[szany]

This is a bit defeatist. Parsing the definition in your head is only the first level of understanding you can have about a mathematical structure. You don't really understand something until you can reinvent it (and in particular give a plausible answer to "why these axioms and not others?")

For example, to motivate groups, you could introduce the concept of a symmetry as a mapping from an object to itself that doesn't change its properties, introduce the idea of an isomorphism as a mapping with an inverse (where f and g being inverse means they compose to identity maps), put them together and postulate that a reasonable formalization of a symmetry is an automorphism (isomorphism from an object to itself), note that isomorphisms are closed under composition, and arrive at the definition of a group by considering sets of symmetries closed under finite composition (thinking of identity maps as the composition of 0 morphisms).

I'm sure there's a similarly conceptual way to motivate monads in functional programming. Hyland and Power have papers on algebraic theories of "effectful" operations and how they give rise to (finitary) monads, as one starting point.

[nhaliday]

Indeed the typical wisdom in math is that the best way to understand a group (and the reason we usually care about one) is its actions.

There's some quote to the effect of: "Just as with men, a group is known by its actions." This was the only reference I could find off the top of my head https://gowers.wordpress.com/2011/11/06/group-actions-i/, but I've definitely heard the sentiment repeated in a couple different texts.

The sentiment expressed in this article might useful to hear if you're solely interested in learning how to use monad transformers or whatever in Haskell, eg, I know a fair amount of math, but very little category theory, and manage use monads just fine without any deep intuition as to why they're an important formalism. But if you want to develop your problem-solving skills, for example, I think "quiet contemplation" of axioms is pretty much always worse than working through several examples and absorbing some good exposition.

Here's a recent article that I found very though-provoking: http://cognitivemedium.com/invisible_visible/invisible_visib.... They quote some mathematicians discussing their visual intuition for one property of groups.

[mrkgnao]

If I remember correctly, you could forget the "inverses" bit and define a group action with only the monoid corresponding to the group.

So not all the "groupiness" is captured by actions.

Also, Bill Thurston's beautiful "On Proof and Progress in Mathematics" [1] contains several ways to understand the derivative, in addition to it being a look into what a very distinguished mathematician thought of the nature of the mathematical pursuit. Pretty similar ideas, explored in depth.

Edit: Apparently this is linked to in TFA :)

[1]: https://arxiv.org/abs/math/9404236