Pierre Menard, pixel art, and entropy

In Jorge Luis Borges’ short story “Pierre Menard, Author of the Quixote,” we are introduced to the character of Pierre Menard, recently deceased French novelist and essayist, and are given a brief cataloguing of his works. Chief among them are

“the ninth and thirty-eighth chapters of Part I of Don Quixote and a fragment of Chapter XXII.”

Menard, we learn, had set himself the task of reproducing the Quixote, not from memory, but rather from something like sheer force of will, and had succeeded insofar as he was able to write (or rewrite) those chapters. Setting aside the fact that such a task is for all intents and purposes impossible, we are left with an interesting philosophical quandary regarding the status of Menard’s Quixote: namely, what relationship does it bear to Cervantes’ Quixote?

If you haven’t read the story, do; it’s delightful, and not very long. Borges implicitly (and quite cleverly) argues a version of the thesis that Menard’s Quixote and Cervantes’ Quixote are totally different works, no more than superficially similar, despite being textually identical. For one, a whole host of things are contingently true of Menard’s Quixote – the author is French, it was written in the 20th century – that are not true of Cervantes’ Quixote, with corresponding implications for the reader, and perhaps thus for the meaning of the text (and for its identity/​uniqueness). I think one can coherently argue against this thesis, but here I’ll take it to be true.

Indeed, for ease of argument I’ll take to be true a stronger version of the thesis: that any two people who independently produce the same artwork have in fact produced different artworks. Suppose, for example, that my friend Afonso (my discussions with whom inspired this essay, and to whom I credit many of the following ideas) and I agree to each write a brief poem, and by chance produce the same text. The mere fact that my native language is English and his is Portuguese would lend radically different contexts to the two poems; enough (so I claim) to make each text its own unique work, as opposed to two copies of the same work.

Suppose instead, however, that Afonso and I are pixel artists, and we agree to each produce a black-and-white pixel artwork consisting of a single pixel. We again produce the same output: perhaps a black pixel. This time, I’m much less inclined to say that we’ve produced different works. How can we reconcile this intuition with Borges’ thesis?

One important difference between the poem and pixel art cases might be that, in the latter, it feels like Afonso and I have merely chosen from some relatively small set of possible “solutions,” whereas in the former the solution space is much (much) bigger, and so it feels more like we’ve each “created” something. We can formalize this notion by referring to the entropy of the source of a work; a source producing 1-pixel black-and-white “artworks” has 1 bit of entropy, while a source producing 100-character English-language strings has about 800 bits. This difference in entropy explains the further intuition that in the poem case, as with the two Quixotes, the chance of two writers producing the same text is miniscule, while in the pixel art case a duplicate work is quite likely. The probability of randomly generating two identical 1-bit sequences is ½; for two identical 800-bit sequences, it’s 1/​2^800.

Of course, Afonso and I probably aren’t generating our sequences randomly when we write our poems (unless we’re especially avant-garde poets), so the actual entropy is a bit lower, but it is still high enough that the chances of us writing the same poem of any appreciable length are functionally zero. A first pass at a criterion by which two textually (or physically, or some other appropriate qualifier depending on the medium; I’ll lump all these notions together as “semiotic” equivalence) identical works can be considered as meta-textually (metaphysically, etc; “semantically” henceforth) different could be that they are each the result of a process with sufficient entropy. At the risk of pressing one more word into service as jargon, let’s call two works that are semiotically equivalent but semantically different “Quixotic pairs.” I’ll equivocate momentarily between the semiotic and semantic notions of sameness to allow for a nicely memetic statement of this criterion: two identical works can form a Quixotic pair only if it’s functionally impossible for them to be identical.

The criterion distinguishes smoothly between the cases of the Quioxtes and the pixel artworks, as well as correctly classifying some other hypotheticals (suppose Menard had simply copied Cervantes’ Quixote out of a manuscript; [faithfully] copying is a zero-entropy process, and accordingly the copy is clearly not a separate work). Still though, it’d be nice if we could connect this idea back to the original intuition that Menard’s Quixote and Cervantes’ Quixote are different works owing to the differences between their authors and the differing cultural contexts in which they were produced, or something like that. To do this, it might be helpful to think a bit about what makes something an (art)work in the first place. Without attempting to be especially precise, we might think that, in order to count as an artwork, the thing must instantiate or transmit some sort of cultural content, and/​or must evince some degree or combination of both technical skill and intentionality. Entropy as described seems at least roughly correlated with technical skill, insofar as it’s some sort of measure of complexity, and instantiating more complex works (maybe) requires more technical skill. Can we also link the idea of entropy to intentionality or transmission of cultural content?

A related concept to entropy called channel capacity serves as a measure of how much information can be transmitted per use of some communication channel. Channel capacity depends on, among other things, how much noise is introduced by the channel, but is upper-bounded by the entropy of the channel output. Thinking this way, with artists as channels for cultural information, it seems clear that, regardless of what specific aspects of culture or intention need to be transmitted by a work for it to qualify as an artwork, one bit of entropy is almost certainly not enough capacity to transmit them in a single go! A book chapter, definitely; a 100-character poem, maybe; a single-pixel “artwork” chosen from two possibilities, no. Returning finally to the Quixotes, we conclude that they are separate works because (or at least partially because; everything here has claimed necessity, not sufficiency) they are each the output of a source with enough entropy to transmit the different cultural elements underlying them and differing authorial intentions, and that the two pixel artworks are not separate works because they are not.

I’ll address two objections, and include a caveat. First, one might point to works like Untitled Blue Monochrome, or remark that a single-pixel black-and-white work wouldn’t look all that out of place alongside works like Comedian, as a reductio against my entropy criterion for artworks (or, conversely, to wield the criterion as evidence that these or other non-traditional works shouldn’t be considered as art). To address this I return to my original description of the pixel artworks as elements of some solution space. The source that samples pixel artworks from the space of possible single-pixel black-and-white pixel artworks is necessarily very low-entropy, since the size of the solution space is so small. In contrast, the size of the general solution space for “possible artworks” is enormous. An artist who is tasked simply with creating an artwork and replies with a solid blue canvas or single black pixel is likely a fantastically high-entropy source, regardless of the complexity of the output. Of course, there can be low-entropy processes sampling from large spaces (like if Afonso and I agree beforehand to always write the first stanza of “The Raven” when tasked with poem-writing), but the mere fact that the artwork itself consists of something unusually simple is not entropic grounds for concluding that it is not an artwork (it might still be that there is some minimum technical skill floor that works of this kind do not clear, but that’s a question for another essay).

Second, one might object with the following counterexample. Suppose Afonso and I each write two poems, according to the following procedures and with the following outcomes. The first poems are written independently and from the heart, and end up semiotically equivalent. The second poems are written by independently invoking a random poem generator, and also end up semiotically equivalent. An observer relying on the proposed entropy criterion would have no way to distinguish between these two cases, despite the fact that the first two are semantically distinct, and so a Quixotic pair, while the second two are (prima facie) semantically equivalent, and so not a Quixotic pair.

This objection is fair, as far it goes, but I remind the objector that I only claim a necessary criterion for Quixotic pairs, not a sufficient one. The task of determining whether or not two works actually form a Quixotic pair seems to require additional information about the intended sematic content of the works. My goal is not to correctly classify all potential Quixotic pairs with one fell swoop, but rather to hypothesize why identical one-bit pixel artworks don’t seem like Quixotic pairs despite their relevant properties.

Finally, I admit to neglecting a crucial element of the meaning-making process in my discussion thus far: priors. I’ve suggested that low-entropy sources don’t output complex cultural or intentional content, and that semiotically equivalent outputs from low-entropy sources don’t differ semantically in artistically relevant ways, but this is not necessarily so. Consider Paul Revere managing to transmit “The British are coming (by sea)!” using only a single-bit message, or the difference in content between my sommelier friend giving me a thumbs-up over my choice of wine vs. an identical review from my other friend to whom all wines are the same. In both cases, most of the semantic legwork is being done by extensive previous knowledge about the intended meaning or the sender, rather than by the message itself. While I think the lesson here is that the minimum entropy required to meet my criterion ought to be evaluated relative to the strength of priors of the audience, I’ll leave the details of this as an open question, and instead qualify that my preceding arguments should be taken to apply primarily in the communicative context of artworks created for public consumption by non-anonymous artists.