I recently watched this Coursera course on learning how to learn and your post uses different words for some of the same things.
The course described what you call "shower-thoughts" as "diffuse mode" thinking, with an opposite called "focused mode" thinking and the brain only able to do one at a time. Focused mode uses ideas that are already clustered together to solve familiar problems while diffuse mode attempts to find useful connections between unclustered ideas to solve new problems in new ways. Not sure if these are the formal/correct terms from the literature that was behind the class, but if so it might be worth using them instead of making up our own jargon.
As for the class it definitely had some stuff that I still try to keep in mind, but it also had some things that I haven't quite figured out how to incorporate (chunking) or didn't find useful (some of the interviews). There is some overlap with what CFAR seems to be trying to teach. Overall I'd recommend taking a look if you have an hour or so per week over a month for it.
Great course. I took it a few years back and got a lot out of it. I think as it relates to understanding the actual underlying physical processes of the brain and how they relate to making conscious meaning, it was super helpful to me.
Related to: Compartmentalization in epistemic and instrumental rationality; That other kind of status.
Attempted telekinesis
The case of the munching noises
The ad copy writer who doesn’t know if she’s “good enough”
Useful “telekinesis”: Separating babies from bathwater
How to distinguish?
Task type:
Type 1: Problems that System 1 can solve by itself:
Examples: Making breakfast; causing someone to know you care about them.
Suggested response: This sort of wishing is healthy, and may prompt actions that make a lot more sense than those system 2 would plan (e.g., your nonverbals as you apologize are likely to be far better if you viscerally care about your interlocutor). Leave system 1 be.
Type 2: Problems that are worth solving, but that need help from System 2:
Examples: “There’s nothing good to eat” (situation: you notice that several times, over the last hour, you’ve gone to the fridge, opened it, stared inside, closed it... and then opened it again a few minutes later -- as though to see if something good has magically materialized into the closed fridge); Feeling 'stuck' at one's job (or in a relationship); Not having enough money. (The distinguishing feature here is that system 1 has been looping on the problem for a while to no effect, and that system 2 has not yet taken a good look at the problem.)
Suggested response: Raise the problem to conscious attention; then, try to figure out what is bothering system 1; finally, decide what to do about it. As you do this, parts of the wishing will naturally shift from the general problem ("Somehow make work less stuck-feeling") to the specific strategy you've chosen ("Figure out how to renegotiate with my manager").[4]
Type 3: “Problems” that should be given up on:
Examples: “Make the munching noises go away” (in a case where you’ve decided not to); “Make San Franciscans be better drivers”; “Let me vanish into the floor.” (The distinguishing feature here is simply that these are "problems" that, on reflection, you do not wish to take action on.)
Suggested response: Find a way to let system 1 know that solving this problem isn't worth the cost, or that keeping this problem on your internal "worry/fume about" list is quite unlikely to have positive effects. For example, you might:
Examples: The problem of locating a workshop venue (during the hour at which I was trying to write the workshops ad, that October); the situation with your roommates and the dishes (while you're at work solving a coding problem).
Suggested response: Designate a particular future-you to do the task. Dialog with your "inner simulator" (your system 1 anticipations) until both system 1 and system 2 are convinced that that specific you will actually do the task, and that there is no additional positive effect to be gained via staying preoccupied now.
Type 5: Problems that System 2 needs "shower-thoughts" help with:
Examples: Archimedes' problem measuring the king's crown; "My relationship with Fred is broken, and I can't figure out what to do about it"; "How the heck can I solve that math riddle?" (The distinguishing feature here is that both: (1) the problem has already been raised to conscious attention at some point (and system 2 failed to instantly solve it); and (2) the problem is a worthy use of your shower-thoughts -- either for what it'll accomplish directly, or for the improvement it may give to your pattern of thought.))
Suggested response: This sort of wishing is healthy. Leave system 1 be.
Emotional tone:
Wishes often seem to me to have emotional tones. Some tones are simple desire (“Breakfast... mmm....”). Others have an overlayed hopelessness or bitter resignation about them (“I just always have to put up with how everyone else is incompetent”); others, still, have a tone (at least in me) of hammed-up flailing, self-pity, or desire for outside help -- as though if I just feel helpless enough, somehow a grown-up will come to the rescue ("Make the workshop crisis not be in this state... Make the workshop crisis not be in this state...").
It seems to me that it's worth installing an "alert" that sounds, in your head, whenever it hears either the hopeless/bitter/resigned tone, or the flailing/save-me tone. Both are often signs of buggy "attempted telekinesis" situations that are worth conscious debugging (a la the schema above). And the emotional tones can be easier to automatically flag.