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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?

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    $\begingroup$ The associativity is crucial for groups, the most important structures in mathematics. $\endgroup$
    – Peter
    Sep 13, 2021 at 15:52
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    $\begingroup$ @Peter True, but so is its invertability. $\endgroup$
    – Cort Ammon
    Sep 13, 2021 at 15:57
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    $\begingroup$ OK, then consider semigroups. They are still associative. $\endgroup$
    – Peter
    Sep 13, 2021 at 15:58
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    $\begingroup$ @Peter If you can identify what is so much more special about a semigroup than a quasigroup or a unital magma, that would be a useful answer. I didn't want to narrow the question down to just properties that groups have, but I admit that those properties are the ones that pointed me towards pondering this quote. $\endgroup$
    – Cort Ammon
    Sep 13, 2021 at 16:02
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    $\begingroup$ (+1) I think this question goes deeper than simply looking a groups or semigroups, which seem a rather restrictive context for something like this, despite the prevalence of these algebraic structures in mathematics, and your phrasing also shows this to some extent (e.g. "we find the associativity of morphisms baked right down into the deepest layers"). The only thing I can think of now is that without associativity, parsing and manipulating algebraic expressions becomes so difficult, due to their much greater "irreducibility of form", (continued) $\endgroup$
    – Dave L. Renfro
    Sep 13, 2021 at 16:34

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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.

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    $\begingroup$ From a certain POV associativity is deeply baked in to mathematics in another way: One may prove $+$ and $\cdot$ associative in Peano Arithmetic, and associativity of function composition is basically true by definition. But every other instance of associativity more or less boils down to "because $(\mathbb N,+,\cdot)$ is associative" or "because composition of maps is associative" after a fashion. Heck, we don't even have to prove that multiplication in $\mathbb N$ is associative (but we should ^_^), because we can realize $(\mathbb Z,+,\cdot)$ as the endomorphism ring of $(\mathbb Z,+)$. $\endgroup$
    – kahen
    Oct 8, 2021 at 17:02
  • $\begingroup$ @kahen Absolutely! And it's possible to probe even deeper by asking "Why is + important to mathematicians"? I would argue that addition of naturals inherits associativity through composition. After all, addition is reducible to composition of successor functions. Combined with your observations, it seems everything inherits associativity from composition. In the video, I demonstrate that "and then" (which is composition when read from right to left) is a linking together of a sequence (termed "gluing" in the video). So, + is important because of associativity and not the other way around! $\endgroup$
    – Meir Shachar
    Oct 8, 2021 at 19:42

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