There are often "empty objects" in maths. For example, sets are vaguely thought of as "objects that contain things", and the empty set $\emptyset$ is the "set containing nothing". However, something I don't quite understand is the empty graph: The graph with no nodes and no edges.
In the case of the empty set, you still have a concrete object. This is because $a \neq \{a\}$, so I can extrapolate and understand that $\emptyset \neq$ "the absence of everything". However, a graph is simply defined in terms of its vertices and edges. Without any vertices or edges, I don't see how it's any different to "nothing". If I draw the empty graph on a piece of paper, it's exactly the same as not drawing anything. How do you know the empty graph exists?
One way I thought about it was using the approach where $G = (V, E) = (\emptyset, \emptyset)$. using this approach the empty graph is an "object" distinct from the lack of everything. However, using sets feels more like a "tool" which doesn't capture the essence of what graphs are. Graphs are just relations between objects and I believe they exist independent of sets. Therefore it should be possible to have an intuition for the empty graph completely independent of sets.
Any ideas or explanations as to what the empty graph is will be greatly appreciated!
Your philosophical question makes mathematical sense only in a context in which you'd like some definition to make sense or some theorem to be true for the "empty graph". Then the context would suggest the right definition.I suspect it would usually be the pair of empty sets for vertices and edges.