Children of the Labyrinth

Traveling between worlds isn’t as simple, or as dramatic, as falling down rabbit holes, walking into wardrobes, or getting snatched up by cyclones. I should know: I frequently nip off to a parallel world, often without anyone realizing I’ve left this one.

Before you can visit that other world — my second home — you have to imagine it, which is trickier than it sounds: sometimes you think you’re imagining it but in fact you aren’t. It helps to describe what you see in your mind’s eye to someone who’s been there, who can help you determine whether you’re genuinely imagining the other world or deluding yourself. (For instance, it may look like you’ve squared the circle, but someone with a good grasp of geometry can help you see that you haven’t.) Ultimately, when your imaginings of the other world are properly calibrated, you can go there — though “go there” is a misleading phrase, since all of us who visit the other world are engaged in nothing more than calibrated imaginings. But surely the place is real, for how else can you explain why two visitors, independently exploring precincts of that world that no one has ever visited, will see the same things?

Of course, I am talking about the world of pure mathematical form. One recent visit I took to that world prompted me to write the following in a succession of tweets:

My research life this week, in allegory: I’m exploring a magical landscape with some friends, and we’re teaching each other about the terrain. I say “I think there’s a castle on the other side of that hill,” and we climb it, and sure enough, there’s a castle. They get to work turning on the lights and fixing plumbing with impressive speed. Meanwhile, they wonder how I knew there’d be a castle there. I’d like to be able to say “I’ve been climbing hills for forty years, so I’ve learned to recognize the sorts of hills that lead to castles.” But I haven’t. Most of the hills I climb still don’t have castles behind them. But what I’ve learned is that you don’t find castles if you don’t climb hills.

Some features of the mathematical world are easy to imagine as idealizations of experiences in our physical world: for instance, a dot that shows no sign of internal structure leads you to the idea of a perfect Euclidean point, even if there are no such things in our world.  Other features of the mathematical world can’t be seen on a second, tenth, or even hundredth visit, not because they’re hidden in some remote part of the math world that’s hard to go to, but because they can’t be apprehended without whole new sensoria — sensoria that are peculiar to the nature of the math world, and that individual visitors must painstakingly construct for themselves over a long period of time.

I tweeted about this aspect of math recently, writing in the terse, Figure-out-the-context-yourself style common on Twitter:

Like Narnia, but it takes years of study and practice to learn how to get there, and when you’re there, your earthly body stays behind with a vacant expression on its face and your spouse says “You’re doing Narnia now, aren’t you?”

My tweet was evidently too terse because some readers mistakenly thought I was talking about meditation, not mathematics.


What prompted the tweet was my reading a paragraph from a fantasy novel describing a character’s passage from one world into another, and my sudden conviction that jumping from our world into a very different one couldn’t be as simple as doing a ritual — and my realization a few seconds later that the strength of my conviction arose from my own experience of becoming a mathematician and my over-strong identification with the book’s hero.

Susanna Clarke’s “Piranesi” is about a man who —

Wait, hold on a minute. This essay is full of spoilers, so if at any point my description makes you suspect you’d enjoy “Piranesi”, stop reading immediately, get a copy of Clarke’s book, savor it, and then (and only then) come back to “Children of the Labyrinth”. Essays celebrating the joys of math are a dime a dozen (heck, you get mine once a month for free!), but a book by Susanna Clarke is a once-in-a-decade treat.

Where was I? Oh yes: “Piranesi” is about a man who —

Sorry, my wife says I’m not being clear enough.

Okay, here we go, for real. Piranesi lives in another world, having arrived there from our world but having subsequently lost all memory of where he comes from. (Actually, he knows some things about our world but doesn’t know that he knows them, having no place for this knowledge in his conception of the world.) He dwells alone in an enormous building that he calls the House (or the World). The House is something like a vast abandoned art museum, susceptible to periodic flooding by predictable but mysterious Tides. The story is told through entries in a journal that Piranesi faithfully keeps. To him, the House, despite its dangers, is a place of beauty and repose that provides for all his wants, and though he never compares the World to a womb, I couldn’t help feeling that Piranesi has reverted to the life of the not-yet-born.

The name “Piranesi” isn’t the name his parents gave him in our world, nor is it a name he chose for himself; it has been bestowed upon him by the only other person he sees in the House, whom he calls the Other. The Other knows things about the House that Piranesi doesn’t, but Piranesi isn’t eager to learn what the Other knows; Piranesi is content knowing that he is the Beloved Child of The House, free to wander its Halls, marvel at its Statues, draw fish from its Tides, and lovingly tend the corpses he has found (yes, the book is a bit of a murder mystery as well as a fantasy novel). For his part, the Other has very little interest in any of the above; he only cares about the magical powers he hopes to obtain with Piranesi’s help.

As the tale progresses, other people come into the House, and Piranesi begins to question his own understanding of who and where he is. Before that happens, Clarke has already revealed that Piranesi is living in a “tributary world”, created by a sort of spiritual runoff from our own (which explains why the statues he loves all depict things from our world). One feature of the tributary world is that spending too much time there brings amnesia — hence Piranesi’s inability to recall where he comes from.

At one point, Piranesi, having come by degrees to understand that his current world is not the only world that exists, tells a visitor (whom I’ll call X) what he understands of the relationship between the older world that gave rise to the House and the House itself: “In this World the Statues depict things that exist in the Older World.” The following exchange ensues:

“Yes,” said [X]. “Here you can only see a representation of a river or a mountain, but in our world — the other world — you can see the actual river and the actual mountain.”

This annoyed me. “I do not see why you say I can only see a representation in this World,” I said with some sharpness. “The word ‘only’ suggests a relationship of inferiority. You make it sound as if the Statue was somehow inferior to the thing itself. I do not see that that is the case at all. I would argue that the Statue is superior to the thing itself, the Statue being perfect, eternal and not subject to decay.”

You can see why this makes me think of math.


I don’t find that doing math makes me forget things about the real world, though I do at times wonder whether my mathematical propensities have recruited portions of my brain that evolved for other purposes, and that, as a result of their repurposing, I am under-equipped for this world. For instance, would I be better at remembering people’s names if I didn’t spend so much time learning the particularities of all the mathematical forms that fascinate me?

But there is a kind of amnesia associated with learning mathematics, familiar to many teachers, especially those more devoted to research than to teaching, namely: the forgetting of what it is like not to know the things one knows. This is especially true when, as is so often the case with math, the things one knows are not facts but perspectives and habits of thought. What seems like a straight path to the adept can seem like a tortuous labyrinth to the novice.

The network of Halls that Piranesi inhabits seems like a labyrinth to other visitors to his world, and one of them asks him: “How long did it take you to learn it? The way through the labyrinth?” Here is his reaction:

I opened my mouth to say loudly and boastfully that I have always known it, that it is part of me, that the House and I could not be separated. But I realised, even before I spoke the words, that it was not true. I remembered that I used to mark the Doorways with chalk in exactly the same way that [X] did and I remembered that I used to be afraid of getting lost. I shook my head. “I don’t know,” I said. “I can’t remember.”

All too often, that’s me as a teacher.

There’s a line from the movie “The Paper Chase”, in which the fearsome Professor Kingsfield tells a room of first-year law-school students “You come in here with a skull full of mush … and you leave thinking like a lawyer.” This raises the question, will the newly-credentialed future selves of these first-years still be able to not think like lawyers? Or does the education process take away with one hand even as it gives with the other? Consider how hard it is for you (unless perhaps you recently had a stroke) to see the letters that make up this sentence the way an illiterate person might, as confusing geometric patterns; if you’ve learned how to read fluently, you cannot see these marks without reading them. Maybe one of the secrets of being a good teacher is an ability to swim against the tide of forgetting what it is like to not know, to interrupt the automatic insertion of acquired interpretations, to remember the texture of one’s former mental mush, so that in the classroom one can help other people jell their own mush into the needed cortical structures.

My wife, a psychologist, points out that knowledge overwrites anticipatory imagination: “You can remember what the campus looked like when you arrived at college, but can you remember what you imagined the campus would look like before you saw it?” Of course the answer is no. For that matter, I no longer know how I imagined Susanna Clarke looked before I saw her photograph, or how I pictured Jonathan Strange and Mr. Norrell, the protagonists of Clarke’s debut novel, when my wife and I were listening to the audiobook a decade ago; those mushy imaginings have been replaced by memories of the miniseries that we watched a decade later. The After displaced the Before. I wonder to what extent, years from now, assuming “Piranesi” gets made into a movie or TV series, I’ll be able to remember what the House looks like to me now, in my mind’s eye, in 2020.

The issue of Before versus After comes up not just in pedagogy but also in research. Most of us researchers have “Aha!” moments, when our way of looking at a problem is suddenly transformed, and afterwards we are sometimes tempted to regret all the time we “wasted” stumbling around through the Fog of the Before — forgetting that this stumbling may have been a necessary stage on the way to the Illumination of the After. Mathematician Hermann Weyl wrote about this dichotomy in a passage I encountered in the book “Out of the Labyrinth : Setting Mathematics Free” by Robert Kaplan and Ellen Kaplan.

To begin with, there are definite concrete problems, with all their undivided complexity, and these must be conquered by individuals relying on brute force. Only then can the axiomatizers come and conclude that instead of straining to break in the door and bloodying one’s hands one should have first constructed a magic key of such and such a shape and then the door would have opened quietly, as if by itself. But they can construct the key only because the successful breakthrough enables them to study the lock front and back, from the outside and from the inside.

Cartoon by Jeremy Cote. Check out his website .



The book closes on a hopeful note, or at least I found it hopeful, because I don’t want to have to choose between my two worlds, and Piranesi learns that he doesn’t have to choose between his. Through a wondrous inversion, Piranesi discovers that the World he has inhabited for so long now inhabits him. He writes:

In my mind are all the tides, the seasons, their ebbs and their flows. In my mind are all the halls, the endless procession of them, the intricate pathways. When this world becomes too much for me, when I grow tired of the noise and the dirt and the people, I close my eyes and I name a particular vestibule to myself; then I name a hall. I imagine I am walking the path from the vestibule to the hall. I note with precision the doors I must pass through, the rights and lefts that I must take, the statues on the walls that I must pass.

I carry a lot of my second world around in my head, and when I make my voyages of discovery to that other world, sometimes a voyage requires no paraphernalia at all, not even pencil and paper. Admittedly, this is the exception rather than the rule; my short-term memory is no more capacious than the average person’s, and usually I need pencil and paper to keep track of where I am and where I’m going. Sometimes I even need to enlist the help of a magical servitor that, while lacking imagination, can obediently carry out clearly-specified tasks too arduous for my limited brain. But in the end, my laptop’s assurances about what it sees don’t satisfy my desire for insight; like the “axiomatizers” in Weyl’s passage, I want to use what my laptop tells me so that I can construct a magic key that allows me to truly understand what a brute force computation has merely verified. What I find most satisfying, at each journey’s end, is to understand some part of the mathematical landscape so well that I can fit it inside my mind in its entirety, and I can imagine strolling along it, explaining every beautiful part of it to myself or to an imagined Other.

Early one morning, about ten or twenty years ago, I figured something out in my head while my wife was sleeping (in my first world, she was in bed with me and I didn’t want to risk waking her, so I had to forego pencil and paper and figure out a sequence of mental handholds that would get me to where I wanted to go in my second world). I was proud of my discovery but was chagrined when, a few years later, I found I was unable to reconstruct my thought process. From time to time I’ve returned to the problem and failed to reconstruct what it was that seemed so clear to me back then. Don’t get me wrong: the claim is true, and I can prove it. I just can’t see it the way I once did.

Maybe it’s time for me to set aside pride and ask one of my math-friends, one of my fellow Children of the Labyrinth, to help me find again the path that seemed so straight to me before.

The Beauty of the House is immeasurable; its Kindness infinite.

Thanks to Jeremy Cote, Sandi Gubin, Joe Malkevitch, and Evan Romer.

Next month: My Life with Aztec Diamonds.

7 thoughts on “Children of the Labyrinth

  1. jamespropp Post author

    I was able to reconstruct what I figured out fifteen years ago! The key is the formula E[(1+X)(1+Y)] = 1 + E[X] + E[Y] + E[XY], where X and Y are random variables and E[…] denotes expected value. I thought I’d mention this happy ending, though this particular key may not open any of the doors that any of you are trying to open. (Then again, the underlying principle — linearity of expectation — is a very potent spell, useful for fashioning many magic keys.)


  2. David Jacobi

    It is a common observation that studying science changes the way you see the world. Astronomers see stars as thermonuclear furnaces. Geologists see the Grand Canyon is a snapshot of eons of differential erosion. Are their common mathematical landscapes or objects who’s perception is radically changed by studying math. Any particular examples come to mind? David Jacobi


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