On Sun, Dec 30, 2012 at 12:05 PM, kirby urner <firstname.lastname@example.org> wrote: ... >> 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. >
Let's not forget that even in the area of conjecture, there are still crackpots and crackpot thinking - it's not true that anything goes, since if one is not careful, one can end up via conjecturing denying already established theorems and proofs without even knowing it at first and when it is pointed out that one's conjectures do negate prior definitions, or the theorems and proofs that ensue, and one persists, then one no longer avoids crackpot-ism.
If one wants to redefine one term, then one may have to redefine a whole lot of other terms as well (such as perhaps what a given numeral symbol means) to avoid end up denying already established fact, or one will easily end up falling into crackpot-ism.
That is, one needs to know and understand enough the relevant mathematics to avoid being a crackpot making crackpot conjectures or crackpot systems.
>> 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. >
There's more to making mathematical conjectures or creating new theorems and proofs based on alternative definitions than simply making or creating them.
There was some talk earlier about creating a system in which -(ab) = (-a)(-b) was true for all a,b in the set rather than ab = (-a)(-b) being true for all a,b in the set.
But I pointed out that since it is already established fact that ab = (-a)(-b) is true for all a,b in any ring, then any system that negates ab = (-a)(-b) being true for all a,b in the set and replaces it in which -(ab) = (-a)(-b) was true for all a,b in the set cannot be an example of a ring - one has to go outside of a ring to create such a system. And since the real numbers and even the integers are examples of rings, then this new theorem cannot be true in the real numbers or integers without redefining even these terms "real numbers" or "integers" themselves into meaning objects that are not examples of a ring.
By the way, to generalize this:
In order to negate ab = (-a)(-b) being true for all a,b in the set and replace it with -(ab) = (-a)(-b) being true for all a,b in the set, one has to go all the way outside of a ringoid that is either a left-cancellative loop under addition or a right-cancellative loop under addition, since in either of these we can prove ab = (-a)(-b) being true for all a,b in the set. No, we do not need either the associative or commutative property under addition and we do not need even one of the algebraic properties under multiplication save closure. (See http://en.wikipedia.org/wiki/Loop_(algebra)#Loop for more. For a set being left- or right-cancellative, see http://en.wikipedia.org/wiki/Cancellation_property for more.)
Two such proofs:
For all a,b in a ringoid that is a left-cancellative loop under addition,
0 = 0 a0 = 0(-b) a(-b + b) = (a + (-a))(-b) (a)(-b) + ab = (a)(-b) + (-a)(-b) ab = (-a)(-b).
For all a,b in a ringoid that is a right-cancellative loop under addition,