I have hit upon major (for *me*—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending through the night into the morning.
But I have never actually solved a problem in my sleep.
I have awakened thinking *That's it!*, but never was it actually *it*.


**Q**. Can anyone report an actual significant advance in their research
that occurred during and emerged from their sleep?


Of course this is entirely subjective, but you would know it if it happened to you.


Poincaré's famous step onto the bus in 1908
("At the moment when I put my foot on the step the idea came to me...")
indicates significant unconscious processing,
and his insomnia account (quoted below) adds further credence to
such "background" processing.
But I am not aware of first-hand reports of
significant and accurate reasoning
occurring *during* sleep.




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> 
> One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. ... ([Link](http://web.archive.org/web/20160609203836/http://www.is.wayne.edu/DRBOWEN/CRTVYW99/POINCARE.HTM))
> 
> 
> 




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Have you solved problems in your sleep?
On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep.


For example, one of the critical ideas in my paper


* *Joel David Hamkins*, [**Gap forcing**](https://doi.org/10.1007/BF02773382), *Israel J. Math.* **125** (2001), 237--252,


came to me this way, and waking up with the mathematical idea, I tore myself out of bed to work it out more fully on paper. It was totally right and formed the basis of later work. I remember sitting in my night attire in the bare moonlight at the table in my apartment, looking out at the empty sidewalk at Wall Street and Williams, where I lived at the time, pondering the approximation property applied to ultrafilters.


Because this has now happened several times, I now quite regularly try to prime myself, by intentionally focusing on a particular mathematical issue just as I am going to sleep. My mind floods with mathematical ideas just as I drift off. On welcome rare occasions, the problem is solved in the hypnagogic state, and having awoken I lay in bed pondering it, trying to check it, and wondering if it really is right (sometimes, of course, what seems right is later found to be mistaken). More often, though, when there is welcome news it consists not of a full solution but rather of a new perspective, which later forms the framework of a solution. That is, the result of the unconscious thought is a new way of thinking about the problem, rather than a complete logical proof.


At times, naturally, it is an interesting (or obsessive) MathOverflow question that I set myself to thinking about as I lay myself down. But let me say categorically that it has never been the case (*ahem, cough, cough*) that an hour or two after going to bed, I would wake with an answer and crawl out to my computer to type up an MO answer in the dark, while the rest of the household is sleeping, only to realize at that point, right before clicking "Post Your Answer" that the solution was totally flawed or wrong. What a downer that would be, to be sitting in the dark in the middle of the night, tired, with nothing to show for it but a wrong mathematical idea. That has NEVER happened... :-)


