[FOM] philosophical literature on intuitionism

[hendrik@topoi.pooq.com]

On Mon, Oct 13, 2008 at 11:27:15PM +0100, Thomas Forster wrote:
> 
> I'm curious to know what the people who dreamt this stuff up actually 
> thought they were doing.
> 
> Where is the best place to start?  Is it Dummett's book? Did Brouwer write
> anything one might want to read? I seem to remember there is an essay in
> one of the collections (Benacerraf and Putnam?).  One of my colleagues
> here says that Intuitionism is really a form of solipsism, and that for an
> intuitionist to countenance any kind of interpretation into classical
> logic (or vice versa) is to undermine the solipsism and would not be
> welcomed by the true believers.  I do remember reading that Brouwer was
> hostile to attempts to axiomatise constructive logic..

Intuitionism and constructivism are not the same thing, though in 
pracctice they have more-or-less the same mathematics.

Constructivism is, roughly speaking, the philosophy that mathematics 
should limit itself to the things that can actualy be constructed.  This 
is opposed to so-called "classical" mathematics that allows existence 
proofs that do not manage to construct the things proved to exist (and 
leave one with no idea how to find one).

Intuitionisn is the philosophy that mathematics deals with mental 
constructions, and asserts that the creativity of the human mind makes 
it impossible to enumerate for once and for all the ways in which mental 
constructions can be carried out.  This is the way Brouwer was hostile 
to attempts to formalize intuitionistic logic.

In practice, intuitionists are to be constructive mathematicians, 
probably because the original intuitionist was, and made such a point of 
it.

But in theory, nothing would prevent an intuitionist from having an 
intuition that, for him, at least, would construct, say, large 
cardinals.  Though other intuitionists would probably ridicule the idea.

Nowadays the term "constructive" tends to be used to describe 
the mathamatics, because it is relatively objective.  Intuition is 
subjective.

-- hendrik