[FOM] philosophical literature on intuitionism

William Tait williamtait at mac.com
Thu Oct 23 14:29:35 EDT 2008

On Oct 16, 2008, at 8:09 PM, mark van atten wrote:

> Bill Tait Thu Oct 16 11:31:57 EDT 2008 wrote:
>> 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.
> No so-called elimination theorem has ever been produced for the
> general notion of choice sequence; only for very specific classes. The
> creating subject is wholly free to generate sequences that fall
> outside of them.

I did not mention any 'elimination theorems' in my posting, and I  
agree completely with this part of what Mark wrote. I merely wanted to  
point out that a part of intuitionistic mathematics can, like Bishop- 
style constructive mathematics, be formulated without reference to the  
creating subject. For me, that part of intuitionistic/constructive  
mathematics is well-motivated although not absolutely compelling. I  
was suggesting that the other part, involving 'objects' (choice  
sequences) that do essentially involve the creating subject, motivated  
Thomas Forster's "dream up" question. From Thomas' response (Oct 18),  
it seems that I was wrong about this and that he was referring to  
constructive mathematics generally. But I do think that a discussion  
of the distinctively Brouwerian element of intuitionistic math would  
be interesting.

In Weyl's version of intuitionism is there such an essential reference  
to the creating subject? My impression is that there is not, at least  
in "On the new foundational crisis of mathematics", but I haven't   
gone back to check.

Incidentally, I possess lecture notes "Inleiding tot de  
Intuitionistische Wiskunde", for a course by Heyting in 1952-3, edited  
by Johann de Iongh. I got them when I was a student in Amsterdam in  
1954-5. They are typed in Dutch with symbols drawn in by hand. There  
doesn't seem to be much more than is in Heyting's book *Intuitionism:  
An Introduction*, but my now almost nonexistent Dutch does not admit  
of accurate skimming. I expect that there are other copies of the  
notes around, but I mention mine just in case. (I don't think these  
notes were mentioned by anyone in listing the literature on  

Bill Tait

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