The real mathematics is the one that we dream

The ancient taboo at the heart of rationality

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.

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.

Here’s how Srinivasa Ramanujan described his thought process:

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:

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?

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 He left an unpublished manuscript, The Key to Dreams, where he expressed his belief that God himself—whom he called “the Dreamer”—was dreaming inside of his head.

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:

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:

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?

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?

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:

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.

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?

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.

The spiritualist worldview

In an earlier post, I argued that we’ve been wrong about math for 2300 years, operating under a flawed metaphysics that is incompatible with the practical aspects of doing math.

Our fantastic assumptions on the nature of math reflect our continued adherence to the spiritualist worldview, the system of beliefs that we inherited from our hunter-gatherer past.

The spiritualist worldview posits that there are two “layers” in reality, the physical world, inhabited by material objects, and the spiritual world, inhabited by spiritual entities, whichever way you want to categorize them: gods, spirits, sprites, souls, ideas, abstractions, “truth” and “meaning”, qualia…

Remarkably, many people who view themselves as “rationalists” stick to this archaic worldview. Most of them are oblivious to their own spiritualist beliefs, which they have conveniently swept under a very large rug which they call “mathematics”.

This particular tradition goes back to Galileo’s famous pronouncement that the universe is written in the language of mathematics.

We’re so used to this framework that it takes a while to defocus and notice its bizarre implications. Language is a behavioral trait of Homo Sapiens and other animals—is Galileo trying to tell us the universe is sentient and speaks mathematics? Or that it was created by a mathematical demiurge? Except for the substitution of mathematics in place of Word, his pronouncement isn’t much different from the opening line of the Gospel of John: “In the beginning was the Word, and the Word was with God, and the Word was God.”

Galileo’s vision enables scientists and laypeople to go about their daily lives undisturbed by spirits and sprites, except for the few people whose job is to operate under the rug: mathematicians. This is where the framework breaks down, and it breaks down spectacularly.

The issue is that Galileo never explained how humans can develop this curious ability to figure out and manipulate the language of the universe. If you take his words at face value, mathematicians are modern shamans who interpret nature by entering a dialogue with cosmic entities. No wonder, mathematical talent is presumed to be innate and manifests itself through trance-like illuminations.

This also leads to Descartes’s dualism, his belief that the human brain is both a physical organ and a spiritual organ created by God in his image. Descartes’s method, in his own words, amounts to paying attention to the “seeds of Truth” that God has planted in each of us.

For the working mathematician, this creates a really awkward situation. Mathematics is the pinnacle of rationality, and yet the mathematical experience itself feels supernatural.

Some mathematicians go full-blown mystics and attribute their creativity to divine intervention, such as Ramanujan who claimed that his results were revealed to him by his personal goddess and, as we’ve seen, Grothendieck who thought that God himself was dreaming inside his head.

Platonism, the notion that mathematical objects “exist” in the ethereal realm of ideas, is the soft-core and socially acceptable version of this belief. It is extremely prevalent in the mathematical community, although most mathematicians are vaguely aware that it is somehow problematic. Of course, no-one has a clue as to how mathematicians’ brains can magically “access” the ethereal world of ideal entities.

Contrary to a widespread misconception, formalism doesn’t offer a viable alternative. Formalism is a methodological framework for unifying mathematics and analyzing proofs,⁶ but it isn’t a credible ontology for mathematics. No-one studies ZFC set theory without projecting “meaning” onto it: why would you care about specific axioms, if they are meaningless garbles? In the real world, self-proclaimed formalists are just Platonists in the closet.

The conceptualist worldview

Once we give up on magical entities, the situation becomes much clearer:

If we stick to what we are certain of, we can observe that:

1. Mathematics is written using formal systems, whether legacy ones (numerals, symbolic expressions, geometric constructions) or modern ones (axiomatic set theory, category theory, proof assistants).

2. To make sense of these formal systems, we project “meaning” onto them: we imagine that symbols represent “real objects”, just like we think of a person when we read their name.

3. To develop our intuitive ability to “perceive” the mathematical “reality”, we have to spend considerable time interacting with it with an active exploration mindset, playing with examples, following reasonings and calculations, and thinking intensely about it in a meditative, dreamlike fashion.

This leads to my proposed conceptualist characterization of mathematics as a mental activity based on manipulating formal systems and imagining that they hold absolute and immutable truths about real objects.

Many mathematicians, even Platonists, agree that—for all practical matters—doing mathematics is just doing that. But most object to the metaphysical deflationary step of declaring that mathematics is just that.

They refuse to let go of the “transcendence” and “objectivity” of mathematics, which they view as its most precious aspects. As Grothendieck pointed out, there is an ancient taboo against imagination and subjectivity in mathematics, and no serious mathematician is supposed to primarily rely upon them.

But by insisting that the objects they imagine to be real are actually real, they lock themselves in a much broader delusion.

Yet Platonists do have a point: it is humanly impossible to do mathematics without imagining that mathematical objects are real. Grothendieck is absolutely clear about that in Harvests and Sowings:

All my life I’ve been unable to read a mathematical text, however trivial or simple it may be, unless I’m able to give this text a ‘meaning’ in terms of my experience of mathematical things, that is unless the text arouses in me mental images, intuitions that will give it life.

Like formalism, Platonism is an ontological dead-end but a methodological necessity.

Platonists are also right to insist that mathematics isn’t arbitrary. The superhuman attributes of mathematics, its “transcendence” and “objectivity”, aren’t lost in the conceptualist approach: they are merely contained to where they belong, within meaningless formal systems. The meaning of mathematics, our intuitive interpretation of symbolic expressions as “statements” about “mathematical objects” that hold a “truth value”, is a human cognitive phenomenon that can’t possibly have these qualities.

Truth and meaning aren’t objective attributes of the physical reality. They only exist in the intelligibility layer that we project onto it. Through its reliance on consistent formal systems, mathematics enables us to fantastically expand our intelligibility layer and create broad cognitive alignment at species level.

This explains the unique and indisputable Platonic feel of mathematics—but it’s not an excuse for magical thinking.

The reality of mathematics

When we couldn’t make sense of lightning and thunder, we attributed it to the wrath of gods. When we couldn’t make sense of mathematical cognition, we were stuck with the spiritualist worldview.

This is no longer the case and, in my view, time has arrived for career mathematicians to seriously reexamine their position. Mathematical Platonism isn’t just metaphysically wrong, it is also harmful in very practical ways.

First, it obscures the actual cognitive processes involved in mathematical thinking and creates an unnecessary barrier to entry. Imagination shouldn’t be a dirty word. If mathematicians never explain the special mental tactics that drive neuroplasticity and help them solidify their mathematical intuition, if no-one tells kids that they should actively engage their imagination, it shouldn’t come as a surprise that most people don’t get math and just hate it.

The conceptualist perspective makes it possible to have an honest conversation on how one should process math in the secret of their head. This is the approach that I have followed in my book, and the feedback received proves that this conversation can be illuminating at all levels, even for people who always thought that they hated math.

The second reason concerns mathematicians themselves.

I started this post with striking examples of mathematicians flirting with the limits and entertaining a bizarre proximity with dissociative modes of thoughts. These reckless acts of imagination are characteristic of high-level research and often produce stunning breakthroughs.

But the story doesn’t always end well.

There is an inherent danger in suggesting that mathematicians have a direct, shamanic access to higher-level truths. When performed outside of mathematics, without the ultimate guidance of formal systems, reckless acts of imagination rarely produce valuable insights.

When John Nash was first committed to a psychiatric institution for his paranoid delusions, he received the visit of his colleague George Mackey, who asked him this blunt question:

How could you, a mathematician, a man devoted to reason and logical proof… how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world?⁷

Nash had this stunning reply:

Because the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.

As Nash’s case dramatically illustrates, the spiritualist worldview isn’t just confusing to outsiders: it is also confusing to mathematicians themselves.

There should be no secrets and no taboos regarding the underpinnings of rationality. The power of mathematics lies not in supernatural access to eternal truths, nor in unbridled imagination, but in the constant back-and-forth between an apparently sterile formalism and our continuing effort to make intuitive sense of it.

1. Sourced from https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

2. R.W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories, in The Grothendieck Festschrift Volume III, Progress in Mathematics, vol. 88, Boston, MA: Birkhäuser, pp. 247–435, https://doi.org/10.1007%2F978-0-8176-4576-2_10

3. Grothendieck’s Harvests and Sowings hasn’t yet appeared in English. All excerpts quoted in this post were translated by Kevin Frey for the Yale University Press edition of my book, which cites them.

4. Quoted from Thurston’s foreword to The Best Writing on Mathematics 2010, edited by Mircea Pitici, Princeton University Press, 2011. I highly recommend this three-pages essay, one of the finest ever written on the mathematical experience (here is a scanned version).

5. My translation from Adrien Baillet, La Vie de M. Descartes, 1691.

6. As Pierre Deligne puts it: “Zermelo-Fraenkel or any formal system is not a tool for writing mathematics, it’s a tool for analyzing proofs and also, a very useful thing, it gives a common meaning to what it means to have a proof” (from his lecture at the Vladimir Voevodsky Memorial Conference, September 11, 2018.)

7. Mackey’s question and Nash’s reply are quoted from the prologue of A Beautiful Mind, by Sylvia Nasar.