[FOM] Frank Quinn article in January Notices
meskew at math.uci.edu
Sun Dec 25 23:21:31 EST 2011
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.
On Dec 22, 2011, at 11:30 AM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
> Frank Quinn has an article in the current (January 2012) Notices of the
> AMS entitled, "A Revolution in Mathematics? What Really Happened a Century
> Ago and Why It Matters Today."
> Strictly speaking there is not much here that hasn't already been said by
> Quinn, or others, before, but I felt that Quinn has articulated several
> points particularly clearly and forcefully in this article. I personally
> 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 everybody,
> even many professional mathematicians; and that this lack of understanding
> has caused dangerously widening rifts in the mathematical, scientific, and
> educational worlds.
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