John, (01)
Lainaus "John F. Sowa" <sowa@xxxxxxxxxxx>: (02)
> Pat,
>
> On foundational issues, I disagree with Avril who is trying legislate
> a finitistic foundation. But I also disagree with *everyone* who is
> trying to legislate *any* fixed foundation.
>
> But I wouldn't call myself antifoundational, since I am not opposed
> to having foundations. I am just opposed to the idea that there must
> be exactly one or exactly two foundations or exactly N foundations. (03)
finitism is not a foundational approach. You say, no foundations.
Finitism says, no foundations, but if someone imposes foundations,
then do not build subjective interpretations as foundations, such
as: (04)
-Cantor states that there are aleph-1 points in continuum (05)
-Brouwer states that there are less than aleph-1 (06)
-Peirce states that there are as many as is the cardinality
of the greatest conceivable transfinity. (07)
Compare this to Mary Tiles' analogy: (08)
The question ’What was the diameter of the top of the Madhatter’s hat?’
has no answer, for Alice in Wonderland neither includes this directly
in its description nor provides details of other dimensions from which
an answer could be deduced. Given that this is so, we could consistently
add to the story by filling in this detail (in the way that illustrators
have filled in the price of the hat - Tenniel gives it a price tag of
10/6d, whereas that in Rackham’s illustration is 8/4d) and we might do
so in different ways, so continuing the creative process. Of course we
might also argue that dimensions in Wonderland are peculiarly problematic;
given Alice’s tendency to grow and shrink we would have to specify the
frame of reference carefully. It seems that the continuum hypothesis is a
similarly unanswerable question about infinite numbers. (09)
> I strongly agree with Peirce's "First Rule of Reason":
>
> Do not block the way inquiry.
>
> For me to tell a professional mathematician not to pursue his
> or her own intuitions about any mathematical subject would be
> to set up a roadblock on a way of inquiry.
>
> Although I do not intend to pursue that path, I see no reason
> why I should try to stop anyone else. (010)
Also transfinitism can be seen as a roadblock to constructivism,
but I cannot see finitism as any sort of a roadblock. (011)
Why, do you want to have a never-ending as a completed totality? (012)
Avril (013)
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