John Corcoran corcoran at buffalo.edu
Wed Sep 29 15:06:05 EDT 2004

According to Brouwer a "formal deduction" [in principle ignoring
intuitive content and attending only to logical form] can never produce
knowledge of the conclusion [even when the person doing the deduction
has intuitive knowledge of the premises, and even when the person doing
the deduction continually comprehends the premises and intermediate
conclusions as contentful propositions]. For him the process that is
needed to produce knowledge must be fully contentful at every point, and
essentially so, because it needs to transfer the evidentiary basis for
knowledge of the premises so that it makes the conclusion evident, so
that it serves as an evidentiary basis for the conclusion. Moreover, the
premises and intermediate conclusions in this process must not only be
contentful in the way that an open question or mere hypothesis is
contentful, but they must somehow be entertained as intuitively evident
known truths.

For Brouwer, proof cannot be reduced to intuitive knowledge plus
information-processing: intuitive knowledge -- of the premises -- plus
information-processing -- showing that the conclusion's information is
entirely within the information in the premises, analyzing the premises
to find the conclusion within them, so to speak. Rather what is needed
is a way of bringing to bear on the decision to accept the conclusion
the very evidence already found sufficient for the premises.*******Given
this, it would seem natural to call classical mathematicians
"FORMALISTS" , at least those who hold that the content of the premises
is theoretically irrelevant and dispensable in determining whether the
conclusion follows --and further that attending to the content is or can
be a detriment to cogent deduction because it can lead to
premise-smuggling.**** But, what label or labels have been used by
intuitionists to mark this distinction? Of course, I am looking for
descriptive, non-pejorative labels.

Thanks in advance for your suggestions, and for your criticism of the
above statement of one limited aspect of the much richer viewpoint known
as intuitionism.


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