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Topic: intuitionism
Replies: 16   Last Post: Nov 1, 2009 3:33 AM

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Keith Ramsay

Posts: 1,745
Registered: 12/6/04
Re: intuitionism
Posted: Oct 28, 2009 2:42 AM
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On Oct 27, 9:00 am, nukeymusic <nukeymu...@gmail.com> wrote:
|building further on what was said before in this thread:
|1. is the following an example of "p or not(p) is absurd":
|p=(i>0) with i=sqrt(-1)

No, it's not a matter of ill-defined questions. One is
dealing with well-defined questions, where "p or not p"
is not ever absurd.

|2. did Brouwer possibly think of this kind of example?

I don't know. Mathematicians often seem to find the
possibility of inventing nonsense questions to be an
irrelevant distraction. When one claims that "p or not
p" is generally valid, one has in mind propositions p,
which are necessarily actual coherent propositions and
not merely nonsense sentences that sound like
propositions.

|3. can anyone give an example of a p for which: p or not(p) is false

It's never false. It's just that not being false
isn't taken as sufficient to demonstrate that
it is true.

|4. was "not(|=) p or not(p)" only stated for examples which applied
to
|infinite sequences

It depends upon the context. It's possible that this
is simply propositional calculus, i.e. "p or not p"
is not a validity of intuitionistic propositional
calculus. Then it's not a matter of a specific example
of p.

It's important to distinguish here between not affirming
and denying. There are many choices of p where an
intuitionist would not (at present) affirm "p or not p".
But they would also not deny "p or not p" for an individual
p. If it's valid to deny "p or not p" for a specific p,
then it's also valid to deny p separately; but that makes
it valid to affirm "not p".

Intuitionism tends to differ from some other kinds of
constructivism in that intuitionists have made assumptions
that contradict the law of excluded middle as a general
statement (that "for every p, either p or not p"). Others
have worked with sets of assumptions that simply leave
the question (whether the law of excluded middle holds)
open. One can assume that it is true, or assume that it
is false, or just not assume either way.

Brouwer would offer illustrative examples, but not
claiming that the p in the example was a case where
"p or not p" was unacceptable in a permanent way, but
simply where it was clear that we did not at the time
have a basis for accepting it, if one interprets the
concepts (such as "or" and "not") as he wanted to. His
examples often had to do with infinite sequences. To say
at the time that pi either has 9 consecutive 9s in it or
does not have 9 consecutive 9s in it, as he would take it,
would mean being assured of having a way to determine
one or the other as true. We have since then of course
computed many digits of pi and have settled the specific
question he offered as an example. But the specific case
is not crucial; it just illustrates the fact that in
general, one doesn't have a method for answering all such
questions.

One could as well use other kinds of number-theoretical
questions, like whether there exists an odd perfect
number.

Keith Ramsay



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