Actually, AvA’ is an axiom or a consequence of admitting A’'=>A. It’s only a tautology if you accept this axiom. Otherwise it cannot be proven or disproven. Excluded middle is, in reality, an axiom rather than a theorem.
The question lies not in the third option, but in what it means for there to be an option. To the intuitionist, existance of a disjunct requires a construct that allocates objects to the disjunct. A disjunct is, in essence, decidable to the intuitionist.
The classical mathematician states “it’s one or the other, it is not my job to say which”.
You have an apple or you don’t, god exists or it doesn’t, you have a number greater than 0 or you don’t. Trouble is, you don’t know which, and you may never know (decidability is not a condition for classical disjuncts), and that rather defeats the purpose! Yes we can divide the universe into having an apple or not, but unless you can decide between the two, what is the point?
So, obviously there’s a big overlap between maths and philosophy, but this conversation feels very solidly more on the side of philosophy than actual maths, to me. Which isn’t to say that there’s anything wrong with it. I love philosophy as a field. But when trying to look at it mathematically, ¬¬P⇒P is an axiom so basic that even if you can’t prove it, I just can’t accept working in a mathematical model that doesn’t include it. It would be like one where 1+1≠2 in the reals.
But on the philosophy, I still also come back to the issue of the name. You say this point of view is called “intuitionist”, but it runs completely counter to basic human intuition. Intuition says “I might not know if you have an apple, but for sure either you do, or you don’t. Only one of those two is possible.” And I think where feasible, any good approach to philosophy should aim to match human intuition, unless there is something very beneficial to be gained by moving away from intuition, or some serious cost to sticking with it. And I don’t see what could possibly be gained by going against intuition in this instance.
It might be an interesting space to explore for the sake of exploring, but even then, what actually comes out of it? (I mean this sincerely: are there any interesting insights that have come from exploring in this space?)
Pardon the slow reply!
Actually, AvA’ is an axiom or a consequence of admitting A’'=>A. It’s only a tautology if you accept this axiom. Otherwise it cannot be proven or disproven. Excluded middle is, in reality, an axiom rather than a theorem.
The question lies not in the third option, but in what it means for there to be an option. To the intuitionist, existance of a disjunct requires a construct that allocates objects to the disjunct. A disjunct is, in essence, decidable to the intuitionist.
The classical mathematician states “it’s one or the other, it is not my job to say which”.
You have an apple or you don’t, god exists or it doesn’t, you have a number greater than 0 or you don’t. Trouble is, you don’t know which, and you may never know (decidability is not a condition for classical disjuncts), and that rather defeats the purpose! Yes we can divide the universe into having an apple or not, but unless you can decide between the two, what is the point?
So, obviously there’s a big overlap between maths and philosophy, but this conversation feels very solidly more on the side of philosophy than actual maths, to me. Which isn’t to say that there’s anything wrong with it. I love philosophy as a field. But when trying to look at it mathematically, ¬¬P⇒P is an axiom so basic that even if you can’t prove it, I just can’t accept working in a mathematical model that doesn’t include it. It would be like one where 1+1≠2 in the reals.
But on the philosophy, I still also come back to the issue of the name. You say this point of view is called “intuitionist”, but it runs completely counter to basic human intuition. Intuition says “I might not know if you have an apple, but for sure either you do, or you don’t. Only one of those two is possible.” And I think where feasible, any good approach to philosophy should aim to match human intuition, unless there is something very beneficial to be gained by moving away from intuition, or some serious cost to sticking with it. And I don’t see what could possibly be gained by going against intuition in this instance.
It might be an interesting space to explore for the sake of exploring, but even then, what actually comes out of it? (I mean this sincerely: are there any interesting insights that have come from exploring in this space?)