Ghosts Exist, According to Math

Alec Sarché
The Startup
Published in
10 min readSep 26, 2019


Fig. 0: Proof of ghosts

There is a lot of very stupid literature in the world about ghosts. There is, of course, plenty of educated, informed (still guesswork-heavy, but rabbis and priests can’t prove God exists and we trust them) work on seeing, interacting with, and learning from ghosts, but what we’ve failed to see so far is that ghosts are an interdisciplinary prize for scholars to win. We can maybe catch glimpses of them with spectrometers or communicate with them through the odd Ouija or suspiciously extinguished candle, but the fact is, ghosts are mathematical creatures, and we have, for millennia, failed them by treating their study as a right-brained effort.

Fig. 1: A scholar, refusing to see what’s right in front of him (a ghost)

Here’s the problem, and my sweeping claim, wrapped into one convenient package to haunt you as your great-aunt may haunt her plantation home or your grandpa haunts his old rocking chair in the living room: we live in the third dimension, and ghosts live in the fourth.

Now, we must first agree that human beings are in fact creatures of three dimensions. We are born in, walk through, and spend our lives learning from a three-dimensional world. Up, down, left, right, forward, and back: three familiar axes in which we can drive cars and cook and listen to podcasts. There’s a whole lot of literature about the third dimension; I would argue more than 99% of human writings concern or are set in it. The third dimension is to humanity what the city of New York is to Sex and the City: the fifth character, who lives and breathes and determines the direction of our lives, while we get used to its sights and sounds and smells.

This is axiomatic. Inarguable. If you told me you were a two-dimensional being, I would assume you had just gone on a Flat Stanley bender and were busy writing a fanfic. If you told me you were a four-dimensional one, however, I would reluctantly agree, and I would ask you if you were a ghost. You would likely say no. This is my loss. We would probably not speak after.

Let’s talk about that special dimension above ours for a minute, though. It deserves as much attention as the third, though it’s harder to see why. In a very, very dumb book called Flatland (a treatise on topology and geometry that somehow manages to spend more time focusing on how dumb women are than on the fourth dimension), the main character, a Square (capitalization sic), is unceremoniously plucked from his comfortable two-dimensional country, Flatland, and forced by a wise Sphere to see his world as it is: a sheet of paper with a bunch of shapes on it. He then attempts to preach the Gospel of the Third Dimension and is taken to jail, where he spends the rest of his days.

Did I mention this book is dumb? Also, it was written by a guy named Edwin Abbott Abbott? Come on.

However, Mr. Abbott Abbott and his woman-hating Square make some good points about dimensional geometry that I will attempt to explain and expand on without myself being imprisoned for life. Let’s see how it goes.

Picture a sheet of paper, with a bunch of shapes on it, all moving around in their flat two-dimensional way, bumping into each other and Feeling (capitalization again sic — this book is kind of gross, honestly) each other to determine how many angles each has and therefore how much respect each deserves (Circles are noble, Isosceles Triangles are peons, and Line Segments are women).

Now picture that you, in all your Three-Dimensional wisdom, are able to put your hand through this sheet of paper. As you do so, the shapes can’t see your hand, but rather a scary set of four cross-sections of your fingers, followed then by a fifth as your thumb crosses through the paper, and then the five cross-sections join into one larger one as your wrist, then your arm pass through the sheet. You can’t go much further, though, so you pull your arm back — and as you do, the line representing your bicep, then your elbow, then your forearm, then your wrist gets smaller and smaller, then splits again into five, then four, then one, as the very tip of your middle finger escapes the second dimension and rejoins the third.

Fig. 2: You, following the parable and learning as you go (fun!)

In all regards what just happened is physically impossible. You are a flesh and blood being (I assume), incapable of passing like vapor through a sheet of paper. The shapes therein are not sentient, and could care less what size cross-section your arm presents to them. However, this is an essay about ghosts. Anything can happen in this world, and I posit that those Squares and Triangles and poor, witless Lines did in fact just have a paranormal experience they’ll tell stories of around two-dimensional campfires for years to come.

However, we’ve only done the easy stuff so far. That shit was just a parable. Let’s get to work now.

Imagine, if you will, a being one dimension above us. Whether it’s a ghost or not is irrelevant at this juncture — we’ll do all the fun stuff later. Right now, let’s call it an OSHCOSH, or “Omniscient Sentient HyperCube (Oh Shit, How cool is that).” As a cube is bounded on all sides by squares, each of which is composed of four right-angled vertices, OSHCOSH is bounded on all sides by cubes, each composed of eight right-angled vertices.

I can’t really picture it either, don’t worry.

This cube, now, in its infinite wisdom and benevolence, chooses to push one of its arm-analogous limbs through the third dimension as we just did through the second. As it does so, we see first a line, then a square, then a cube — as it pushes through our dimension, it increases in size, just as we did in the second, then it thinks better of its venture and withdraws, just as we did, first growing smaller, then changing to a square, then just a single line in space, and then it disappears from our sight, leaving us to wonder why, of all things to hallucinate, our confused and desperate brain seems to have chosen a sentient cube.

Fig. 3: You, agog at your fleeting glimpse of a world beyond your own

Let’s step back here for a minute. What did we just see?

The reason a stupid book like Flatland exists is because the easiest way to digest this kind of information is through analogy. Whether you turn it into a misogynistic, classist manifesto about the importance of aristocracy is up to you (looking at you, Edwin). I’m into ghosts, so that’s not going to happen. Let’s focus on the analogy and keep it harmlessly paranormal.

Earlier, you put your hand through a sheet of paper. You effectively were cross-sectioned through it, infinitesimally thin layer by infinitesimally thin layer, like a living MRI scan of yourself. The stunt OSHCOSH just pulled is the same: he/she/they (let’s assume OSHCOSH is as unbounded by gender as they are by space) put their arm-cube-thing through our dimension, which produces a similar effect to your adventure through Flatland. The salient difference is, of course, that OSHCOSH’s cross-section appears to us in three dimensions, the same way yours did in two.

Again, I can’t really imagine this very well either. You’re fine. Don’t worry about it too much.

We’ve done a lot of work here. Let’s back up and start establishing some other theses about the nature of humanity to eventually link up to this fourth-dimensional exposé.

The whole theory here hinges on metaphysics. “Meta-,” from the Greek, meaning beyond, and “-physics,” meaning, of course, the shit you recognize because it makes sense, like gravity, or you not being able to touch things that are far away because your arm isn’t long enough. Does the concept of metaphysics sound at all familiar? Like a certain HyperCube appearing in your living room, taking a look around, and then vanishing into thin air?

There are many parts of being human that seem to go beyond physics, but let’s focus on one: the soul. Be it a biological phenomenon, the product of synapses firing electronic impulses at impeccably timed rates in order to produce what we know as thoughts and feelings, or be it something a little holier, something given to us by a higher power, we can all agree there’s something a little mysterious going on inside our heads. There’s just more to us than our flesh and blood. And even though those electrical impulses stop when we die, I believe there is something of us that remains. The brain may have stopped in this dimension, but in the fourth, I believe the soul goes on.

Yes, I’m getting a little woo-woo. Soul-talk is not easy to pull off in an aggressively titled essay about ghosts. But it had to get here eventually. Why do you think I spent the whole first half of this manifesto talking about how dumb another one is? I’m making myself look good so you buy into it. Let me repeat: Flatland, by Edwin Abbott Abbott, sucks. Ghosts Exist, According to Math is good. Remember that. Also, my middle name is not the same as my last. Huge advantage.

A shortcut taught in grade-school level math, if I recall correctly, is to remember that the “fourth dimension is time.” Einstein’s enduring legacy of long-tongued mathematical genius forces us to talk this way. Space, the first three dimensions, and time, the fourth, combine forces to form spacetime. Who cares. This is elementary shit. The A-bomb is descended directly from thought along this line. It is not a noble school.

Fig. 4: cross section of Albert Einstein’s head. Yes, he had a brain. But more importantly, he had a soul (represented here by a candle)

The fourth dimension is not time. It is the movement OF everything in existence in the third dimension. Back, forth, north, south, up, down — directions don’t change in the fourth dimension. They are simply exponentiated. Your body in the fourth dimension is just the summation of all the directions your body has ever moved since conception in the third dimension. Consider the cross-sections OSHCOSH expressed their arm in when they appeared to you before. If we were to add up all those three-dimensional cross-sections and line them up, one after another, we would have recreated OSHCOSH’s arm; 4-D printed it, in a way.

Now consider your own body. You are, in the same way OSHCOSH was, a cross-section of yourself through the 4th dimension. If we were to add up all the three-dimensional cross-sections of you and line them up, one after another, we would end up with very, very long, skin-toned four-dimensional worm, whose tail is located inside your mother’s uterus at your date of conception and whose nose is located in the hospital bed you died in at a ripe old age of 205 (modern medicine really picks up in about ten years — OSHCOSH told me not to spoil the surprise but I can’t help myself). You are a long, long, four-dimensional human-size accordion, composed of fold after fold of snapshots of yourself each nanosecond you are alive. And this is what is happening to your body while you are alive. It is accordioning out, becoming longer by the year, getting a little grayer on top and eventually taking up knitting and mah jongg.

When you die, your soul separates from your body. This is established in many widely-renowned peer-reviewed texts. My bibliography includes no prize winners that I can think of (by the way, isn’t it weird that the Bible has never won, like, a Hugo prize? or a Pulitzer or anything? seems like an oversight).

Even if you’re not into the soul thing, I think it’s almost inarguable that when you die, something happens to you. Something. Period. Your body stops moving around, you stop checking emails, and despite urban myth, your hair and nails also stop growing. The thing in your head that made you talk and cry and make grocery lists is no longer there. It’s gone. At least, as far as we, the peons of the third dimension, know.

This is when you enter the fourth. This is it.

My whole argument rests on this idea: that that thing in you, the magical thinking thing, the one that feels and loves and animates you, is an extra-dimensional thing constrained to three by its flesh. The fact that you remember things, that you have the ability to look into the future and speculate on life to come, means there are already vestiges of your fourth-dimensional capabilities doing their best to surface from the murky depths of your body.

When you die, this thing is released, allowed to become the ghost it’s been trying to be for eighty years.

Fig. 5: Your liberated ghost body, trailed by a lifetime of four-dimensional movements. Also, you are smiling.

What happens after that? Well. We all just have to endure our boring cubic existence, until one special day when we transcend it and become an infinitely long human worm ghost. It’s just math, folks.

I repeat: this is just math. No bones about it (ghosts do not have bones). Suck a big fat four-dimensional one, Edwin Abbott Abbott.



Alec Sarché
The Startup

My favorite song is America by Simon and Garfunkel.