On Sat, Aug 16, 2008 at 8:32 AM, Bastien <[EMAIL PROTECTED]> wrote: > "Bill Kerr" <[EMAIL PROTECTED]> writes: > >> • intuition > > [...] > >> • different ways of looking at maths (constructive and intuitive compared >> with rule driven and formal)
It turns out that there is no essential difference between the systems constructed under these seemingly quite different programs. Each contains a model of the others, in much the same way that one can find subspaces of Euclidean space (sphere and pseudosphere) with non-Euclidean geometries, and subspaces of non-Euclidean spaces (horospheres, Clifford's surfaces) with Euclidean geometry. > [...] > >> • other mathematicians who hold similar views - Poincare, Brouwer, Godel) In my study of Poincaré, Brouwer, and Gödel, I found little in common among their views. What are you talking about? > I'd be curious on how Cynthia relates mathematical theories (like > intuitionism) to pedagogical theories. Piaget was greatly impressed by Brouwer's Intuitionism, with its rejection of excluded middle and other "non-intuitive" ideas, but mathematicians are not. It turns out that classical mathematics can be completely modeled within Intuitionism. I found the arguments over mathematical philosophy to be quite arid, particularly those about the nature of mathematical objects. Do they have independent existence, do they exist only in our minds, or do they not exist at all, and only the symbols we work with directly have real existence? None of these questions has any bearing on what theorems can be proven from what sets of axioms. None of them has any bearing on the applications of mathematics. The major practical effect that I have seen from these arguments is the refusal of some mathematicians to study certain questions, a result that I consider in general lamentable. But what can you do? Nobody can study everything any more. In some cases, mathematicians have been inspired by ontological arguments to take up questions that otherwise would not have been studied, which is to the good. Math teachers need to be aware of some of these views, because schoolchildren may well discover them, and other conundrums and paradoxes, and may need help at some points to get past the difficulties that they can create. It is useful to distinguish constructive set theory, as a subset of more general set theories, in the same way that it is useful to distinguish problems with algorithmic solutions in linear time from those that are more difficult (requiring higher-order polynomial time, or even exponential time) or are frankly unsolvable (the undecidable, such as the Halting Problem for computer programs, or the consistency of arithmetic, or membership in any recursively-enumerable but non-recursive set). Working mathematicians have nearly all gone over to the non-constructive side. Hardly anything considered worth studying in these days of categories and toposes is constructive in nature. > What is the "similar views" > that Poincaré, Brouwer and Gödel are holding? Is that views about > pedagogy or views about mathematics (namely intuitionism)? > > Can you tell me more about this? (or send me pointers?) > > Thanks! > > -- > Bastien > _______________________________________________ > IAEP -- It's An Education Project (not a laptop project!) > [EMAIL PROTECTED] > http://lists.sugarlabs.org/listinfo/iaep -- Silent Thunder [ 默雷 / शब्दगर्ज ] is my name, And Children are my nation. The Cosmos is my dwelling place, And Truth my destination. _______________________________________________ Grassroots mailing list [email protected] http://lists.laptop.org/listinfo/grassroots
