[IAEP] [Grassroots-l] Concise explanation of Constructionism from the Learning Team

[Edward Cherlin]

On Sat, Aug 16, 2008 at 8:32 AM, Bastien <bastienguerry at googlemail.com> wrote:
> "Bill Kerr" <billkerr at gmail.com> 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
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