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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 ..."
Abstract

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 ..."
Abstract

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.
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 ..."
Aristotle’s prohibition rule on kindcrossing
"... and the definition of mathematics as a science of quantities 1 ..."