A few years back I came across an article on quantum physics in Quanta Magazine. It described the work of Cohl Furey trying to plumb the secrets of the universe using octonions. The article explains:
Much more bizarrely, the octonions are nonassociative, meaning $(a \times b) \times c$ doesn’t equal $a \times (b \times c)$. “Nonassociative things are strongly disliked by mathematicians,” said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions. “Because 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.”
At the time, I didn't think much of it. This was a popular magazine, so its not unusual for writers to put things like that in the article to spice it up. But as I've been exploring, this quote has started to become more significant. I notice that semigroups and monoids have far more content surrounding them than quasigroups and loops. Indeed, I struggle to find much on quasigroups at all. I find non-associative structures embedded in higher order associative structures to study. Even when we push to the ends of the mathematical universe, with fundamental tools like category theory, we find the associativity of morphisms baked right down into the deepest layers.
What is it about the associative property that makes it so important to mathematicians? Is it just an artifact of how math evolved? Is it just a curious illusion of mine, borne of my particular journey through mathematics? Or is there something more fundamental about this property than other properties we learn about?
Associative operations are important because they are precisely those operations that link objects together into a sequence. Since we think of events in time as being linked in a sequence (first $E_1$ happens and then $E_2$ and then $E_3$, etc.), it is natural that the mathematics that describes linking sequences together would be so fundamental. Sometimes this manifests in unexpected ways. For example, since writing occurs as a sequence of actions in time (e.g., write "t" and then "h" and then "e" to spell "the"), the operation that links letters together (concatenation) is associative. As another example, composition of symmetry transformations forms a group, is associative, and thus chains together a sequence (e.g., "flip the square horizontally" and then "rotate the square 90 degrees" and then "flip the square vertically").
How do we know that associative operations are operations that link together objects into a sequence? Because it was proven. There is a theorem in semigroups called Cayley's theorem for semigroups. Unfortunately, Cayley's theorem for semigroups is usually proved and stated in pretty abstract terms. Here is a YouTube video I made that describes the intuitions behind Cayley's theorem in approachable terms.