Algebras we love
I really dislike the standard definition of a group - that it's a set S together with a binary operation * such that a bunch of properties hold. The definition doesn't build intuition, and doesn't motivate the introduction of the concept of "group".<p>For me, a group is the set of isomorphisms of a graph. If you expand the definition of "graph" a little to include continuous spaces, that is sufficient to define all groups. And yet, this nice, intuitive definition of a group <i>might</i> show up at the <i>end</i> of a course in group theory - if you're lucky.<p>It really is a shame how much intuition is stripped from mathematics teaching in the name of formalism.<p>[0] <a href="https://en.wikipedia.org/wiki/Frucht%27s_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Frucht%27s_theorem</a>