[FOM] philosophical literature on intuitionism

Frank Waaldijk frank.waaldijk at hetnet.nl
Sun Oct 26 10:02:48 EDT 2008

Giuseppina Ronzitti wrote:

> This remark seems to suggest that the best *philosophical  introduction* 
> to intuitionism is indeed an introduction to  intuitionistic mathematics 
> (Inleiding tot de Intuitionistische  Wiskunde), and I do agree (I don't 
> intend to say that is prof. Tait's  opinion, though). On this line, more 
> recent lectures notes (still in  Dutch) are Wim Veldman's 
> "Intuitionistische Wiskunde" (Intuitionistic  Mathematics) for the course 
> he gives on the subject in Nijmegen (NL).  Unfortunately, there is no much 
> else (beside Kleene and Vesley  monography it may be worth mentioning 
> Dragalin's useful book,  "Mathematical Intuitionism, introduction to proof 
> theory" which also  discusses different formalizations of intuitionistic 
> analysis).

I agree that Wim Veldman's lecture notes are the best text I have seen to 
introduce someone to intuitionistic mathematics. Perhaps he could be 
persuaded to have them translated into English. I am confident it would 
enlighten many people about the beauty, and the mathematical as well as 
philosophical legitimacy of intuitionistic mathematics. Perhaps Hendrik's 
offer to translate Heyting/De Iongh's lecture notes would stretch that far?

I also agree with the rest of Giuseppina's message. When studying 
intuitionistic mathematics, one should not mix philosophy with mathematics 
in such a way that the mathematics is obscured. One wouldn't do that with 
classical mathematics either. By explaining the mathematics and the ideas 
behind it, enough philosophy becomes clear.

Still, to be complete and perhaps more as an afterthought, I would like to 
address the `creative subject', as I see it. From a mathematical point of 
view (which I prefer for this discussion), I believe it is nothing more than 
a denial of the axiom CT ( from Church's Thesis) which states that all 
infinite sequences of natural numbers are given by a recursive rule. CT 
forms the basis of RUSS, which is an intriguing branch of constructive 
mathematics since (from an axiomatic point of view) it is far more apart 
from CLASS than INT.

I believe Brouwer started out with considering a preliminary version of CT 
(in his own words, and before the notion of `computability' had really 
developed). However, Brouwer later realized -in my humble opinion- that CT 
would not allow the kind of topological foundations that he envisaged for 
intuitionism. Specifically, CT conflicts with the Fan Theorem, or to put it 
more topologically: with CT the natural compactness of the real unit 
interval [0,1] is lost, and continuous functions on [0,1] which are not 
uniformly continuous can be constructed.

Brouwer developed intuitionistic mathematics fully only after his seminal 
work in topology, and this -imho- is not a coincidence. But to motivate his 
non-adoption of CT, he perhaps needed the philosophical image of the 
`creative subject'.

What is interesting mathematically is that in this light, from an axiomatic 
point of view, CLASS also adopts the creative subject. CLASS also states 
that there are non-recursive sequences...which I believe to amount 
axiomatically to the same thing as the creative subject.

>From an axiomatic point of view, the only intuitionistic axiom which 
conflicts with CLASS is the Continuity Principle (CP). And now, in formal 
topology I believe that the essence of CP is adopted and captured in the 
definitions, bringing a very large part of Brouwer's ideas under a 
constructive topological umbrella which axiomatically is in the intersection 
of CLASS and INT (and RUSS, but...see next paragraph on earlier posts).

In some previous posts I therefore asked -using other words- whether 
constructive formal topology was not simply a new marketing for 
intuitionism...;-). But I confess to not knowing enough on this issue to 
really put forward worthwhile arguments. Perhaps some others more 
knowledgeable would like to comment (again).

Hope to have contributed something,
kind regards,

Frank Waaldijk

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