FOM: intuitionistic mathematics and building bridges
wtait@ix.netcom.com
wtait at ix.netcom.com
Wed Mar 4 10:31:24 EST 1998
The following is from Neil Tennant 2/27/98
>Dummett's argument for
>intuitionism is based, rather, on the *publicity* of communication, and
>the so-called Manifestation Requirement. The MR is that grasp of meaning
>should be able to be made manifest in the appropriate exercise of
>recognitional
>capacities relating to the use of sentences. Moreover, the intuitionist is
>demanding a very great degree of objectivity for mathematical
>claims---namely,
>possession of a proof! If any semantics is objectivist, it is intuitionistic
>semantics.
Dummett's argument is for a certain way of understanding logic in
mathematics; but it applies just as well to classical as to
intuitionistic logic. The added condition needed for the restriction to
intuitionism---e.g. that when you construct a number with a certain
property, you must be able read off the construction its place in the
sequence of numerals---does not follow from that conception.
The last time I heard D. on the subject of intuitionism, he had given up
that argument (without explanation) and was arguing that intuitionistic
logic was appropriate because of `indefinite extendability' (or something
like that). Somehow, the essential incompletability of the system of
transfinite numbers extends down to the reals and then to the finite
ordinals. It was a work of magic (i.e. I didn't follow the argument). It
was at a conference some years back in Amherst and the procedings are in
print (_Mathematics and Mind_, ed. Alexander George, Oxford Pree 1994). I
looked briefly at Dummett's paper there and had the impression---but only
that---that there may have been significant revisions.
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