[FOM] philosophical literature on intuitionism

[giuseppina.ronzitti@helsinki.fi]

Quoting "William Tait" <williamtait at mac.com>:

> 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
> intuitionism.)

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).

For my part, I do not see the (philosophical or mathematical) need for  
a "creative" or "creating" subject argument (very unfortunate  
terminology) in justifying intuitionistic reasoning based on the  
adoption of infinitely proceeding sequences of natural numbers as  
*legitimate* mathematical objects - as I do not see that the adoption  
of actual infinite sets as *legitimate* mathematical objects and  
reasoning about them is ever justified on the basis of some  
*philosophical* argument.

The philosophical debate about the adoption of some mathematical  
entity as a *legitimate* mathematical object is of course interesting  
(when (self-)critical and not dogmatic, as often is, at least it seems  
to me, the *philosophical* literature on intuitionism) - but as the  
two things, "intuitionism as a philosophy" and "intuitionism as  
mathematics", are constantly mixed up in a way that a refusal of  
"intuitionism as a philosophy" leads to a refusal of intuitionistic  
mathematics as *legitimate* mathematics, I really think that the best  
way to be introduced to the philosophical conception(s) (and I'm not  
at all speaking of "creating" or "creative" subject's arguments)  
underlying intuitionistic mathematics is by studying the basic notions  
and principles of intuitionistic mathematics: how "entities" and  
"collections of given entities" are defined and which principles of  
reasoning are or aren't allowed on the basis of their constitution,  
what follows from them, etc. etc.

Best regards,
Giuseppina Ronzitti


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