[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

but rather

  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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