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I was hoping to pick the brains of some of the mathematicians and mathematically inclined on this site.

I'm very interested in how mathematicians think about abstract objects that don't seem to be grounded in anything concrete. In particular, how do mathematicians think to themselves about abstract algebraic objects like groups, rings, fields, and such stuff. For vector spaces, there is a nice geometric and intuitive picture of collections of arrows. What about more general "sets with binary operations"? Do mathematician's have tricks to visualize such sets and structures, or is it more like learning rules to manipulate symbols on a page? I'm teaching myself some very basic group theory and Cayley diagrams are neat ways of thinking about the structure of groups, but is this really how mathematician's think or is it just a crutch and eventual hindrance to further learning?

I vaguely remember an interview with Paul Halmos (I think) talking about some group theorist he knew. When the group theorist was asked what he thought of when thinking about a group, he responded by saying "the letter 'G' ". Is this more than just a funny anecdote; do algebraists really not know how they do their magic? Are they just some kind of mental demi-gods?

I mean, in geometry or linear algebra, sometimes you are told something that is so obviously true because you can "see" it, even before you prove it. Is there a way to develop such an instinct and intuition for abstract algebra, or is it more like you have in your mind a catalogue of "if...then..." statements and the best you can do is get really good with using these statements. In other words, does being good at abstract algebra just come down to being good with logic and symbolic manipulation, or is there such a thing as algebraic intuition, analogous to geometric intuition?

I ask because I'm teaching myself some basic group theory now and am enjoying it. But, it's exactly because of the intuitive aids like Cayley diagrams that allow me to "see" groups, that make me like it. I hated high school algebra because I foud it to be just manipulating "x,y,z"s on a page until they are configured in the desired way. If abstract algebra is like that too then I don't think I'll continue to enjoy learning it.

I apologize if some things in my post are vague and unclear; I'm still just a newbie so I don't have many concrete examples to draw on.