I think we have to make a distinction here between cognitive thinking and awareness. One of the examples Joel has in the DVD is if you're pulling, and the thought occurs to you, "wow, this is taking a long time to get to my nose," that thought is cognitive, or conceptual thinking about an action, as opposed to actual physical awareness of an action that is taking place, and is an interruption in your concentration, and means that you should let down the bow and start over. He constantly asks the question, "what were you thinking?" because humans tend to blur the distinction between conceptualizing and doing. We may have conceptual thoughts about pulling the bow that seem real to us, but they're not; the only thing that is real is the physical feeling of pulling the bow. If we can distinguish what our conceptual thoughts are from our physical awareness, then we can recognize the cognitive thoughts that interrupt our concentration in time to do something about it (let down the bow), as opposed to just guessing what went wrong after the shot.
When you say the mantra, "Keep pulling, keep pulling....." the reason you're saying it is to block cognitive thoughts about pulling the bow, and instead, just pull it.
The reason you release the arrow when the feather touches your nose is that you have made a plan to do that in advance. The physical touch of the feather against your nose triggers the release, not the thought of the feather touching your nose. Ideally, you should be thinking about nothing as you pull the bow, you should instead be physically aware of everything that is happening.
I think everything I said so far is in agreement with Joel's DVD. However, I would go one step further and say that the ideal would be to do away with the mantra and just experience the pull of the bow. The mantra is a step designed to do away with cognitive thoughts during the shot, but the goal should be to do away with cognitive thoughts without having to say the mantra. Joel may disagree with what I just said, and if he does, I hope he will chime in and maybe I will learn something.