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 <firstname.lastname@example.org> 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
I communicated so:
"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:
"Every discipline has its crackpots: Stories of mathematics"
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
> 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
Perhaps your students and you need to see this:
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.
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."
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."