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anonymous1 comments on Causal Reference - Less Wrong
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PA proves "2 + 2 = 4" using the associative property. PA does not prove "2 + 2 = 3". "2 + 2 = 4" is actually shorthand for "((1+1) + (1+1)) = (((1+1)+1)+1)". Moving stuff next to other stuff in our universe happens to follow the associative property; this is why the belief is useful.
I have myself usually seen Peano arithmetic described with 0 and the successor operation (such as in the context of actually implementing it in a computer). in this case,
where the two theorems needed are that x + S(y) = S(x) + y and that x + 0 = x. I find this to have less incidental complexity (given that we are interested in working up from axioms, not down from conventional arithmetic) perhaps because the tree of the final expression has no branches. The first theorem can be looked at as expressing that “moving stuff results in the same stuff”, i.e. a conservation law; note that the expression has precisely the same number of nodes.
(I like that! The idea that it follows just from the associative property and no other features of PA is quite elegant.)
This post feel surprisingly insight-bringing to me because the "associative property" can in a sense be considered the "insignificance of parentheses"... and hence both the insignificance of groupings, and the lack of the need to define a starting point of calculation... and in turn these concepts feel connected to the concepts of reductionism and relativity...
I would go further and say that without the associative property the concept of numbers, math, and "2 + 2 = 4" does not make sense.
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.