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

frank waaldijk fwaaldijk at gmail.com
Wed Dec 28 18:41:12 EST 2011

Frank Quinn's article (http://www.ams.org/notices/201201/rtx120100031p.pdf, as
put forward by Timothy Chow in a previous posting) to me is a strange
combination of insight and blindness.

I find Quinn's thoughts on modern mathematical methods vs. older
mathematical methods sufficiently convincing, in the sense that the modern
`precise´ foundational style (building with axioms and precise definitions)
has strong advantages over the older more intuitive styles.

I also agree that this modern style should be more present when teaching
mathematics, at least when I consider how mathematics is usually taught in
the Netherlands (apart from university math degrees). In fact I find our
way of teaching mathematics in secondary school generally appalling,
notwithstanding good teachers who try to overcome the methods and didactics
prescribed. I´m sure I will irritate many by saying this, but I see many
high school students receiving 3-4 hours of math a week for 6 years who
cannot even reliably perform addition/subtraction and
multiplication/divisions with fractions or simple symbolic terms. Let alone
understand what they are doing when performing these arithmetical

What in heaven's name can be the use of this? These students will need
basic arithmetic in their lives far more than trigonometry and differential
equations. They do not really understand what they are supposed to
`perform´ in higher math and they generally don´t get the impression that
understanding and logical precision are valued. There is simply no time to
give them a feel for real precision, and this leads to a quite large number
of students who are confused because of insufficient precision.

They leave high school with a general sense of hatred of mathematics, but
for the wrong reasons! Because they have been saddled with the idea that
they are too stupid to understand mathematics, whereas in fact the math
taught in high school is often highly imprecise and badly phrased.

So I would like just once to put forward my view that it would be far
better to teach these kids arithmetic well, and mathematics only as far as
their understanding speed allows. And enough of this mathematics should
also be highly precise, so that students develop  a real feel for logical
precision and mathematical methods. (Of course for the more mathematically
gifted students, more can be achieved). The result will be, in my perhaps
not so humble opinion, that the general population will have better skills
in arithmetic (which is highly needed) and also an appreciation of
mathematics (instead of a misplaced hatred).

But enough on this. Now to what I consider blindness in Quinn's article.

He writes on the issue of `excluded middle´, for instance:

As it was, this issue was a constant pain for
> Hilbert; Brouwer’s Intuitionist school kept it alive
> into the 1930s; and it died out more from lack of
> interest than any clear resolution.

Of course it did not die out, as we are well aware on this forum. Also in
the rest of the article it seems to me that Quinn has largely missed the
repercussions of the foundational crisis of the previous century. He is
sadly not alone in this. But his own conclusion therefore applies to him
and the large number of similar unaware mathematicians. Namely that one
should not go on ignoring these foundational issues, just because one's
peers also do so with blissful insouciance. The revolution in mathematics
spreads much further than `just´ methodology. The revolution is about the
concepts underlying all of our thinking about math, science and reality.

And my preliminary conclusion is: we still know nothing for sure. We are
stumbling in the dark.

I therefore cannot take Quinn's stance on the role of `excluded middle´ in
mathematics very seriously. 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

This will also help dispel the notion of many high school graduates that
mathematics is a done-and-finished thing. Where in fact we are almost
completely ignorant about really fundamental questions, perhaps not as
badly as in physics, but these disciplines are entwined also in the debates
on finiteness-infinity, determinism-indeterminism, etc.

So if we could present our own ignorance as a challenge to the next
generation, would that not be far more attractive for young and eager minds?

Anyway, to wrap it up, I find Frank Quinn's article thought-provoking and
well written, and can only wish for a continuation of dialogue on these
matters. That´s all I guess, I hope it contributes something...

kind regards,

Frank Waaldijk
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