I've read (in Griffith's text) that it is "possible, though cumbersome" to dispense with the field concept in electrodynamics entirely and instead use an action-at-a-distance theory.

What exactly is meant here? Since fields physically exist, why is this permissible?

I've read (in Griffith's text) that it is "possible, though cumbersome" to dispense with the field concept in electrodynamics entirely and instead use an action-at-a-distance theory.

What exactly is meant here? Since fields physically exist, why is this permissible?

Edit: In light of the answers below, I'm curious about the apparent contradiction here - if fields physically exist, then how can one dispense with the field concept?