If I say “two plus two equals four”, I’m being objective. My statement should be true regardless of who is saying it, who’s doing the maths, etc.
Even this is quite subjective, as it builds on the (subjective) acceptance of axioms. To most reading this, they would’ve been educated using the 8 Zermelo-Fraenkel (ZF) axioms, with the controversial 9th axiom of choice.
I disagree that this is subjective. Even if someone hypothetically doesn’t accept the ZF[C], the statement still accurately describes reality, in a way that doesn’t depend on the subject. For example, you can’t start with two apples and two oranges and have five or tree fruits.
The volume of a mixture cannot be described by a simple sum of the volume of its components. As such, this does not make the statement “1+1=2” false in this situation; it’s still true but irrelevant, there’s no “+” here on first place.
Additionally, let us suppose for a moment that the reasoning above is invalid. Even then, it’s still an objective matter - because then the truth value of “1+1=2” would vary depending on the object (are we dealing with apples, or liquid mixtures?), not on the subject (who’s mixing the liquids - you or me?).
It’s subjective as in: imagine a different society/species constructing a sense of reality and computation, based on liquid mixtures. Their basis of computation, their axiom is 1l of alcohol + 1l of water = 2l of mixture.
They meet us, and we exchange ideas.
They go: of course 1 + 1 = 2, look at our mixture. For fruits, apples and pears? That’s outside of normal arithmetics, it’s an exception. There’s no + there, as you’re not mixing. You have to correct for the non mixture nature, the answer will be larger than 2.
Even this is quite subjective, as it builds on the (subjective) acceptance of axioms. To most reading this, they would’ve been educated using the 8 Zermelo-Fraenkel (ZF) axioms, with the controversial 9th axiom of choice.
I disagree that this is subjective. Even if someone hypothetically doesn’t accept the ZF[C], the statement still accurately describes reality, in a way that doesn’t depend on the subject. For example, you can’t start with two apples and two oranges and have five or tree fruits.
Yet in some contexts it isn’t as easy as that. You can combine 1 liter of water with 1 liter of alcohol, and get less than 2 liters of fluids. (1)
The volume of a mixture cannot be described by a simple sum of the volume of its components. As such, this does not make the statement “1+1=2” false in this situation; it’s still true but irrelevant, there’s no “+” here on first place.
Additionally, let us suppose for a moment that the reasoning above is invalid. Even then, it’s still an objective matter - because then the truth value of “1+1=2” would vary depending on the object (are we dealing with apples, or liquid mixtures?), not on the subject (who’s mixing the liquids - you or me?).
It’s subjective as in: imagine a different society/species constructing a sense of reality and computation, based on liquid mixtures. Their basis of computation, their axiom is 1l of alcohol + 1l of water = 2l of mixture.
They meet us, and we exchange ideas.
They go: of course 1 + 1 = 2, look at our mixture. For fruits, apples and pears? That’s outside of normal arithmetics, it’s an exception. There’s no + there, as you’re not mixing. You have to correct for the non mixture nature, the answer will be larger than 2.
But at that point you are out of realm of simple arithmetics any way.
1+1=2, except for when it’s not :)