In a recent article, John Baez is quoted as making a nice point about how non-commutativity is common in the world around us, whereas non-associativity is not:
“[...] while it’s very easy to imagine noncommutative situations—putting on shoes then socks is different from socks then shoes—it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”
This thus begs the question: are there any examples of non-associative processes in nature/the physical universe? Note that I emphasize that it be an actual physical process (like putting on shoes) and not just a description of reality (cause there are plenty of examples like that to be found already on the wikipedia page , which are somehow artefacts of the language we use to describe things).
I think half the struggle is to figure out how to ask the question in physical terms. It makes sense to ask for two physical processes A and B such that 'first A then B' does not give the same outcome as 'first B then A', hence giving an example of non-commutativity. I don't quite know how to phrase the desired non-associative property for three physical processes A, B and C in a purely physical language. I guess my question is perhaps: are there parentheses in nature?