[FOM] Frank Quinn article in January Notices
Timothy Y. Chow
tchow at alum.mit.edu
Tue Dec 27 10:44:15 EST 2011
On Sun, 25 Dec 2011, Monroe Eskew wrote:
> 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.
> 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.
Monroe Eskew makes some good points and I agree that Quinn makes some
inaccurate statements here. However, in Quinn's defense, I think that
some of what he says can be salvaged by careful rephrasing.
By "excluded-middle reasoning," I think Quinn does *not* mean
the principle that if we know that "not P" is false, then we know
that "P" is true
the principle that we cannot know that "P" is true unless we can
prove that "not P" cannot be true.
With this definition, I think we can recognize the principle as being
something that is standard among mathematicians but foreign to most
Similarly, instead of defining mathematical *truth* as being that which is
"impossible to contradict," Quinn probably meant to say that mathematical
*knowledge* is that which is impossible to contradict, i.e., that we
cannot claim to "know" some mathematical assertion until we have proved
that it cannot be otherwise.
Now, those who have thought about f.o.m. carefully may not agree with the
above characterization of mathematical knowledge, but I think it is clear
that it is the currently-prevailing view. Just imagine what the average
mathematician thinks about questions such as, "Is it known that ZFC is
consistent?" or "Is it known that strongly inaccessible cardinals exist?"
F.o.m. experts of course realize that the answers to these questions
depend on what you mean by "known," but the typical mathematician has a
lot of trouble understanding that the word "known" is problematic here,
because the currently-prevailing view of mathematical knowledge is so
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