[FOM] philosophical literature on intuitionism

Vaughan Pratt pratt at cs.stanford.edu
Wed Oct 15 16:01:02 EDT 2008

Frank Waaldijk wrote:
> So why choose classical mathematics 
> over intuitionism? As a formal game, what would be the higher worth of one 
> over the other?

I have more confidence in my classical understanding of Heyting algebras 
than in my intuitionistic understanding of Boolean algebras.  In 
particular I find it very hard to be sure that I'm maintaining a "clean 
room" environment when trying to reason strictly intuitionistically 
about any sort of mathematics, that is, sticking to thought processes 
that haven't been contaminated by some classical principal that is 
intuitionistically unsound.

The essential asymmetry here is that while classical reasoning can 
contaminate intuitionistic, the converse cannot happen.  The benefits of 
sterility are greatly overblown.  Living the intuitionistic mathematical 
life is like living the germ-free life in a six-foot bubble.  Humans 
evolved to get along fine with germs by and large, long before they'd 
heard of sterilization, and mathematicians evolved to get along nicely 
with reasoning that is not necessarily intuitionistically sound, long 
before they'd heard of intuitionism.

Vaughan Pratt

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