[FOM] R: Intuitionists and excluded middle

Antonino Drago drago at unina.it
Fri Oct 28 20:08:47 EDT 2005

About the LEM validity I would like to advance some points.

1) I wonder why in previous postings nobody remembered that an exact
distinction between constructive mathematics and classical mathematics is
given by the principle of omniscience (E. Bishop: Constructive Mathematics,
Mc Graw-Hill 1967, 1-10) "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'." For E the set of prime numbers,
say, we have an algorithm deciding such a question in a finite number of
steps; but being the power of the subsets of N the continuum and the power
of algorithms the denumerable, it is clear that we cannot have enough
algorithms for deciding such a question for all E. To think to be capable to
decide the truth or the falsity of the question for any E is attributing to
himself omniscience.
That shows that to hold LEM means to attribute to himself mental
capabilities beyond any constructive (and operative) bound.
However, in constructive mathematics any
undecidable question gives the same conclusion;
likely as even in the antiquity, when some statements were undecidable
(squaring the circle, trisecting an angle, doubling a cube, etc.) according
to the bound to use rule and compass only; then LEM did not apply to the
correspondent statement.

2) In fact previous debate turned the attention from the LEM to Double
negation law . I seem this fact as a significant though unaware event in the
correct direction. In 70's Prawitz and Dummettt identified the very
borderline between
classical logic and intuitionistic logic in the DNL rather
than in the LEM. The question why DNL is not always valid is more easily
approachable than the same question about LEM.  In particular, one may
easily discover that original texts of logicians, mathematicians and
physicists appropriately made use of the failure of DNL for expressing their
theories; for instance  for introducing his theory Lobachevsky started from
the statement "In the uncertainty that there are not two
parallel lines... " (i.e. "Being not true that there are not two parallel
lines...": a double negation statement). .

3) The debate addressed itself to inquire according to intuitionism about
the reductio theorem, a kind of arguing which incvolves DNL . A distinction
was made between weak and strong reductio in intuitionism. In his posting of
October 22th Ron Rood explained why intuitionism can allow weak reductio
only. I agree with him. I add that justification of weak reductio can be
found out also in a more general context.  Strong reduction, by changing the
ending double negated sentence in a  positive sentence binds to choice
classical logic; instead weak reductio ends just with the double negated
statement, which in intuitionism is not equivalent to a positive statement;
then one can pursue intuitionistically the wider argument by
using this statement as methodological principle, likely as previous
Lobachevsky's statement; a new theory can exit (surely, by induction, not by
deduction). A fine example is ialso n sect. V.B of the celebrated 1925 paper
Kolmogoroff (p. 431). He disproves, as he had already announced in
"Introduction", a Brouwer's thesis; i.e. he proves that never the use of LEM
leads to a contradiction. Notice that his argument, playing a crucial role
in K.'s paper, relies upon a sequence of double negated statements. He
obtains the result by means of an weak reductio theorem - although verbally
given -, just as a sequence of DNS's can conclude. Notice that pseudo-x is
defined by Kolmogoroff as the double negation of x.
"The use of the principle of excluded middle never leads to a contradiction
[not equal to: leads to compatible conclusions]. In fact, if a false formula
were obtained with its help, then the corresponding formula of
pseudomathematics [not equal to: mathematics] would be proved without its
help and would also lead to a contradiction [not equal to: with its help.
would lead to compatible conclusions]."

More details in my paper: A.N. Kolmorogoff and
the Relevance of the Double Negation Law in Science, in G. Sica (ed.):
Essays on the Foundations of Mathematics and Logic, Polimetrica, Milano,
2005, 57-81.
Best regards,

Antonino Drago
via Benvenuti 5
Castelmaggiore Calci Pisa 56110
tel. 050 937493
fax 06 233242218

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