this post was submitted on 26 Mar 2024
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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, Ā¬Ā¬PāP 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+1ā 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?)
I would say mathematics is a consequence of, or branch of philosophy in its own right. The name intuitionism derives from the source of this branch of mathematics - "2 primal intuitions".
Twoity - we are able to perceived time, and are thus able to split the universe into two, three, four etc parts. Counting is not something we just learn, it is something built into us as humans.
Repetition - we can repeat operations and not stop, just as we can never stop counting.
From these two (heavily paraphrased) ideas we can derive all of mathematics.
The first is actually enough to give us everything up to the rationals, the second grants us the reals and beyond.
While we lose excluded middle, we gain things such as "all total real functions are uniformly continuous on the unit interval" (Brouwer), the removal of the information paradox in physics (someone used Posey's take on intuitionism to rewrite all physics to see where it led), and the wonder of lawless sequences (objects we cannot predict entirely, but still work with).
The intuitionist is very very formal "you are either alive or not alive" is a very nice statement to make, but entirely worthless if one cannot tell which you are! Excluded middle is not universally false in intuitionism; it is true for decidable statements, of which having an apple or not does seem to fall within (though here we can question how "apple-like" must something be to be considered an apple if we wish to be peverse). However, to argue it is true for any statement means your disjunct (or) must be very weak indeed - the classical mathematician is happy with this, the intuitionist demands that a disjunct not only present two options, but provide a way of determining which if the two applies on a case to case basis (hence excluded middle applying for decidable things).
Simplifying your example of an apple, you can think of it as a Platonist just having the statement that everything is either and apple or not. Meanwhile, the Intuitionist also demands there be a guide on how to sort everything into "apple" or "not apple" before they make that statement.
Classical mathematics does also have a huge unintuitive step - mathematics must exist independently of humanity. Every theorem ever proved, and ever to be proved, exists somewhere. Where you ask? The platonic plane of ideal forms beckons, with all the madness it entails!