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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
An algorithmic proof of the MotzkinRabin theorem on monochrome lines
, 2002
"... We present a new proof of the following theorem originally due to Motzkin and Rabin (see [9, 1, 6, 7]). MotzkinRabin Theorem. Let S be a finite noncollinear set of points in the plane, each colored red or blue. Then there exists a line l passing through at least two points of S, all points of S on ..."
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We present a new proof of the following theorem originally due to Motzkin and Rabin (see [9, 1, 6, 7]). MotzkinRabin Theorem. Let S be a finite noncollinear set of points in the plane, each colored red or blue. Then there exists a line l passing through at least two points of S, all points of S on l being of the same color. We say that a set of points S is twocolored if each point in S is assigned one of the colors red or blue. A line passing through at least two points of S with all points of S on the line assigned the same color is called a monochrome line. It makes no di erence whether the plane in the theorem is the Euclidean or the projective plane, since if we are in the projective plane we can always find a line disjoint from the finite set S, project it to infinity, and then the set S can be considered to be in the Euclidean plane. There are two essentially different proofs of the MotzkinRabin theorem in the literature, both proving the projective dual of the theore...
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
TeachingGeometry According to Euclid
"... itions, from the first printed edition of 1482 up to about 1900. Billingsley, in his preface to the first English translation of the Elements (1570) [1], writes, "Without the diligent studie of Euclides Elements, it is impossible to attaine unto the perfecte knowledge of Geometrie, and consequently ..."
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itions, from the first printed edition of 1482 up to about 1900. Billingsley, in his preface to the first English translation of the Elements (1570) [1], writes, "Without the diligent studie of Euclides Elements, it is impossible to attaine unto the perfecte knowledge of Geometrie, and consequently of any of the other Mathematical Sciences." Bonnycastle, in the preface to his edition of the Elements [4], says, "Of all the works of antiquity which have been transmitted to the present time, none are more universally and deservedly esteemed than the Elements of Geometry which go under the name of Euclid. In many other branches of science the moderns have far surpassed their masters; but, after a lapse of more than two thousand years, this performance still retains its original preeminence, and has even acquired additional celebrity for the fruitless attempts which have been made to establish a different system." Todhunter, in the preface to his edition [18]
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
QUANTUM MECHANICS AS A SPACETIME THEORY
, 2005
"... Abstract. We show how quantum mechanics can be understood as a spacetime theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects ..."
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Abstract. We show how quantum mechanics can be understood as a spacetime theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects differs from that of classical objects. The systems that are nonlocal when measured in the classical spacetime continuum may be localized in the quantum continuum. We compare this new description of spacetime with the Bohmian picture of quantum mechanics. 1. What is quantum spacetime? Both modern mathematics and modern physics underwent serious foundational crises during the 20th century. The crisis in mathematics occured at the beginning of the century and the main problem was to deal with certain infinities that are directly related to the concept of real number. Poincaré [31] explained this crisis in terms of different attitudes to infinity, related to Aristotle’s actual infinity and the potential infinity (the first attitude believes that the actual infinity exists, we begin with the collection in which we find the preexisting objects, the second holds that a collection is formed by successively adding new members, it is infinite because we can see no reason why this process should stop). It led finally to the emergence of new, nonstandard definitions of real numbers. The crisis in physics concerns the interpretation of the quantum theory, the measurement problem and the question of nonlocality. In previous works we showed how in principle certain paradoxes of the quantum theory can be explained provided we enlarge our conception of number [10].Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of nonstandard real numbers that we call qrumbers. It is our goal in the present paper to analyze nonlocality and the concept of spacetime at the light of the new conceptual tools that we developed in the past.
Proof and Computation in Geometry
"... Abstract. We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computercheckable firstorder proofs in geometry. We might try to produce such proofs ..."
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Abstract. We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computercheckable firstorder proofs in geometry. We might try to produce such proofs directly, or we might try to develop a “backtranslation” from algebra to geometry, following Descartes but with computer in hand. This paper discusses the relations between the two approaches, the attempts that have been made, and the obstacles remaining. On the theoretical side we give a new firstorder theory of “vector geometry”, suitable for formalizing geometry and algebra and the relations between them. On the practical side we report on some experiments in automated deduction in these areas.
FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY
"... Abstract. Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics” leads to the development of a first ..."
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Abstract. Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics” leads to the development of a firstorder theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. Using Gentzen’s cutelimination theorem, we show that when ECG proves an existential theorem, then the things proved to exist can be constructed by Euclidean rulerandcompass constructions. In the second part of the paper we take up the formal relationships between three versions of Euclid’s parallel postulate: Euclid’s own formulation in his Postulate 5, Playfair’s 1795 version, which is the one usually used in modern axiomatizations, and the version used in ECG. We completely settle the questions about which versions imply which others using only constructive logic: ECG’s version implies Euclid 5, which implies Playfair, and none of the reverse implications are provable. The proofs use Kripke models based on carefully constructed rings of realvalued functions. “Points ” in these models are realvalued functions. We also characterize these theories in
Improved Bounds for Incidences between Points and Circles ∗
"... We establish an improved upper bound for the number of incidencesbetweenmpointsandnarbitrarycirclesinthreedimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O ∗ m 2/3 n 2/3 +m 6/11 n 9/11 +m+n (where the O ∗ (·) notation ..."
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We establish an improved upper bound for the number of incidencesbetweenmpointsandnarbitrarycirclesinthreedimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O ∗ m 2/3 n 2/3 +m 6/11 n 9/11 +m+n (where the O ∗ (·) notation hides subpolynomial factors). Since all the points and circles may lie on a common plane or sphere, it is impossible to improve the bound in R 3 without first improving it in the plane. Nevertheless, we show that if the set of circles is required to be “truly threedimensional”in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n, then the bound can be improved to O ∗ ( m 3/7 n 6/7 +m 2/3 n 1/2 q 1/6 +m 6/11 n 15/22 q 3/22 +m+n). For various ranges of parameters (e.g., when m = Θ(n) and q = o(n 7/9)), this bound is smaller than the best known twodimensional worstcase lower bound Ω ∗ (m 2/3 n 2/3 +m+n). We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound
Contributions on Hybrid Localization Techniques For Heterogeneous Wireless Networks
"... présentée par ..."