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Möbius transformations and ellipses
 The Pi Mu Epsilon Journal
, 2007
"... This expository article considers noncircular ellipses in the Riemann
sphere, and the action of the group of Mobius transformations. In
particular, we find which Mobius transformations are symmetries of an
ellipse, and which take one ellipse to another. We also survey some
of the ``special plane ..."
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Cited by 5 (3 self)
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This expository article considers noncircular ellipses in the Riemann
sphere, and the action of the group of Mobius transformations. In
particular, we find which Mobius transformations are symmetries of an
ellipse, and which take one ellipse to another. We also survey some
of the ``special plane curves'' which appear as inversive images of
the ellipse.
Rethinking geometrical exactness
 Historia Mathematica
"... A crucial concern of earlymodern geometry was that of fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed in it. According to Bos, this is the exactness concern. I argue that Descartes ’ way to respond to this concern was to suggest a ..."
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Cited by 2 (2 self)
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A crucial concern of earlymodern geometry was that of fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed in it. According to Bos, this is the exactness concern. I argue that Descartes ’ way to respond to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In section 1, I outline the exactness concern as, I think, it appeared to Descartes. In section 2, I account for Descartes ’ views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’ geometry and EPG. Une question cruciale pour la geométrie à l’âge classique fut celle de décider si certains objets, procédures ou arguments devaient ou non être admis au sein de ses limites. Selon Bos, c’est la question de l’exactitude. J’avance que Descartes répondit à cette question en suggérant une extension conservative de la géomètrie plane d’Euclide (EPG). Dans la section 1, je reconstruis la question de l’exactitude ainsi que, selon moi, elle se présentait d’abord aux yeux de Descartes. Dans la section 2, je rends compte des vues de Descartes sur la question de l’exactitude et de son attitude face au types de constructions plus communes dans la geométrie classique. Je montre aussi en quel sens sa geométrie peut se concevoir comme une extension conservative de EPG. Je conclue en discutant brièvement certaines analogies et différences structurales entre la geométrie de Desacrtes et EPG.
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
Aristotle’s prohibition rule on kindcrossing
"... and the definition of mathematics as a science of quantities 1 ..."
iv Promotor: prof.dr. W.R. de Jong
"... Proofs, intuitions and diagrams Kant and the mathematical method of proof ..."
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Proofs, intuitions and diagrams Kant and the mathematical method of proof
Ellipses in the inversive plane
"... This expository article considers noncircular ellipses in the Riemann
sphere, and the action of the group of Mobius transformations. In
particular, we find which Mobius transformations are symmetries of an
ellipse, and which take one ellipse to another. We also survey some
of the ``special plane ..."
Abstract
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This expository article considers noncircular ellipses in the Riemann
sphere, and the action of the group of Mobius transformations. In
particular, we find which Mobius transformations are symmetries of an
ellipse, and which take one ellipse to another. We also survey some
of the ``special plane curves'' which appear as inversive images of
the ellipse.
ARISTOTELIAN REALISM
"... Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as rat ..."
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Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,
Indirect proofs and proofs from assumptions
"... In his valuable book on mathematics and its philosophy in the ..."