[FOM] philosophical literature on intuitionism

William Tait williamtait at mac.com
Thu Oct 16 11:31:57 EDT 2008

On Oct 14, 2008, at 1:08 PM, Frank Waaldijk wrote:

> dear Thomas,
> you wrote about intuitionism:
>> 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..
> Perhaps I could pose a counterquestion about classical mathematics?
> You see, I'm curious to know what the people who dreamt up classical  
> math,
> especially unrestricted ZFC, actually thought they were doing.

Thomas might have expressed himself more respectfully, but I think  
that his question has a point which Frank's question about classical  
math does not.

If we leave aside ZFC and just focus on the development of mathematics  
in the nineteenth century, classical math was not dreamed up by  
anyone: It just happened---say starting with Bolzano's proof of the  
intermediate value theorem in 1817. ZFC itself maybe does trace back  
to something one person, namely Cantor, dreamed up: namely the  
transfinite numbers. But the nonconstructive development of math, in  
function theory for example, was independent of that.

Incidentally, what is interesting historically to me is the origin and  
delay of the constructive critique: It was a long time between Bolzano  
(or perhaps one should say Cauchy, whose 1822 lectures were better  
known than Bolzano's paper) and Kronecker's reaction. A good  
historical research project for someone would be the origins of the  
constructive critique of mathematics as it developed in that century.

Nevertheless, the critique itself is well motivated by the thought  
that, when we prove existence, the proof should yield an algorithm. I  
think one can understand this motivation whether or not one agrees  
that it should dictate the development of math. So I don't see that  
Thomas's question is justified if it is simply aimed at constructive  
mathematics, say as it is developed by Bishop and his school. But I do  
think that someone, coming from a training in ordinary (and so, lets  
face it, nonconstructive) mathematics, might very well wonder at the  
apparent subjectivist and intentional elements in the version of  
constructive math that Brouwer introduced with his theory of choice  
sequences. Of course, this subjectivist element in Brouwer's theory  
can be eliminated and the theory of choice sequences can be tamed--- 
this is done in the Kleene-Vesley paper on intuitionistic analysis and  
in Dummett's book on intuitionism. But, as I understand it---and some  
of the responses to Thomas's submission are evidence for this, there  
are many people who remain committed to Brouwer's subjectivism, which  
for many of us is alien to mathematics. From such people, I think that  
Thomas's question needs more of an answer than just counterattack.

Kind Regards,

Bill Tait

More information about the FOM mailing list