Equivalence between definitions of simple connectedness.

Question

Wikipedia gives two definitions of simple connectedness:

1. A topological space $X$ is simply connected iff it is path-connected and any loop in $X$ can be contracted to a point.
2. A topological space $X$ is simply connected iff it is path-connected and any two paths with the same start-point and end-point are homotopic.

I do not see how these are equivalent. What is the connection between loops being contractable and and paths being homotopic? I am mostly looking for intuition here, as opposed to a formal proof.

By the way, I am aware of another definition: A topological space $X$ is simply connected iff its fundamental group is trivial. I would like to avoid that characterization here.

Answer 1

The intuition to have here, is to think that $X$ has no holes in it. Both definitions make perfect sense from this point on.