These discussions about whether 2+2=4 are confusing because 2+2 is 4 by definition of this operation of addition, and then we see in what cases real world phenomena are described by this operation. If you define 2+2 as anything but 4, then you’re just describing a different operation. There are many operations where 2 and 2 give 5.
The problem, always, is that there’s no causal connection between mathematics and reality. Suppose you try to force one and say that however many rocks you have in a cup when you combine two cups (each with two rocks), that is going to be THE operation of addition. Then asking what 2+2 has to be is asking how many rocks you can have in the final cup. Well, it’s not logically impossible for rocks to follow a rule that every four rocks in a certain small area will make a new rock and increase their number to 5. It just happens, in our reality, that rocks satisfy Peano arithmetic (and don’t resonate daughter rocks).
It just happens, in our reality, that rocks satisfy Peano arithmetic (and don’t resonate daughter rocks).
Well, it’s complicated by the fact that rocks can break, which means that you need to go into a lot more detail to say the extent to which the standard axioms of math map to rocks. This is why simplistic proofs of 2+2=4 by reference to rock behavior are so misleading and unhelpful.
It’s important to unpack exactly what is meant by “2+2=4”. The most charitable unpacking I can give is that it means both:
a) There exists an axiom set under which (by implication, not definition), 2+2=4. b) That axiom set has extremely frequent isomorphisms to (our observations of) physical phenomena.
But most people’s brains, for reasons of simplicity, truncate this to “2+2=4”. The problem arises when you try to take this representation and locate it somewhere in the territory, in which case … well, you get royalties from The Big Questions, but you’re still committing the mind-projection fallacy :-P
Under a sufficiently high temperature, they will coalesce into one rock.
I don’t quite have the argument framed, but it’s something like arithmetic applies in our world, but only under circumstances which have to be specified separately from arithmetic.
These discussions about whether 2+2=4 are confusing because 2+2 is 4 by definition of this operation of addition, and then we see in what cases real world phenomena are described by this operation. If you define 2+2 as anything but 4, then you’re just describing a different operation. There are many operations where 2 and 2 give 5.
The problem, always, is that there’s no causal connection between mathematics and reality. Suppose you try to force one and say that however many rocks you have in a cup when you combine two cups (each with two rocks), that is going to be THE operation of addition. Then asking what 2+2 has to be is asking how many rocks you can have in the final cup. Well, it’s not logically impossible for rocks to follow a rule that every four rocks in a certain small area will make a new rock and increase their number to 5. It just happens, in our reality, that rocks satisfy Peano arithmetic (and don’t resonate daughter rocks).
Well, it’s complicated by the fact that rocks can break, which means that you need to go into a lot more detail to say the extent to which the standard axioms of math map to rocks. This is why simplistic proofs of 2+2=4 by reference to rock behavior are so misleading and unhelpful.
It’s important to unpack exactly what is meant by “2+2=4”. The most charitable unpacking I can give is that it means both:
a) There exists an axiom set under which (by implication, not definition), 2+2=4.
b) That axiom set has extremely frequent isomorphisms to (our observations of) physical phenomena.
But most people’s brains, for reasons of simplicity, truncate this to “2+2=4”. The problem arises when you try to take this representation and locate it somewhere in the territory, in which case … well, you get royalties from The Big Questions, but you’re still committing the mind-projection fallacy :-P
Under a sufficiently high temperature, they will coalesce into one rock.
I don’t quite have the argument framed, but it’s something like arithmetic applies in our world, but only under circumstances which have to be specified separately from arithmetic.