[FOM] Quinn article, teaching math, excluded middle and constructive mathematics

[Nick Nielsen]

On Wed, Dec 28, 2011 at 3:41 PM, frank waaldijk <fwaaldijk at gmail.com> wrote:

> ...we still know nothing for sure. We are stumbling in the dark.
>
> ...Classical mathematics, in its fullfledged embrace of excluded middle, can be compared to science fiction...or dreamland if you would like a stronger metaphor. It's nice to dream, and nice > to be able to conjure battlestars and time travel and black hole mining and...
>
> But it is also important to return to reality from time to time. This is where constructive mathematics comes in. Constructive mathematics and classical mathematics are not always at odds
> per se...it is `just' a major difference of focus and perspective. But I am personally convinced that we need constructive mathematics for a better understanding of our physical world and
> physical reality. And constructive views on excluded middle should already be taught in high school, not exclusively but at least for comparison.

Perhaps a more charitable conception of the relationship between
classical eclecticism and its tolerance of non-constructive modes of
reasoning on the one hand, and on the other hand the many species of
constructivism that have been proposed to place limits on classical
eclecticism, is to be found in an image proposed by Alain Connes:

"Constructivism may be compared to mountain climbers who proudly scale
a peak with their bare hands, and formalists to climbers who permit
themselves the luxury of hiring a helicopter to fly over the summit."

(Conversations on Mind, Matter, and Mathematics, Changeux and Connes,
Princeton, 1995, p. 42)

On the next page Connes says, continuing the image, "...the
uncountable axiom of choice gives an aerial view of mathematical
reality -- inevitably, therefore, a simplified view."

If we think of the constructivist perspective very roughly as a
"bottom up" approach, like a mountain climber who starts at the base
and clambers over every cliff and every ledge on the way up, and
non-constructive methods as a "top down" approach, an aerial view of
mathematics, perhaps lacking in definite detail, but giving the big
picture of the scene, then the two approaches are complementary. An
adequate conception of mathematical reality must include both
constructive and non-constructive approaches, rather than dismiss
classical mathematics as science fiction or dreamland.

I suggest that the top-down perspective of classical mathematics and
the bottom-up perspective of constructivism meet in the middle, and
that middle is constituted by macroscopic mathematical intuitions --
the familiar instances of mathematical knowledge like counting with
cardinal numbers. The classical foundationalist project plunges down
from the heights and seeks to immerse itself in the details of
mathematical knowledge from above; the constructivist seeks to build
from below only what can be built step-by-step, and neglecting the big
picture and therefore blind to the landscape in which he patiently
builds.

To put it in a quasi-scientific idiom, constructivism "explains"
macroscopic mathematical intuitions as being constructed from
ur-intuitions (as, for example, from Brouwer's first and second acts
of intuitionism), while classical eclecticism "explains" macroscopic
mathematical intuitions from the top down, with reference to the
abstract mathematical entities, from which flow macroscopic
mathematical intuitions when the mind directly "perceives"
mathematical reality.

The constructivist, who is a mathematical fox and knows many little
things, and the classical mathematician, who is tolerant of
non-constructive methods and as a mathematical hedgehog knows one big
thing, need each other.

Best wishes,

Nick Nielsen