[FOM] philosophical literature on intuitionism

[mark van atten]

Thomas Forster Mon Oct 13 18:27:15 EDT 2008 wrote:

> I do remember reading that Brouwer was
> hostile to attempts to axiomatise constructive logic..

When Heyting showed Brouwer the manuscript of his (partial)  
formalisation of intuitionistic logic, Brouwer wrote to him on July  
17, 1928, that he found this `extraordinarily interesting', and  
continued:

"By now I have already begun to appreciate your work so much, that I  
should like to request that you revise it in German for the  
Mathematische Annalen (preferably somewhat extended rather than  
shortened)." (The paper would instead be published in the proceedings  
of the Prussian Academy of Sciences.)

Interestingly, Brouwer also suggested (with an eye on the  
formalization of the theory of choice sequences)

"And, with an eye on §13, perhaps also the notion of `law' can be formalized."


Also, in papers from the 1920s Brouwer was quite optimistic about  
Hilbert's program for a consistency proof of formalized classical  
mathematics.

But, on his conception of the relation between logic and mathematics,  
work in logic cannot, as such, lead to new contentual mathematics.

On the other hand, Brouwer was well aware of the practical need for  
language, both in order to communicate mathematical results to others  
and to help ourselves in remembering and reconstructing our previous  
results. Only an ideal mathematician with perfect and unlimited memory  
would be able to practice pure mathematics without recourse to  
language. Clearly, given these two practical functions of language,  
the more precise the language is, the better.

For references and further details, see
http://plato.stanford.edu/entries/intuitionistic-logic-development/
in particular sections 2.1, 3.5, and 4.2.

Best wishes,
Mark.


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