Octonion multiplication is not associative. Exponentation isn’t either; (2^2)^3 = 4^3 = 64, 2^(2^3) = 2^8 = 256. There’s likely some kind of useful math with numberlike objects where no interesting operation is associative.
How can this math refer to the real world without the associative property? If you can’t count the initial “2” then you can’t multiply it or anything else, right? Plus, how could we arrive at a concept of exponentation that didn’t entail the concept of the associative property?
The pure math might just be too much of an inferential leap for me. I need to see how the math would be created from observations in the real world before I can really understand what you are saying.
Well, addition of positive integers is associative and has an obvious real-world analogue, so the associophobic math isn’t a good choice for describing reality. But if you lived in Bejeweled, addition wouldn’t make much sense as a concept—sometimes pushing a thing close to another thing yields two things, sometimes zero. The most fundamental operation would be “flip a pair of adjacent things”, which is not associative. (It’s sort of a transposition, which would give you group theory, which is full of associative operations, but I don’t think you can factor in disappearing rows while preserving associativity—it destroys the bijectivity.)
Octonion multiplication is not associative. Exponentation isn’t either; (2^2)^3 = 4^3 = 64, 2^(2^3) = 2^8 = 256. There’s likely some kind of useful math with numberlike objects where no interesting operation is associative.
How can this math refer to the real world without the associative property? If you can’t count the initial “2” then you can’t multiply it or anything else, right? Plus, how could we arrive at a concept of exponentation that didn’t entail the concept of the associative property?
The pure math might just be too much of an inferential leap for me. I need to see how the math would be created from observations in the real world before I can really understand what you are saying.
Well, addition of positive integers is associative and has an obvious real-world analogue, so the associophobic math isn’t a good choice for describing reality. But if you lived in Bejeweled, addition wouldn’t make much sense as a concept—sometimes pushing a thing close to another thing yields two things, sometimes zero. The most fundamental operation would be “flip a pair of adjacent things”, which is not associative. (It’s sort of a transposition, which would give you group theory, which is full of associative operations, but I don’t think you can factor in disappearing rows while preserving associativity—it destroys the bijectivity.)
Cool, great example.