[FOM] philosophical literature on intuitionism
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
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
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,
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