In classical logic, trichotomy on the reals (any given numbers is either >0, <0 or =0) is provably true; in intuitionistic logic it is probably false. Thanks to Godel’s incompleteness theorem, we’ll never know which is right!
Ok, so let’s start with the following number, I need you to tell me if it is greater than, or equal to, 0:
0.0000000000000000000000000000…
Do you know yet? Ok, let’s keep going:
…000000000000000000000000000000…
How about now?
Will a non-zero digit ever appear?
The Platonist (classical mathematician) would argue “we can know”, as all numbers are completed objects to them; if a non-zero digit were to turn up they’d know by some oracular power. The intuitionist argues that we can only decide when the number is complete (which it may never be, it could be 0s forever), or when a non-zero digit appears (which may or may not happen); so they must wait ever onwards to decide.
Such numbers do exist beyond me just chanting “0”.
A fun number to consider is a number whose nth decimal digit is 0 if n isn’t an odd perfect number, and 1 of it is. This number being greater than 0 is contingent upon the existance of an odd perfect number (and we still don’t know if they exist). The classical mathematician asserts we “discover mathematics”, so the question is already decided (i.e. we can definitely say it must be one or the other, but we do not know which until we find it). The intuitionist, on the other hand, sees mathematics as a series of mental constructs (i.e. we “create” mathematics), to them the question is only decided once the construct has been made. Given that some problems can be proven unsolvable (axiomatic), it isn’t too far fetched to assert some numbers contingent upon results like this may well not be 0 or >0!
It’s a really deep rabbit hole to explore, and one which has consumed a large chunk of my life XD
I’m gonna be honest, I just don’t see how a non-Platonic interpretation makes sense. The number exists, either way. Our knowledge about it is immaterial to the question of what its value is.
To the Platonist, the number exists in a complete state “somewhere”. From this your argument follows naturally, as we simply look at the complete number and can easily spot a non-zero digit.
To the intuitionist, the number is still being created, and thus exists only as far as it has been created. Here your argument doesn’t work since the number that exists at that point in the construction is indeterminate as we cannot survey the “whole thing”.
Both points of view are valid, my bias is to the latter - Browser’s conception of mathematics as a tool based on human perception, rather than some notion of divine truth, just felt more accurate.
Actually I’ve done some more reading and frankly, the more I read the dumber this idea sounds.
If a statement P is provable, then P certainly cannot be refutable. But even if it can be shown that P cannot be refuted, this does not constitute a proof of P. Thus P is a stronger statement than not-not-P.
This reads like utter deranged nonsense. P ∨ ¬P is a tautology. To assert otherwise should not be done without done extraordinary evidence, and it certainly should not be done in a system called “intuitionist”. Basic human intuition says “either I have an apple or I do not have an apple”. It cannot be a third option. Whether you believe maths is an inherent universal property or something humans invented to aid their intuitionistic understanding of the world, that fact holds.
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?)
In classical logic, trichotomy on the reals (any given numbers is either >0, <0 or =0) is provably true; in intuitionistic logic it is probably false. Thanks to Godel’s incompleteness theorem, we’ll never know which is right!
I don’t understand, where’s the problem here? If course every number is either greater than zero, less than zero, or zero. That’s highly intuitive.
Ok, so let’s start with the following number, I need you to tell me if it is greater than, or equal to, 0:
0.0000000000000000000000000000…
Do you know yet? Ok, let’s keep going:
…000000000000000000000000000000…
How about now?
Will a non-zero digit ever appear?
The Platonist (classical mathematician) would argue “we can know”, as all numbers are completed objects to them; if a non-zero digit were to turn up they’d know by some oracular power. The intuitionist argues that we can only decide when the number is complete (which it may never be, it could be 0s forever), or when a non-zero digit appears (which may or may not happen); so they must wait ever onwards to decide.
Such numbers do exist beyond me just chanting “0”.
A fun number to consider is a number whose nth decimal digit is 0 if n isn’t an odd perfect number, and 1 of it is. This number being greater than 0 is contingent upon the existance of an odd perfect number (and we still don’t know if they exist). The classical mathematician asserts we “discover mathematics”, so the question is already decided (i.e. we can definitely say it must be one or the other, but we do not know which until we find it). The intuitionist, on the other hand, sees mathematics as a series of mental constructs (i.e. we “create” mathematics), to them the question is only decided once the construct has been made. Given that some problems can be proven unsolvable (axiomatic), it isn’t too far fetched to assert some numbers contingent upon results like this may well not be 0 or >0!
It’s a really deep rabbit hole to explore, and one which has consumed a large chunk of my life XD
I’m gonna be honest, I just don’t see how a non-Platonic interpretation makes sense. The number exists, either way. Our knowledge about it is immaterial to the question of what its value is.
Ah, and therein lies the heart of the matter!
To the Platonist, the number exists in a complete state “somewhere”. From this your argument follows naturally, as we simply look at the complete number and can easily spot a non-zero digit.
To the intuitionist, the number is still being created, and thus exists only as far as it has been created. Here your argument doesn’t work since the number that exists at that point in the construction is indeterminate as we cannot survey the “whole thing”.
Both points of view are valid, my bias is to the latter - Browser’s conception of mathematics as a tool based on human perception, rather than some notion of divine truth, just felt more accurate.
Actually I’ve done some more reading and frankly, the more I read the dumber this idea sounds.
This reads like utter deranged nonsense. P ∨ ¬P is a tautology. To assert otherwise should not be done without done extraordinary evidence, and it certainly should not be done in a system called “intuitionist”. Basic human intuition says “either I have an apple or I do not have an apple”. It cannot be a third option. Whether you believe maths is an inherent universal property or something humans invented to aid their intuitionistic understanding of the world, that fact holds.
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?)