I don't see where I've denied any theorems in setting up a tension between the deficit at each point and an argument that gets rid of the deficit at some limit, using an epsilon / delta approach. Rather, I've introduced several new theorems and suggested we prove them.
> One of the responsibilities of [a] teacher [is] to teach his or her > students how to think such that they can avoid becoming cranks. >
Lets not forget that mathematics is more than a scaffolding of proved theorems. It consists of conjectures (unproved) upon which other scaffolding may be contingent, and it depends on definitions which alter what's proved and/or provable.
For example, per standard definitions, a cube with edges SQRT(2) must have a volume of SQRT(8) because our definition of 3rd powering incorporates the geometric hexahedron as a model of 3rd powering.
In contrast, a Martian civilization might consider the regular tetrahedron their 3rd powering model. There'd be a conversion constant and all volume ratios would remain the same (e.g. cube to tetrahedron inscribed as face diagonals = 3:1).
The argument *against* teaching both Earthling and Martian mathematics side by side would be we don't want to confuse students. We want them to reflexively think and say "squared" and "cubed" in connection with 2nd and 3rd powering, not "triangled" and "tetrahedroned". PhDs might explore alternatives, but such forays into non-orthogonal (aka unorthodox) thinking would only scramble the brains of the young.
The argument *for* teaching the two paradigms in close conjunction is this approach retains more flexibility in thought and allows the role of definition and cultural choice to remain in the foreground. I'd consider such multi-paradigm training more useful for students planning a career in diplomacy / international relations, where rigid adherence to the "one right way" is too close to mindless fundamentalism for comfort. Better to scramble their brains a little than to encourage strait-jacketed habits of mind.
So here's another case where there's legitimate debate and yet no one is denying any proved theorems. There's no need to fling any accusations that one side or the other are just crackpots.
In my view, any "one size fits all" approach to math pedagogy is ill advised. This is different from saying we should have a fast track for the talented and gifted and various slower tracks for those less good at math. I'm saying not all who are stellar at math need the same training, even in early childhood. Those trying to impose the "one right way" (what topics to include or exclude, and in what sequence) are the closest to crackpots in my vista.
> 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. >
There's more to mathematics than theorems and proofs.