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 by Jacques Fleuriot Pdf
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 by Jürgen Richter-Gebert,Dongming Wang Pdf
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 by Jacques D. Fleuriot Pdf
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 by Claude Kirchner,Helene Kirchner Pdf
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 by Mark Aagaard,John Harrison Pdf
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.
Automated Deduction in Geometry by Pascal Schreck,Julien Narboux,Jürgen Richter-Gebert Pdf
This book constitutes the thoroughly refereed post-workshop proceedings of the 8th International Workshop on Automated Deduction in Geometry, ADG 2010, held in Munich, Germany in July 2010. The 13 revised full papers presented were carefully selected during two rounds of reviewing and improvement from the lectures given at the workshop. Topics addressed by the papers are incidence geometry using some kind of combinatoric argument; computer algebra; software implementation; as well as logic and proof assistants.
A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton's Principia by Jacques D. Fleuriot Pdf
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 by Xiao-Shan Gao,Dongming Wang,Lu Yang Pdf
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 by Xiao-lu Gao,Dongming Wang,Lu Yang Pdf
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 by Danny A. J. Gómez Ramírez Pdf
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 by Gregory Michaelson Pdf
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.
Abraham Lincoln and the Structure of Reason by David Hirsch,Dan Van Haften Pdf
The secrets of one of history’s greatest orators are revealed in “one of the most stunningly original works on Abraham Lincoln to appear in years” (John Stauffer, professor of English and history, Harvard University). For more than 150 years, historians have speculated about what made Abraham Lincoln truly great. How did Lincoln create his compelling arguments, his convincing oratory, and his unforgettable writing? Some point to Lincoln’s study of grammar, literature, and poetry. Others believe it was the deep national crisis that gave import to his words. Most agree that he honed his persuasive technique in his work as an Illinois attorney. Here, the authors argue that it was Lincoln’s in-depth study of geometry that made the president’s verbal structure so effective. In fact, as the authors demonstrate, Lincoln embedded the ancient structure of geometric proof into the Gettysburg Address, the Cooper Union speech, the first and second inaugurals, his legal practice, and much of his substantive post-1853 communication. Also included are Lincoln’s preparatory notes and drafts of some of his most famous speeches as well as his revisions and personal thoughts on public speaking and grammar. With in-depth research and provocative insight, Abraham Lincoln and the Structure of Reason “offers a whole new angle on Lincoln’s brilliance” (James M. Cornelius, Curator, Lincoln Collection, Abraham Lincoln Presidential Library and Museum).
Automated Reasoning by Nicolas Peltier,Viorica Sofronie-Stokkermans Pdf
This two-volume set LNAI 12166 and 12167 constitutes the refereed proceedings of the 10th International Joint Conference on Automated Reasoning, IJCAR 2020, held in Paris, France, in July 2020.* In 2020, IJCAR was a merger of the following leading events, namely CADE (International Conference on Automated Deduction), FroCoS (International Symposium on Frontiers of Combining Systems), ITP (International Conference on Interactive Theorem Proving), and TABLEAUX (International Conference on Analytic Tableaux and Related Methods). The 46 full research papers, 5 short papers, and 11 system descriptions presented together with two invited talks were carefully reviewed and selected from 150 submissions. The papers focus on the following topics: Part I: SAT; SMT and QBF; decision procedures and combination of theories; superposition; proof procedures; non classical logics Part II: interactive theorem proving/ HOL; formalizations; verification; reasoning systems and tools *The conference was held virtually due to the COVID-19 pandemic. Chapter ‘Constructive Hybrid Games’ is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Exploration of Space, Technology, and Spatiality: Interdisciplinary Perspectives by Turner, Phil,Turner, Susan,Davenport, Elisabeth Pdf
"For researchers and scholars working at the intersection of physical, social, and technological space, this book provides critical research from leading experts in the space technology domain"--Provided by the publisher.
Most aspects of our private and social lives—our safety, the integrity of the financial system, the functioning of utilities and other services, and national security—now depend on computing. But how can we know that this computing is trustworthy? In Mechanizing Proof, Donald MacKenzie addresses this key issue by investigating the interrelations of computing, risk, and mathematical proof over the last half century from the perspectives of history and sociology. His discussion draws on the technical literature of computer science and artificial intelligence and on extensive interviews with participants. MacKenzie argues that our culture now contains two ideals of proof: proof as traditionally conducted by human mathematicians, and formal, mechanized proof. He describes the systems constructed by those committed to the latter ideal and the many questions those systems raise about the nature of proof. He looks at the primary social influence on the development of automated proof—the need to predict the behavior of the computer systems upon which human life and security depend—and explores the involvement of powerful organizations such as the National Security Agency. He concludes that in mechanizing proof, and in pursuing dependable computer systems, we do not obviate the need for trust in our collective human judgment.