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.

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."

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.

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.

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.

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."

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.