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Re: intuitionism
Posted:
Oct 28, 2009 2:42 AM


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 illdefined questions. One is dealing with welldefined 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 numbertheoretical questions, like whether there exists an odd perfect number.
Keith Ramsay



