Inspired by the "revelation " thread, i am wondering about revelations which result in a breakthrough. This has been noted by many famous matematicians…such as Karl Gauss, who reported wrestling with an equation, and getting nowhere, despite days of concentrated effors…then he experienced a breakthrough. he reported that all of a sudden, the answer appeared to him…as if in a dream, with no concsious effort. Lagrange (the french expert in orbital mechanics) also reported a similar happening-he was tired and burnt out, and could not get anywhere-then the answer was revealed to him!
My question: does the human brain function at some subconscious level, and provide answers for us 9when our own reasoning fails)?
I’m not sure about the whole subconscious reasoning thing - I have a feeling it’s been rather exaggerated by hypnotists and the like in the past.
Edward De Bono coined the phrase ‘lateral thinking’ back in the 60s - the gist of it being that in order to find your way around a dead-end of inquiry or reasoning, you need to back up and find a place to branch off, but continue in the same general direction.
I (and if it isn’t obvious, I’m not any kind of expert in any authoritative field here) would say that a similar kind of process can happen spontaneously in the scenario you describe - you give up and stop, and in so doing, you retreat from the most recent (and thus most resoundingly dead-ended) parts of the puzzle - and this permits new lines of inquiry - which have been there all along - to be noticed.
At least that’s how it feels when that sort of thing has happened to me, but as I say, I’m not an authority on the subject.
I awoke from a sound sleep with the solution of a Mate in Two problem I’d unsuccessfully wrestled with the day before.
I would bet the farm that geniuses have similarly solved knotty problems in the same way.
Somebody I read about (Edison?) kept a notepad on his bedside table for just that purpose.
And didn’t Archimedes shout “BarnOwl!!” when he stepped in bird shit and noticed how he’d displaced stuff equal to the volume of his sole?
I’ll bet Dopers will follow with a whole list of other examples (less elegant than mine, of course). 
Our brains do a great variety of things that we are not aware of. Our awareness is a product of some of these processes. I don’t understand why you seem to consider them as alien to us. (“provide answers for us”, “our own reasoning fails”)
PC
Let me use a truck metaphor* to explain how the process works.
Imagine yourself driving a truck on the causal highway of knowledge. Then at some point, your headlights go out and you can’t see the road anymore. You stop the truck in frustration and you get so mad that “lightening” bolts come out of you and strike at random spots in the dark around you, briefly illuminating them.
If you have enough lightning bolts *and there is a new road “branching off” (as Mangetout described), one of your lightning strikes will eventually illuminate it. Merry and giddy, you can go on driving towards your destination. That’s a breakthrough.
I hope this clarified things for you. If not, get mad and maybe you’ll get it.
*Not a good idea, surprisingly. And for my next number, watch as I explain economy using nothing but ants and screwdrivers!
One of the most famous ‘I had a dream’ ideas.
Effective thinking is nimble. You get a lot of ideas floating about, like when you’re looking at one of those “find the pattern” IQ problems and start listing and checking the possible patterns in your head. ‘Nimble’ is maybe similar to ‘subconscious’ in the sense that you can’t be trying too hard and really spelling out each idea else the brain kind of stops working. Exactly like on a multiple-choice test where the more time you spend on a problem the murkier it all becomes.
When Lagrange, for example, spends too much time thinking on the same problem he really can’t get new ideas. He needs to take a break, relax, and restore that light-handedness in thinking. The successful results of this nimble thinking may seem like spontaneous “revelations” but really it’s the way thinking works all the time: mostly a ‘below’ conscious bubbling of ideas, interrupted by a more conscious introspection when something interesting shows up.
A lightness of the hand is very important. Yoda’s words are very true. When you “try” you’re really just stressing and locking up your mind or your hands (ie, the whole thing is just as true for athletic performance as mental). A lot of people don’t understand that and think thinking is a process of deeply concerted effort. And then they’re surprised when the best ideas don’t come from that.
Isaac Asimov has an interesting read about how he broke through when stuck in that mental rut: he went to see a movie. The most mind numbingly entertaining yet vapid popcorn movie he could find. Most of the time when he came out, he’d have his answer. His theory was that, as has been said already, his neurons were firing in a circuit that was self-reinforcing, and he just had to get them out of that pattern and onto something else. “Something else”, for him, was explosions and heaving bosoms. Then, with seemingly little or no effort, he could pursue his previous train of though but take it to a new station.
Damon Knight discussed this in his book on short story writing. He called his subconscious Fred. You present Fred with the data required (in his case the characters and background for a story) let Fred work, and he eventually deposits the answer in your conscious mind.
This has always worked for me. From puzzle answers to ideas that were breakthrough enough to be worth a couple of papers, I’ve found my subconscious is much smarter than my conscious mind. I don’t usually wake up with ideas, they pop up when I’m taking a shower.
Another personal example would be video games. I can try a mission in Grand Theft Auto, for example, until I’m so frustrated I turn the system off. When I come back to it several hours later, I’ll beat it on the first try. That’s gotta be at least as profound as Lagrange. All he did was multiply some stuff, right?
Any problem that doesn’t involve my immediate response (i.e. if it doesn’t involve blood or fire), I put it from my mind for 24 hours. When I put it back in, I often come up with a solution.
It’s called the “Scarlett O’Hara method of problem solving.”
Kekule is popularly credited with visualising the structure of benzene as he lay dreaming of a snake swallowing his own tail. It is less widely known that he almost certainly ripped the idea off another contemporary chemist called Loschmidt, who had published the structure a few years earlier.
I guess this speaks to the OP in some respects - watch your competitors like a hawk , their ideas may lead to profound breakthroughs on your behalf.
My take is that frustration is a killer of inspiration. I have always said that I earn my living during my morning shower…that being most often the time when I resolve my most vexing challenges.
Most of my solutions to problems and inventions happen while I’m showering. Don’t know why, but I suspect the combination of white noise from the water spray and performing a function that I don’t have to expend any conscious thought upon allows me to “what if”. Then the solution appears fully formed in my mind’s eye.
I actually don’t understand why this was apparently such a long-standing problem of renown. Surely, once you know the chemical formula of benzene to be C6H6, you sit around and play with black and red balls and sticks for a little while (or whatever analogous visualization aid), and you’ll get it in within the afternoon. [Well, it and the few alternative structures matching that formula]. Educate me, folks; what was such a breakthrough about the idea that it could have a ring structure, that took so long to figure out?
Flashing on Dr. Grant Swinger of the Breakthrough Institute. Omni magazine published several interviews with him. (Not to be confused with the contemporary, non-self-consciously-satirical Breakthrough Institute, of which the less said the better.)
With me, it’s cryptic crossword clues - I’m not very good at cryptic crosswords, but really enjoy the feeling of success when I do solve them. I have often had the experience of putting a crossword down for a few days/hours, frustrated because I’m getting nowhere, and then within minutes of picking it up again, solving four or five clues in a row…
It was a deep problem for a couple of reasons. Firstly, each and every structural question was a deep problem at that time. The idea of tetravalent carbon was first published in 1858 by Archibald Scott Couper, Loschmidt published his structural work on organic molecules (including benzene) in 1861, and Kekule described benzene in 1865. So it was a primordial time in chemistry. There would simply not have been a concensus as to the rules for joining up the aforementioned balls and sticks in structural models.
A second reason for the mythic status of the benzene problem is that most of the evidence for structure at that time came from doing chemical reactions on the molecule in question. Benzene does not behave like the structure Kekule proposed in many respects. Reconciling the differences in reactivity between benzene and the formal ‘cyclohexatriene’ structure of Kekule became one of the formative concepts of organic chemistry - aromaticity.
Ah. But did Kekule himself work on the reconciliation? The way the story is told, it always seems as though his acclaim is for simply discovering the structure in the first place. (Did he simply propose a ring of alternating single and double bonds, or did he think in terms of resonance as I gather is more accurate?)