versions were checked again by external reviewers. We hope that these papers cover some of the most recent and signi?cant research results and developments and re?ect the current state-of-the-art of ADG.
Artificial Mathematical Intelligence
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Author : Danny A. J. Gómez Ramírez
language : en
Publisher: Springer Nature
Release Date : 2020-10-23
Artificial Mathematical Intelligence written by Danny A. J. Gómez Ramírez and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-23 with Mathematics categories.
This volume discusses the theoretical foundations of a new inter- and intra-disciplinary meta-research discipline, which can be succinctly called cognitive metamathematics, with the ultimate goal of achieving a global instance of concrete Artificial Mathematical Intelligence (AMI). In other words, AMI looks for the construction of an (ideal) global artificial agent being able to (co-)solve interactively formal problems with a conceptual mathematical description in a human-style way. It first gives formal guidelines from the philosophical, logical, meta-mathematical, cognitive, and computational points of view supporting the formal existence of such a global AMI framework, examining how much of current mathematics can be completely generated by an interactive computer program and how close we are to constructing a machine that would be able to simulate the way a modern working mathematician handles solvable mathematical conjectures from a conceptual point of view. The thesis that it is possible to meta-model the intellectual job of a working mathematician is heuristically supported by the computational theory of mind, which posits that the mind is in fact a computational system, and by the meta-fact that genuine mathematical proofs are, in principle, algorithmically verifiable, at least theoretically. The introduction to this volume provides then the grounding multifaceted principles of cognitive metamathematics, and, at the same time gives an overview of some of the most outstanding results in this direction, keeping in mind that the main focus is human-style proofs, and not simply formal verification. The first part of the book presents the new cognitive foundations of mathematics’ program dealing with the construction of formal refinements of seminal (meta-)mathematical notions and facts. The second develops positions and formalizations of a global taxonomy of classic and new cognitive abilities, and computational tools allowing for calculation of formal conceptual blends are described. In particular, a new cognitive characterization of the Church-Turing Thesis is presented. In the last part, classic and new results concerning the co-generation of a vast amount of old and new mathematical concepts and the key parts of several standard proofs in Hilbert-style deductive systems are shown as well, filling explicitly a well-known gap in the mechanization of mathematics concerning artificial conceptual generation.
Mathematical Reasoning The History And Impact Of The Dream Group
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Author : Gregory Michaelson
language : en
Publisher: Springer Nature
Release Date : 2021-11-20
Mathematical Reasoning The History And Impact Of The Dream Group written by Gregory Michaelson and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-11-20 with Computers categories.
This collection of essays examines the key achievements and likely developments in the area of automated reasoning. In keeping with the group ethos, Automated Reasoning is interpreted liberally, spanning underpinning theory, tools for reasoning, argumentation, explanation, computational creativity, and pedagogy. Wider applications including secure and trustworthy software, and health care and emergency management. The book starts with a technically oriented history of the Edinburgh Automated Reasoning Group, written by Alan Bundy, which is followed by chapters from leading researchers associated with the group. Mathematical Reasoning: The History and Impact of the DReaM Group will attract considerable interest from researchers and practitioners of Automated Reasoning, including postgraduates. It should also be of interest to those researching the history of AI.