07-31-2020 | math, category theory, philosophy

In the summer of 2017 A friend sent me a job posting from The Orange Website and said that it looked like a thing I’d be interested in. It was about 9 p.m. on a Wednesday, and being that you can find me on just about any Wednesday at 9 p.m. sitting here, at the bar in my kitchen, typing on this computer, with a beer or two in my belly, I took it upon myself to whip up a CV and apply.

Thanks to those beers, my CV was snarky, even silly in some places. It landed in the lap of a person with exactly the right disposition for this levity, and in a matter of a few months I was writing a valedictions email to soon-to-be ex-coworkers and starting a completely new adventure.

At this job, I spent my days reading, writing, and researching ways to integrate software verification techniques, particularly dependent type theories, into the practice of writing programs in a language whose ontology was to run, not to verify. It was the closest I ever got to research-as-work—to the experience of a PhD recipient.

While I was there, I met an incredible group of people, some of whom have come to be amongst my closest friends. Outside of work, we’d quibble about literature or philosophy, and they’d humor me when I wanted to soliloquy about profunctors.

Like all good things, this had to come to an end. We were unexpectedly sent our separate ways, and I lost touch with this new research-as-work identity-appendage I’d grown. The next job left it to atrophy and I’ve all but forgotten what the beautiful experience of connecting neurons for money felt like. Until this week.

This week, all at once, a friend told me about he and his partner talking with one another about when I was finally going to get a PhD, which prompted me to reëxplore the work of Peli Grietzer, a mathematician and philosopher—literally mathematized philosophy and literary criticism—who’s responsible for the theme of this month’s newsletter. Simultaneously, I read a chapter of the home-schooling book The Call of the Wild and Free about cultivating a family culture, and it hit me. It leveled me like a falling tree. I’m lost. I’ve lost touch with my own vibe.

Peli Grietzer’s The Theory of Vibe equates the semioticSemiotics is the study of signs and symbols. We talk about semiotics here because what’s fed into the autoencoder signifies something, for instance a handwritten word that signifies an object like “tree”, and what comes out signifies that originally input signifier.

system derived from a type of AI called an autoencoder to the semiotics of a universe created in a work of literature:

Suppose that when a person grasps a style or vibe in a set of worldly phenomena, part of what she grasps can be compared to the formulae of an autoencoder trained on this collection.

An autoencoder is a part of a neural network that reduces the patterns in its training set to a discretized and necessarily less precise representation and then back again with the goal of there being zero errors on the “back again” side. Those from that training set that it can roundtrip without error are called its canon.

Grietzer wants to make the connection that this “canon” of an autoencoder—which is the same thing as talking about the autoencoder itself—is the same idea that one gets from an aesthetically consistent universe created in a work of literature. He calls this a vibe.

That is, a vibe is the set of things the reader can recognize, without error, as aesthetically consistent.

This, to me—both the autoencoder and the literary vibe—sound related to Deleuze and Guattari’s smooth and striated spaces dichotomy. D&G wrote about this dichotomy in A Thousand Plateaus, the second volume of their Capitalism and Schizophrenia series. To them, smooth spaces are heterogeneous and continuous, and also untainted by a State Apparatus; while a striated space is homogeneous and discretized, and that discretization is, often times, done by the state. As an example, consider the idea of property “ownership”. Land itself is continuous, but a State Apparatus parcels it out and identifies it. Hell, it even homogenizes it with zoning laws.

I’d like to apply this notion of smooth and striated to both autoencoders and vibes. For an autoencoder, mapping smoothness and striation onto its two ends is obvious: an autoencoder’s canon is the set of “smooth” objects it encodes into a “striated” representation and then decodes without loss. However, the “vibe” of a work of art—particularly a work of literature—is more subtle.

Earlier I mentioned that an autoencoder is, mathematically speaking, the same thing as its canon. Grietzer covers this point early in The Theory of Vibe:

A trained autoencoder and its canon are effectively mathematically equivalent: not only are they roughly logically equivalent, it is also fast and easy to compute one from the other.

So let’s look at the autoencoder as a function, one that takes representations of smooth spaces and in return outputs striated ones. In this light, what is the vehicle for turning an artist’s brain into a vibe?


Language is the function that translates the smooth space of unarticulated thought into an essence that an independent being—the reader—can interpret. That essence is meaning. I’d posit that Grietzer’s “vibe”, that canon of an artist-recognizable sort, is the lexicon that that artist employs to articulate their thoughts.

Mathematically, we’d think of this as, rather than a function’s codomain or range, its image—the specific subset of possible results of a function, as opposed to the class of objects that those results lie in. When we detect that vibe it’s because we’re familiar with, well, that artist’s canon.

The space (this is a math pun) we’ve reached is rich with study, this idea of roundtripping from higher to lower and back to higher “resolutions” or dimensionality has many names: section / retraction, Galois connection, but the most general notion is a categorical idea called adjunction. An autoencoder’s input would be right adjoint to its output. And if that’s the case, then by Grietzer’s own line of thinking, a vibe would be left adjoint to the author’s mind. We’re getting into deeply hand-wavy territory now, but it’s not entirely untrodden. Category theory has been used in linguistics for years now, most interestingly (to me) in distributional semantics. Distributional semantics’ raison lies in using the distribution of a word amongst others to determine its meaning or synonyms. Words that are often seen in similar positions are thought to have similar meaningsIt’s more subtle than this and it involves not just position but also predicates that can make distinctions in the face of polysemy.

. Maybe a distributional model over an artist’s canon could suss out that artist’s vibe? Maybe there’s meaning to be given to adjunction in the categorical model of distributional semantics?

Maybe it’s actually that a vibe, as relayed through the striated space of language, is right adjoint to language itself, capturing the author’s mind at a point and relayed to you, reader, in a form that you can internalize into your own smooth space, only to map it again to your own vibe when you relay it to a friend, and so on and so on, always and forever, composed eternally.

At least until you forget.