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A formal system for Euclid’s elements
 THE REVIEW OF SYMBOLIC LOGIC
, 2009
"... We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. ..."
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We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.
Emergent spacetime and empirical (in)coherence
 Studies in History and Philosophy of Modern Physics
, 2013
"... Numerous approaches to a quantum theory of gravity posit fundamental ontologies that exclude spacetime, either partially or wholly. This situation raises deep questions about how such theories could relate to the empirical realm, since arguably only entities localized in spacetime can ever be observ ..."
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Numerous approaches to a quantum theory of gravity posit fundamental ontologies that exclude spacetime, either partially or wholly. This situation raises deep questions about how such theories could relate to the empirical realm, since arguably only entities localized in spacetime can ever be observed. Are such entities even possible in a theory without fundamental spacetime? How might they be derived, formally speaking? Moreover, since by assumption the fundamental entities can’t be smaller than the derived (since relative size is a spatiotemporal notion) and so can’t ‘compose ’ them in any ordinary sense, would a formal derivation actually show the physical reality of localized entities? We address these questions via a survey of a range of theories of quantum gravity, and generally sketch how they may be answered positively.
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...
Improved Bounds for Incidences between Points and Circles
, 2013
"... We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. 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 ∗ (·) ..."
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We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. 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
PROOF STYLE AND UNDERSTANDING IN MATHEMATICS I: VISUALIZATION, UNIFICATION AND AXIOM CHOICE
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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
NonEuclidean III.36
 Amer. Math. Monthly
"... is held by those that are true in neutral geometry, that is, without either assuming or denying the parallel postulate. Such, for example, are the first twentyeight propositions of Euclid’s Elements, including the triangle congruences SAS (I.4),1 SSS (I.8), ASA (I.26), and the base angles of an is ..."
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is held by those that are true in neutral geometry, that is, without either assuming or denying the parallel postulate. Such, for example, are the first twentyeight propositions of Euclid’s Elements, including the triangle congruences SAS (I.4),1 SSS (I.8), ASA (I.26), and the base angles of an isosceles triangle are equal (I.5). Some concur
The twofold role of diagrams in Euclid’s plane geometry
"... Proposition I.1 of Euclid’s Elements requires to “construct ” an equilateral triangle on a “given finite straight line”, or on a given segment, in modern parlance1. To achieve this, Euclid takes this segment to be AB (fig. 1), then describes two circles with its two extremities A and B as centres, a ..."
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Proposition I.1 of Euclid’s Elements requires to “construct ” an equilateral triangle on a “given finite straight line”, or on a given segment, in modern parlance1. To achieve this, Euclid takes this segment to be AB (fig. 1), then describes two circles with its two extremities A and B as centres, and takes for granted that these circles intersect each other in a point C distinct from A and B. This last step is not warranted by his explicit stipulations (definitions, postulates, common notions). Hence, either his argument is flawed, or it is warranted on other grounds. According to a classical view, “the Principle of Continuity ” provides such another ground, insofar as it ensures “the actual existence of points of intersection ” of lines ([7], I, ∗Some views expounded in the present paper have been previously presented in [30], whose first version was written in 1996, during a visiting professorship at the Universidad Nacional Autónoma de México. I thank all the people who supported me during my stay there. Several preliminary versions of the present paper have circulated in different forms and one of them is available online at
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.
Lines, Circles, Planes and Spheres
, 2009
"... Let S be a set of n points in R 3, no three collinear and not all coplanar. If at most n−k are coplanar and n is sufficiently large, the total number of planes determined is at least 1+k ` ´ ` ´ ` ´ n−k k n−k −. For similar conditions 2 2 and sufficiently large n, (inspired by the work of P. D. ..."
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Let S be a set of n points in R 3, no three collinear and not all coplanar. If at most n−k are coplanar and n is sufficiently large, the total number of planes determined is at least 1+k ` ´ ` ´ ` ´ n−k k n−k −. For similar conditions 2 2 and sufficiently large n, (inspired by the work of P. D. T. A. Elliott in [1]) we also show that the number of spheres determined by n points is at least 1 + ` ´ n−1 orchard − t 3 3 (n − 1), and this bound is best possible under its hypothesis. (By t orchard 3 (n), we are denoting the maximum number of threepoint lines attainable by a configuration of n points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles. 2