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 post

http://mathforum.org/kb/message.jspa?messageID=7944754

I communicated so:

Quote:

"And I'm talking denial of theorems here, not some "I don't understand

how this theorem can be true". One's ability or inability to

understand how a theorem is true is not a legitimate measure of

whether a theorem is in fact a theorem. Using the latter to try to

justify the former is what a mathematical crank (crackpot) does - and

oh yeah, there are a lot of them out there, and yes, some of them can

even 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.aspx

http://en.wikipedia.org/wiki/Crank_(person)

One of the responsibilities of [a] teacher [is] to teach his or her

students how to think such that they can avoid becoming cranks.

Sure, one can engage in free inquiry and "let us reason together" all

one wants, but there is a responsibility here, and that is to have a

respect for fact, for what is really true, and to accept it, and not

to deny it.

Unfortunately, since the Internet is one whale of a breeding ground

for this bane of humanity, this crackpot-ism, fact-denial is becoming

more and more prevalent, and this denial of fact is happening in all

forms whether it is mathematics denial or science denial or whatever.

People more and more seem to think that making up one's own facts is

part of what it means to be a thinker."

This "arguing pro and con" is fine as long as theorems and proofs are

not denied - if it's only a pedagogical technique for introducing

theorems and proofs, then OK. But if the agenda is to deny

mathematical theorems and proofs, then it's not fine - it's not fine

if this "pedagogical technique" is a smokescreen for denying these

facts.

> 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 and

thus the theorems are established. Crackpots deny the theorems and

proofs, while wrongly thinking that whatever controversy existed still

exists.

Perhaps your students and you need to see this:

http://en.wikipedia.org/wiki/Defect_(geometry)

Quote:

"Descartes' theorem

Descartes' theorem on the "total defect" of a polyhedron states that

if the polyhedron is homeomorphic to a sphere (i.e. topologically

equivalent to a sphere, so that it may be deformed into a sphere by

stretching without tearing), the "total defect", i.e. the sum of the

defects 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 equals

the Euler characteristic of the polyhedron. This is a special case of

the Gauss-Bonnet theorem which relates the integral of the Gaussian

curvature to the Euler characteristic. Here the Gaussian curvature is

concentrated at the vertices: on the faces and edges the Gaussian

curvature is zero and the Gaussian curvature at a vertex is equal to

the defect there.

This can be used to calculate the number V of vertices of a polyhedron

by totaling the angles of all the faces, and adding the total defect.

This total will have one complete circle for every vertex in the

polyhedron. Care has to be taken to use the correct Euler

characteristic for the polyhedron."

http://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem

Quote:

"Polyhedra

Main article: Descartes' theorem on total angular defect

Descartes' theorem on total angular defect of a polyhedron is the

polyhedral analog: it states that the sum of the defect at all the

vertices 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 the

defect is 2(pi)X. This is the special case of Gauss-Bonnet, where the

curvature 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 discrete

measure, and Gauss-Bonnet for measures generalizes both Gauss-Bonnet

for smooth manifolds and Descartes' theorem."