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Little Theories
 Automated DeductionCADE11, volume 607 of Lecture Notes in Computer Science
, 1992
"... In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable wa ..."
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Cited by 59 (18 self)
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In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
Theory Interpretation in Simple Type Theory
 HIGHERORDER ALGEBRA, LOGIC, AND TERM REWRITING, VOLUME 816 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
"... Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admit ..."
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Cited by 37 (17 self)
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Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admits partial functions and subtypes. The method is patterned on the standard approach to theory interpretation in rstorder logic. Although the method is based on a nonclassical version of simple type theory, it is intended as a guide for theory interpretation in classical simple type theories as well as in predicate logics with partial functions.
Symplectic Geometry and Hilbert's Fourth Problem
"... Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an arealength duality between a class of metric spaces and a class of symplectic manifolds. In particular, it is shown that there is a twistorlike corr ..."
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Cited by 11 (1 self)
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Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an arealength duality between a class of metric spaces and a class of symplectic manifolds. In particular, it is shown that there is a twistorlike correspondence between Finsler metrics on RP^n whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2planes in R^n+l. Symplectic and integral geometry are also used to show that if the geodesics of a Finsler metric on an open convex subset of RP^n are projective line segments, then hyperplanes are areaminimizing.
Sense and sidedness in the graphics pipeline via a passage through a separable space
 THE VISUAL COMPUTER
, 2009
"... Computer graphics is ostensibly based on projective geometry. The graphics pipeline—the sequence of functions applied to 3D geometric primitives to determine a 2D image—is described in the graphics literature as taking the primitives from Euclidean to projective space, and then back to Euclidean spa ..."
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Computer graphics is ostensibly based on projective geometry. The graphics pipeline—the sequence of functions applied to 3D geometric primitives to determine a 2D image—is described in the graphics literature as taking the primitives from Euclidean to projective space, and then back to Euclidean space. This is a weak foundation for computer graphics. An instructor is at a loss: one day entering the classroom and invoking the established and venerable theory of projective geometry while asserting that projective spaces are not separable, and then entering the classroom the following week to tell the students that the standard graphics pipeline performs clipping not in Euclidean, but in projective space—precisely the operation (deciding sidedness, which depends on separability) that was deemed nonsensical. But there is no need to present Blinn and Newell’s algorithm [4, 24]—the crucial clipping step in the graphics pipeline and, perhaps, the most original knowledge a student learns in a fourthyear computer graphics class—as a clever trick that just works. Jorge Stolfi described in 1991 oriented projective geometry. By declaring the two vectors (x, y, z, w) T and (−x, −y, −z, −w) T distinct, Blinn and Newell were already unknowingly working in oriented projective space. This paper presents the graphics pipeline on this stronger foundation.
HILBERT’S FOURTH PROBLEM IN TWO DIMENSIONS I
"... Abstract. Hilbert’s fourth problems asks to construct and study the geometries in which the straight line segment is the shortest connection between two points. In this paper the reader shall find an elementary introduction to the problem and its solutions in dimension two by Busemann, Pogorelov, a ..."
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Abstract. Hilbert’s fourth problems asks to construct and study the geometries in which the straight line segment is the shortest connection between two points. In this paper the reader shall find an elementary introduction to the problem and its solutions in dimension two by Busemann, Pogorelov, and Ambartzumian. The relationship between integral geometry and inverse problems in variational calculus is emphasized. Contents
Geometry and Proof
, 2006
"... We discuss the relation between the specific axiomatizations, specifically Hilbert’s [Hil71] reformulation of geometry at the beginning of the last century, and the way elementary geometry has been expounded in high schools in the United States. Further we discuss the connections among formal logic, ..."
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We discuss the relation between the specific axiomatizations, specifically Hilbert’s [Hil71] reformulation of geometry at the beginning of the last century, and the way elementary geometry has been expounded in high schools in the United States. Further we discuss the connections among formal logic, the teaching of logic and the preparation of high school teachers. In part our goal is to describe how high school geometry instruction developed in the United States during the 20th century in hopes of learning of the development in other countries. We conclude with some recommendations concerning teaching reasoning to high school students and preparing future teachers for this task. We view Hilbert’s geometry as a critique of Euclid and focus on three aspects of it: a) the need for undefined terms, b) continuity axioms, c) the mobility postulate. (We are at the moment being historically cavalier and using ‘Hilbert’ as a surrogate for an analysis by many contributors including in particular
On the nongeneric TzitzeicaJohnson’s Configuration
"... Working in the context of Hilbert’s axiomatic system, we prove that the TzitzeicaJohnson configuration has sense in neutral geometry. The elements of a TzitzeicaJohnson configuration allow us to analyse the Euclidean or the nonEuclidean character of the metric of the plane. 1 ..."
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Working in the context of Hilbert’s axiomatic system, we prove that the TzitzeicaJohnson configuration has sense in neutral geometry. The elements of a TzitzeicaJohnson configuration allow us to analyse the Euclidean or the nonEuclidean character of the metric of the plane. 1
MATHEMATICAL LOGIC: WHAT HAS IT DONE FOR THE PHILOSOPHY OF MATHEMATICS?
"... to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community a ..."
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to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics. Kreisel’sviewsgreatlyinfluencedmeintheSixtiesandthe Seventies. His critical remarks on the foundational programs taught me that one could and should have an approach to the subject of mathematical logic less dogmatic, corporative and even thoughtless than the one the logical community sometimes used to have. This is even more true today when the professionalization of mathematical logic generates a flood of results but few new ideas and the lack of ideas leads to the sheer byzantinism of most current production in mathematical logic. In the past few years,however,IhavecometotheconclusionthatKreisel’scriticism hasnot been radical enough: his main worry seems to have been to preserve as much as possible to save the savable of the tradition of mathematical logic. His critical remarks have focused on the defects of the foundational schools, thus drawing attention away from the intrinsic defects of mathematical logic itself