The real mathematics is the one that we dream
David Bessis
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The real mathematics is the one that we dream<br>The ancient taboo at the heart of rationality
David Bessis<br>Feb 19, 2025
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With the possible exception of a few lost tribes somewhere in the Amazon or Andaman Islands, career mathematicians are the last true animists in this world.<br>It’s no secret that something weird is going on with mathematical research. But when you experience it from the inside, it’s orders of magnitude weirder than anything you could have imagined.<br>Here’s how Srinivasa Ramanujan described his thought process:<br>While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.1
Maybe Ramanujan doesn’t count. He was an outlier and his story may have been exaggerated. So let’s take another example. This influential research article, published in 1990 and cited over a thousand times, was coauthored by Bob Thomason and a ghost:<br>The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, “The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.” Awaking with a start, I knew this idea had to be wrong… I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument.2
OK, but what if Thomason doesn’t count either? What about the truly famous mathematicians, those who went beyond proving great theorems and redefined mathematics itself?<br>Alexander Grothendieck, often regarded as the greatest mathematician of the 20th century, wrote extensively about his obsession with dreams. He left an unpublished manuscript, The Key to Dreams, where he expressed his belief that God himself—whom he called “the Dreamer”—was dreaming inside of his head.<br>In Harvests and Sowings, his 1000 pages autobiographical essay, he insisted on the importance of dreaming and daydreaming in mathematical discovery. He attributed his unique creativity not to extraordinary intellectual abilities, but to his uncommon dreamlike approach, “gathering intangible mists from out of an apparent void”, which he cultivated in transgression of an ancient taboo:<br>It would seem that among all the natural sciences, it is only in mathematics that what I call ‘the dream’ or ‘the daydream’ is struck with an apparently absolute interdiction, more than two millennia old.3
But in all honesty, Grothendieck was also some kind of weirdo. Let’s try a more grounded mathematician, Bill Thurston, the 1982 Fields medallist who wrote several influential essays on the nature of mathematics. He too was an avid dreamer:<br>I have decided that daydreaming is not a bug but a feature.4
What is going on here? Are we observing the deliquescence of mathematics, a postmodern collapse fuelled by New Age and psychedelics?<br>What about classical mathematicians? What about the 17th century? What about René Descartes, the greatest mathematician of his time and the father of modern rationality?<br>His early biographer Adrien Baillet had access to a manuscript, now lost, where Descartes explained how his philosophy—what we call rationalism—was revealed to him directly by the Spirit of Truth. The event took place on the fateful night of November 10, 1619, when 23 year old Descartes fell asleep near an overheated stove and had a series of three dreams. After a first dream involving a melon, Descartes woke up and fell asleep again:<br>A new dream immediately came to him, in which he thought he heard a sharp and resounding noise, which he took for a clap of thunder. The fright he felt woke him up immediately; and having opened his eyes, he perceived many sparks of fire scattered throughout the room… The thunder, the clap of which he heard, was the signal of the Spirit of Truth descending upon him to possess him.5
Ramanujan, Thomason, Grothendieck, Thurston, Descartes: it looks like we have a pattern.<br>But what could drive serious mathematicians to view their dreams as a credible source of insight? Aren’t dreams supposed to be the opposite of rationality, the opposite of what mathematics relies upon?<br>As we will see, there is a way to resolve the apparent contradiction between the rationality of mathematics and the irrationality of mathematical creativity, but it will require us to reset our fundamental beliefs about reality, human cognition, and language.<br>The spiritualist worldview
In an earlier post, I argued that we’ve been wrong about math for 2300 years,...