To be clear: i am a gnostic atheist; to me the answer to the question “does God exist?” is a clear no.
I am very empirical even about mathematical claims—you prove the existence of (say) a nonabelian group of order 8 by carrying out the construction, which is a laborious real-world computation with real-world results.
You prove the existence of a certain pathogen in a patient's body by running a certain lab test.
Maybe i'm using a funny notion of “prove”.