[FOM] R: Intuitionists and excluded middle

Michael Kremer 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
himself omniscience."

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.

--Michael Kremer

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