```Date: Dec 30, 2012 3:16 AM
Author: Paul A. Tanner III
Subject: Re: A Point on Understanding

On Sat, Dec 29, 2012 at 9:52 PM, kirby urner <kirby.urner@gmail.com> wrote:>>>> This feeble attempt to change the subject won't work. The context here>> is denial of mathematical theorems, pure and simple. There is no>> debate in the denial of a theorem.>>>> So my I conclude you're against this strategy of arguing pro and con a> mathematical proposition?No, and in my last posthttp://mathforum.org/kb/message.jspa?messageID=7944754I communicated so:Quote:"And I'm talking denial of theorems here, not some "I don't understandhow this theorem can be true". One's ability or inability tounderstand how a theorem is true is not a legitimate measure ofwhether a theorem is in fact a theorem. Using the latter to try tojustify the former is what a mathematical crank (crackpot) does - andoh yeah, there are a lot of them out there, and yes, some of them caneven be trained in science and mathematics:http://books.google.com/books/about/Mathematical_Cranks.html?id=HqeoWPsIH6EC"Every discipline has its crackpots: Stories of mathematics"http://blogs.msdn.com/b/oldnewthing/archive/2006/05/29/610090.aspxhttp://en.wikipedia.org/wiki/Crank_(person)One of the responsibilities of [a] teacher [is] to teach his or herstudents how to think such that they can avoid becoming cranks.Sure, one can engage in free inquiry and "let us reason together" allone wants, but there is a responsibility here, and that is to have arespect for fact, for what is really true, and to accept it, and notto deny it.Unfortunately, since the Internet is one whale of a breeding groundfor this bane of humanity, this crackpot-ism, fact-denial is becomingmore and more prevalent, and this denial of fact is happening in allforms whether it is mathematics denial or science denial or whatever.People more and more seem to think that making up one's own facts ispart of what it means to be a thinker."This "arguing pro and con" is fine as long as theorems and proofs arenot denied - if it's only a pedagogical technique for introducingtheorems and proofs, then OK. But if the agenda is to denymathematical theorems and proofs, then it's not fine - it's not fineif this "pedagogical technique" is a smokescreen for denying thesefacts.> Surely you're not suggesting mathematics should be presented as free> from controversy, as that would go against the historical facts.There is no more controversy once the issue is settled via proof andthus the theorems are established. Crackpots deny the theorems andproofs, while wrongly thinking that whatever controversy existed stillexists.Perhaps your students and you need to see this:http://en.wikipedia.org/wiki/Defect_(geometry)Quote:"Descartes' theoremDescartes' theorem on the "total defect" of a polyhedron states thatif the polyhedron is homeomorphic to a sphere (i.e. topologicallyequivalent to a sphere, so that it may be deformed into a sphere bystretching without tearing), the "total defect", i.e. the sum of thedefects of all of the vertices, is two full circles (or 720[degrees]or 4pi radians). The polyhedron need not be convex.[1]A generalization says the number of circles in the total defect equalsthe Euler characteristic of the polyhedron. This is a special case ofthe Gauss-Bonnet theorem which relates the integral of the Gaussiancurvature to the Euler characteristic. Here the Gaussian curvature isconcentrated at the vertices: on the faces and edges the Gaussiancurvature is zero and the Gaussian curvature at a vertex is equal tothe defect there.This can be used to calculate the number V of vertices of a polyhedronby totaling the angles of all the faces, and adding the total defect.This total will have one complete circle for every vertex in thepolyhedron. Care has to be taken to use the correct Eulercharacteristic for the polyhedron."http://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theoremQuote:"PolyhedraMain article: Descartes' theorem on total angular defectDescartes' theorem on total angular defect of a polyhedron is thepolyhedral analog: it states that the sum of the defect at all thevertices of a polyhedron which is homeomorphic to the sphere is 4(pi).More generally, if the polyhedron has Euler characteristic X = 2 - 2g(where g is the genus, meaning "number of holes"), then the sum of thedefect is 2(pi)X. This is the special case of Gauss-Bonnet, where thecurvature is concentrated at discrete points (the vertices).Thinking of curvature as a measure, rather than as a function,Descartes' theorem is Gauss-Bonnet where the curvature is a discretemeasure, and Gauss-Bonnet for measures generalizes both Gauss-Bonnetfor smooth manifolds and Descartes' theorem."
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