[FOM] Frank Quinn article in January Notices

David Roberts david.roberts at adelaide.edu.au
Mon Dec 26 19:30:29 EST 2011

Dear Monroe,

I thought I should point out that whenever statistical reasoning is
involved in exact sciences (and some inexact ones), one inherently cannot
assume excluded middle. Hypothesis testing - in its simplest form asking
whether a measurement yields a null result - is full of phrases like 'fail
to reject the null hypothesis at x level of uncertainty', which is
definitely *not* the same as accepting the null hypothesis. This is one
area where beginning students of statistics trip up all the time, mostly
because the are expecting, implicitly, EM to hold.

Witness two recent examples, namely faster-than-light neutrinos and the
not-quite-discovery of the Higgs particle. Everything is stated in
statistical terms, levels of significance and so on. And when they analyze
things like the so-called look elsewhere effect in the latter, one can see
vestiges of intuitionistic reasoning.

David Roberts

On Dec 27, 2011 6:35 AM, "Monroe Eskew" <meskew at math.uci.edu> wrote:
> Dear all,
> I appreciate Quinn's goal of arguing for the importance of "core"
mathematics, but I disagree with him on an important point, which if
approached differently, would better serve the big goal.  The idea that
"excluded middle reasoning" is somehow uniquely applicable to mathematics
seems quite bizarre to me, and if true, would seem to count against the
relevance of pure mathematics.  I think I remember reading Brouwer arguing
for the exact opposite view-- that physical reality is well-determined and
so statements about it either hold or fail, but mathematics doesn't have
this property because truth means provability.  Quinn claims that the
excluded middle cannot hold for reality because it is impossible to
describe reality precisely.  This is a strong claim for which he gives
almost no argument.  Why does excluded middle require complete precision in
description (or sense, or reference)?  Why is it always impossible to
describe relevant slices of reality with enough precision?!
>   Anyway, a small amount of historical evidence shows that it's not as
though excluded middle reasoning is something unique to modern mathematics
or the modern era. His statement about what Hilbert should have said
regarding the excluded middle is, in my opinion, totally misguided, and I'm
glad Hilbert never said such a thing.
> Also, Quinn says something that I think is really an error rather than
something on which reasonable people could disagree.  He says that
mathematical truth means "impossible to contradict."  Not only is this an
inaccurate description of logic, it is inconsistent-- There are of course
statements P for which P and ~P are both impossible to contradict from
given axioms.
> Best,
> Monroe
> On Dec 22, 2011, at 11:30 AM, "Timothy Y. Chow" <tchow at alum.mit.edu>
> > Frank Quinn has an article in the current (January 2012) Notices of the
> > AMS entitled, "A Revolution in Mathematics? What Really Happened a
> > Ago and Why It Matters Today."
> >
> > http://www.ams.org/notices/201201/rtx120100031p.pdf
> >
> > Strictly speaking there is not much here that hasn't already been said
> > Quinn, or others, before, but I felt that Quinn has articulated several
> > points particularly clearly and forcefully in this article.  I
> > believe that his main thesis is correct and important: that mathematics
> > underwent a revolution in precision and rigor about a century ago; that
> > the reasons for this revolution are poorly understood by almost
> > even many professional mathematicians; and that this lack of
> > has caused dangerously widening rifts in the mathematical, scientific,
> > educational worlds.
> >
> > Tim
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