A similar question is found below:
Scientific progress claimed to be caused by dreaming
Has anyone ever made similar claims regarding other altered states of consciousness, besides dreaming? I know similar claims have been found in the sciences, but what about mathematics specifically?
Let us exclude disease, as there are cases of mathematicians with mental illnesses that have certainly given much!
Scientists and mathematicians rarely self-report on psychological circumstances of their creative process, at best presenting a rationalization of their path to discovery. Kekule's dream and Kepler's vivid struggles described in Astronomia Nova are exceptions (and there are doubts that Kepler's descriptions are factual, see How did Kepler "guess" his third law from data?). Third person accounts often take on anecdotal character, like Eureka or Newton's apple.
Two first hand works on psychology of discovery by mathematicians are a chapter in Poincare's Science and Method and Hadamard's Psychology of Invention in the Mathematical Field. In particular, Poincare gives a famous report of his "altered state" when discovering what he called Fuchsian functions (now automorphic forms):
"It is time to penetrate further and to see what goes on in the very soul of the mathematician. For this, I believe, I can do best by recalling memories of my own. But I shall limit myself to telling how I wrote my first memoir on Fuchsian functions. I beg the reader's pardon; I am about to use some technical expressions, but they need not frighten him, for he is not obliged to understand them. I shall say, for example, that I have found the demonstration of such a theorem under such circumstances. This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem but the circumstances.
For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. 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. By the next morning I had established the existence of a class of Fuchsian functions, those which came from the hypergeometric series; I had only to write out the results, which took but a few hours."
Paul Erdős was a mathematician noted for his prolific contributions to the science of mathematics as well as a proclivity for the use of amphetamines. His friend Ronald Graham famously bet him $500 (in the 1970's) that he could not stop taking amphetamine for a month. He successfully completed the challenge, however begrudgingly, saying, "Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper."
Erdős resumed his amphetamine use after the month was over, allegedly asserting that the science of mathematics had been set back by a month due to his abstinence. While it is not immediately clear which of his mathematical works were completed under the influence of amphetamines and which were not, it is quite likely that at least some of his work, especially in the last quarter-century of his life, was created under the influence of stimulants.
A mention should also be given to a commonly-related story about Drs. Watson and Crick that they discovered the structure of DNA under the influence of LSD; while although they have been historically credited for the discovery, credit has been more recently given to Dr. Rosalind Franklin, whose unpublished work on DNA structure was evidently shared with Watson and Crick some months before their historic article was published in Nature. Therefore, the story that they first imagined the double-helix structure of DNA molecules while under the influence of the psychedelic must be at least somewhat apocryphal.
Kary Mullis was awarded a 1994 Nobel Prize in Chemistry for his invention of the polymerase chain reaction (PCR) technique.
Mullis reputedly said:
Back in the 1960s and early 1970s I took plenty of LSD [...] and I found it to be a mind-opening experience. It was certainly much more important than any courses I ever took.
During a symposium held for centenarian Albert Hofmann, Hofmann said Mullis had told him that LSD had "helped him develop the polymerase chain reaction that helps amplify specific DNA sequences".
In a Notices of the AMS interview with Mikhail Gromov [PDF], Gromov says:
I realized that there had been a huge development in molecular biology in the 1980s, after the discoveries of genetic engineering and of PCR (polymerase chain reaction). It was really mathematical procedures applied to living cells. Mathematicians could invent PCR. It didn’t happen, but mathematicians could have invented PCR.
So perhaps this is a mathematical breakthrough while in an altered state of consciousness.
Bob Thomason attributed of co-authorship to his deceased (former??) collagorator Tom Trobaugh for a significant paper, due to Trobaugh's coming to him in a dream with critical ideas. (Thomason has a Wikipedia page. I'm not sure which paper this was, as it all happened 25+ years ago. Perhaps it was the "Higher algebraic K-theory of schemes and of derived categories", in the Grothendieck Festschrift.)
As Conifold noticed, only occasionally scientists report the circumstances of their creative process. The following story does not describe an altered state of consciousness induced by psychotropic substances, but rather a connection between ideas belonging to different domains made in a reduced level of consciousness due to tiredness, similar to Poincaré's altered state.
Lindenmayer systems are a special type of formal grammar developed by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist.
L. especially studied yeast and fungi and wanted to develop a general formal model of the structure and development of such simple organisms. He found a perfect basis [...] in the theory of formal languages [...] Before long, L. might have found formal languages as the model for his system by himself, but a nice coincidence made it happen earlier: L. was absorbed in studyng photosynthetic organism called "algae" as he passed the lecture hall where a lecture on formal languages was taking place. The lecturer repeatedly used the term "L(G)", which cause the biologist to immediately experience an aha moment: "Algae, that's it!" (cited from Arto Salooma: Mathematician, Computer Scientist, and Teacher, Jukka Paakki, p. 111.)
I recently watched a Ted-Ed video, and it talked about how Van Gogh supposedly painted turbulence in a mathematically correct way when he was in a period of intense suffering. This is a feat because turbulence had not been discovered yet, and to this day is not fully understood.
