The proof of the following theorem:
A graph can be embedded on the surface of a sphere without crossings if and only if it can be embedded in the plane without crossings.
is very short-
The plane is topologically a sphere with a missing point at the North pole.
Now, before I start: I do believe this theorem is true. My goal was to do some reflection on why I believe it's true. It's a good practice, we should always ourselves 'what makes me believe it?' and 'should I believe it?'. Similarly, we should ask: how much abstract thinking and intuition is allowed in mathematics? What if we reach wrong conclusions by using too much intuition to prove theorems - if so, where is the boundary? We take certain statements as axioms, use logical thinking and derive conclusions which we call theorems. And so on. This is the only proper way of doing mathematics.
Why should I consider this as a convincing argument? Topology is just a purely mathematical construct. Here, we have a real-world problem and want to solve it using the formal methods of mathematics. I don't trust simplified proofs based on intuition - what if I'm being fooled into thinking this is true for every planar graph? Even if the smartest person in the world "believed it", there may still exist a counterexample.
A proof is valid if it shows that, in this example, there doesn't exist a planar graph that cannot be drawn on a sphere.
Can we make it a bit more precise and formal? I believe the first step is to state the define the theorem in formal terms.
- How to define edges drawn on a sphere, their intersections?
- What allows us to model this problem in topological terms?
- Why homeomorphism is believed to guarantee that if there are no intersections in topological space of the plane, then there aren't in topological space of the sphere without a point? Should we believe it?
What I'm concerned about is the transition from natural description of the problem to topological, formal one. We solve the problem in the domain of topology and assume that the problem stated in topological terms is equivalent to the original problem. In other words, we are assuming that topological formulation of the problem is equivalent to the original problem - this assumption is based on intuition!. If not, what axioms or theorems (which is what maths is made of) tell us we can do that - e.g. make a topological space out of a graph, solve the problem in topology domain, and come back to our graph-theory domain?