Is the gospel story (Matthew 22:15-21) speaking about the separation of church and state?
No, this is not a story about politics. This story always reminds me of the old 1966 alert that the Surgeon General of the USA started placing on packs of cigarettes: Caution: Smoking may be hazardous to your health. Just so, whenever the Pharisees engage Jesus in dialogue, you see those flags go up immediately: Danger Pharisees hazardous to your health are approaching. The Pharisees had long ago decided to trick or trap Jesus into either committing a blasphemy against God or voicing a protest against the Emperor. Either way would lead to death: the Jewish penalty for blaspheming was death by stoning; and the Roman penalty for revolutionaries who fought Rome tax issues was death by crucifixion. Jesus may have just entered the teenage ranks at the time when Judas the Galilean revolted against a Roman census. Sepphoris was his stronghold, only four miles from Jesus hometown of Nazareth. The historians say that hundreds of his group were crucified as an example and warning to Israelites w