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

frank waaldijk fwaaldijk at gmail.com
Wed Jan 4 06:01:32 EST 2012

I wrote earlier:

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.

which prompted Nick Nielsen to write:

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 think that to compare classical mathematics to science fiction is not the
same as to dismiss it. Actually I'm quite appreciative of good science
fiction. And I also think it is important to dream. I just stated that it
is also important to return to reality from time to time.

Connes' image above suggests that constructive math and classical math
actually study the same mathematical earth, from a different point of view.
For very important parts of mathematics, I can more or less live with this
image although I also consider it too simplified and too charitable.
However, the unrestricted axiom of choice gives rise to whole galaxies
which are not studied in detail by constructive mathematicians (because to
them these galaxies are rather wild science fiction, with little realistic
content). Perhaps a nice short paper to also read is `Reality and Virtual
Reality in Mathematics' by Douglas Bridges (

Apart from that, my more important point was and is that there could be a
better balance between the large number of classical mathematicians which
swarm in the sky and the handful of constructive mathematicians who try to
validate what has been glimpsed from above. So my intention was not to
dismiss classical mathematics (since I value its helicopter view also) but
to emphasize that for a better understanding of reality, we need
constructive mathematics. And for this we need more researchers in this
field, and we also should teach constructive views already in high school,
not exclusively like I said, but for comparison. One should not forget that
all constructive mathematicians have had a heavy training in classical
math...but vice versa?

This to me seems the deeper implication of the foundational crisis that
Frank Quinn mentioned. Some progress can be noted, since nowadays
acceptance of constructive mathematics seems much better than in Brouwer's
time. On the other hand, one still finds occurrences of Zorn's lemma
(equivalent to axiom of choice) being used in texts on number theory to
prove the existence of a maximal ideal, where a simple constructive proof
can also be given...This means that generally, mathematicians have little
feel for the constructive level/content of what they are doing. Reverse
mathematics then helps only so much.

In reply to my quote above, Panu Raatikainen wrote:

These are strong claims, and we've heard them now and then before, but it
> would be nice to hear some convincing arguments supporting them... I've
> honestly tried hard to find one for some time, but have so far failed...

I would like to turn this around. Classical mathematics with the axiom of
choice for example claims the existence of an non-principal ultrafilter on
N. Now that I consider a strong claim, that I've heard now and then
before...and it would be nice to hear some convincing argument to support
this existence... As a student, I loved ultraproducts and ultrafilters and
non-standard analysis...so it's not like I dismiss all these ideas out of
hand. I just can't find any real description of such a non-principal
ultrafilter. So if anyone can define for me just one non-principal
ultrafilter on N, I might take back my remarks on science fiction and

(By the way, there seem to be some constructive developments emulating
ultrafilters and non-standard analysis, but I'm completely ignorant on this

kind regards,

Frank Waaldijk
http://www.fwaaldijk.nl/mathematics.html (I had the address wrong in the
previous post, sorry)
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