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Note: When this essay was originally written, I relied a lot on my qualitative experience of mental activity, especially when talking about intuition. I think the broad sketches of this theory are generalizable to the vast majority of humans, but the specific aspects of how things feel are unlikely to be generalizable.
Frame hermeneutics is one of those notions that's obvious, but extremely useful to make clear. To summarize: the process of interpreting anything that can be interpreted (the "interpretable") involves breaking that interpretable down into interpretable subcomponents, constructing "frames" for each of these (recursively breaking them down if necessary), and then trying to construct a frame for the entire interpretable which permits and makes legible each of these subcomponent frames. A "frame" here is something like a conceptual perspective held in intuition.
Having developed this basic theory of interpretation, we can begin to articulate a general model of what (in this particular case mathematical) understanding consists of. This brings us to the notion of conceptual perspectives, or large frames which connect with some form of sensory intuition and are legitimized by a web of actual theorems/results.
Why is it possible for two different people to extract two different messages from the same book, both of which appear valid (or at least not obviously invalid)? Why is it possible for the same person to, at two different points in their life, interpret the same book in radically different ways? Here we'll speak of how it's possible for books to have a meaning at all.
Obviously, the meaning that a person ascribes to a book is in part determined by the actual content of the book. Obviously, it is not completely determined by this, as exemplified by the above paragraph; rather, there is a second aspect to the creation of meaning, in which the book provides a mold for the reader to actively construct the book's meaning. In fact, as meaning isn't a physical thing — e.g. you can't buy a used book with "a damaged spine, torn cover, missing meaning" — but entirely inside the reader's head, at least insofar as the reader is able to cognize it, this second aspect can be considered to encapsulate the first.
In other words: books cannot directly transmit mental images to us; we must necessarily construct, through our own effort, these images in response to the content of the book. The book is a mold insofar as its content stimulates certain patterns of mental images within its readers, just as you and I might visualize more or less the same property-of-a-thing upon seeing the word "green". Clear writing, then, is in the skillful and intentional formation of a mold, with the writer choosing their words so as to reliably direct their readers to roughly the same mental images, and good reading is in the skillful determination and filling of the shape of the mold in their mind. (Not to say that good reading involves agreeing).
With this in mind, we may ask: what is the process by which readers create meaning from, i.e. interpret, a book? The model I want to posit is the subject of this essay, frame hermeneutics. It seems more or less obvious, but is useful to articulate and keep in mind.
(Of course, this applies not just to books but to movies, paintings, games, and so on — anything from which one intends to extract a meaning. I'm just using books for the sake of concreteness and non-convoluted grammar.)
Interpretation is a fractal activity: our interpretation of a paragraph comes from the way we combine our interpretations of all of its sentences, each interpretation of which comes from the way we combine our interpretations of all its words. It has this bottom-up direction. But it also has a top-down direction: our interpretation of a word depends on our interpretation of the sentence into which it fits, which depends on the interpretation of the paragraph into which it fits, and so on.
Generally, individual words aren't too open to interpretation aside from homonyms and ambiguity regarding parts of speech (e.g., "using that tape [noun], tape [verb] his mouth shut") — but there are more interesting salient cases in which the interpretation of a word matters.
For instance, take the sentence
"You should get to know him, he's pretty good."
Is he moral? agreeable? cool? skilled? in what sense do you mean the word "good"? But if I expand that to the pair of sentences
"This part is a bit tricky to do by yourself — it helps to have someone around that has a lot of practice. You should get to know him, he's pretty good.",
it becomes clearer that we should interpret "good" as meaning something closer to "competent". However, if we say
"It helps to have role models that you can look up to. You should get to know him, he's pretty good.",
we interpret "good" as meaning something closer to "moral". This is an example of the top-down direction of interpretation.
We clearly employ both the top-down and bottom-up directions of interpretation whenever we read a book, though we're generally not conscious of doing so until we encounter a particularly disorienting sentence and have to look before and after it trying to form a context in which it make sense.
Here's one way of thinking about this: we read a sentence, interpreting it from the bottom-up, and then asking (generally not consciously) whether our interpretation is consistent with the text, throwing up a red flag if it's not.
This isn't that relevant, but is very interesting: this red flag seems to actually physically exist in the form of the P600 (Positively charged spike in brain activity peaking after 600 milliseconds), which seems to be elicited by the exact things you'd expect it to be elicited by if you hypothesized that it were caused by a change in bottom-up interpretations caused by nontrivial top-down interpretations.