[FOM] philosophical literature on intuitionism
Juliette Kennedy
jkennedy at cc.helsinki.fi
Sat Oct 25 11:31:33 EDT 2008
Anne Troelstra gave a very penetrating lecture on the
philosophy of intuitionism in our seminar just yesterday; he spent quite
alot of time on
the notion of the creative subject and creating subject arguments. A
scanned copy of Troelstra's lecture notes
will go up soon at the seminar website www.math.ru.nl/~landsman/PP.html;
see there also
for Wim Weldman's notes for his lecture on intuitionism of 10.10.08.
It occured to me while listening this time that although one can of
course do intuitionistic mathematics without reflecting on
the philosophical basis of it, the notion of a creative subject
seems to be a good way -- perhaps it is the only way, to make
philosophical sense
of a mathematics whose objects and concepts are subjectively constituted.
Otherwise the story one tells, how those objects come about, is going to
involve in an essential way, phenomena exterior to the subject.
Just what the intuitionist wishes to avoid, I would think.
All the best,
Juliette
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On Fri, 24 Oct 2008 giuseppina.ronzitti at helsinki.fi wrote:
> 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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>
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>
>
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