Then again, I thought 1+1=2 is axiomatic (2 being the defined by having a count of one and then another one) So I don't understand why Bertrand Russel had to spend 86 pages proving it from baser fundamentals.

Well, he was trying to derive essentially all of contemporary mathematics from an extremely minimal set of axioms and formalisms. The purpose wasn't really to just prove 1+1=2; that was just something that happened along the way. The goal was to create a consistent foundation for mathematics from which every true statement could be proven.

Of course, then Kurt Gödel came along and threw all of Russell's work in the trash.

Yeah, the four color problem becomes obvious to the brain if you try to place five territories on a plane (or a sphere) that are all adjacent to each other.

I think one of the earliest attempts at the 4 color problem proved exactly that (that C5 graph cannot be planar). Search engines are failing me in finding the source on this though.

But any way, that result is not sufficient to proof the 4-color theorem. A graph doesn’t need to have a C5 subgraph to make it impossible to 4-color. Think of two C4 graphs. Choose one vertex from each- call them A and B. Connect A and B together. Now make a new vertex called C and connect C to every vertex except A and B. The result should be a C5-free graph that cannot be 4-colored.

Then again, I thought 1+1=2 is axiomatic (2 being the defined by having a count of one and then another one) So I don’t understand why Bertrand Russel had to spend 86 pages proving it from baser fundamentals.

It is mathematic. Of course it has to be proved.