FOM: intuitionistic mathematics and building bridges

Stephen G Simpson simpson at
Tue Mar 3 22:38:09 EST 1998

Neil Tennant 27 Feb 1998 14:38:39 writes:

 > I don't agree with Steve that intuitionism has to be based on
 > solipsistic or subjectivist philosophy.  The whole interest in
 > Dummett's meaning- theoretic justification of intuitionism is that
 > it gets away from the earlier solipsistic motivation that Brouwer
 > had.

My account was based on Brouwer's original subjectivistic
philosophical underpinning of intuitionistic mathematics.  Neil seems
to be saying that Dummett has an essentially different philosophical
position with the same mathematical outcome.  My initial reaction to
this is skeptical, because I feel that it's difficult in principle to
separate f.o.m. research from philosophical motivation.  If Dummett's
philosophical position is truly different from Brouwer's, then this
should be reflected in the mathematics.

On the other hand, maybe Dummett's philosophical ideas aren't really
so very different from Brouwer's.  For instance, Neil says:

 > the Dummettian can claim that once one *does* take the physical
 > reality of actual communication into account, one is led along a
 > justificatory path to intuitionism.

and this confusion of physical reality with language does indeed
strike me as subjectivist.  So I'm really not sure what is going on

Neil, instead of asking me to delve into long books by you and
Dummett, could you please summarize the story briefly here on the FOM
list?  I'm pretty ignorant about Dummett's views.  What is the essence
of Dummett's philosophical position, and what kind of mathematics does
it lead to?  Here I am echoing Harvey's request in his posting of 2
Mar 1998 12:02:04.  Harvey said to Neil:

 > You seem to suggest that there is a coherent view of intuitionism -
 > different than Godel, Heyting, and Brouwer. I would like to see you
 > explain *carefully* what this is without sending us to papers and
 > books. If it is good, it can explained on the fom, at least enough
 > so that one can get a feel for it.

and I agree with this.

-- Steve

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