[FOM] Classical logic and the mathematical practice

[Timothy Y. Chow]

On Fri, 13 May 2005, Andrej Bauer wrote:
> Timothy Y. Chow wrote:
>> It generally takes considerable philosophical education before
>> someone seriously entertains the notion that if there is no way to
>> decide whether X happened or not, then maybe X neither happened nor
>> didn't happen.
>
> I suspect Tim did not want to say what he said in this paragraph, namely 
> that "people of considerable philosophical education" might entertain 
> the possibility that "not (X or not X)" is valid.

Right, I was careless.  While it is true that, as you say, some "people
of considerable philosophical education might actually do precisely that,"
it was misleading of me to introduce this concept in the context of 
constructivism.  Nevertheless, the point remains that these sorts of 
distinctions usually elude those without a lot of philosophical training.

Someone else suggested to me offline that commonsense thinking doesn't 
automatically assume bivalence, e.g., in the case of future events.  It's 
true that there are plenty of ostensibly yes/no statements that common 
sense balks at giving either a flat yes or no answer to ("Have you stopped 
beating your wife yet?"  "Surely this either is a heap or it isn't a 
heap?"  "Isn't Eva Longoria the most beautiful woman in the world?" etc.). 
But I would say that in these cases the naive tendency is to blame the 
problem on various malformed or imprecise features of the sentences or 
concepts in question, and to claim that mathematical statements have a 
clarity and precision that avoids these problems.  Again, it takes 
considerable education to articulate objections to this naive view.

Tim