[FOM] R: Intuitionists and excluded middle
kremer at uchicago.edu
Sat Oct 29 06:55:54 EDT 2005
Antonino Drago asks why "in previous postings
nobody remembered that an exact distinction
between constructive mathematics and classical
mathematics is given by the principle of omniscience."
He quotes E. Bishop as formulating the LEM as
"Given a subset of N the set of natural
intergers, E, and given a number n', either there
exists an n° belonging to E such that n°=n' , or
for no n belonging to E n =n'."
He then argues that "to hold LEM means to attribute to himself mental
capabilities beyond any constructive (and operative) bound."
But in his argument for this, he moves from the
statement of LEM above to a statement about the
ability to "decide the truth or the falsity of
the question for any E" which, he says, is "attributing to
But from a classical point of view, the original
statement of LEM says nothing about capabilities
to decide, algorithms, etc. It is a statement
about numbers, not about ourselves. Thus in
Drago's argument, the intuitionist understanding
of the logical connectives involved in LEM is
tacitly presupposed, and so the argument as it
stands simply begs the question against the
classical logician. The classical logician is
quite happy to assert that either A or not-A
without claiming to be able to decide which is the case.
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