Compactness

This post was written by Elisabeth Loveland, Humanities Center Student Fellow

Humanities scholars are more or less challenged to cope, on an existential level, with an infinite volume of factoids and interpretations-on-said-factoids, all of which cannot be neatly jammed into your term paper’s bibliography, let alone your personal knowledge. You don’t have infinite years, you don’t have infinite neurons, and the Library of Alexandria burned down anyway. But awareness of his own pitiable circumstance is no consolation to the listless bookworm inching around the library in search of hidden zingers … after all, somewhere could be that juicy factoid that turns all scholarship on its head …

Alas, alas, alas, alas, it’s not just research that’s creeped on by too-muchness—last summer, considerations of infinity made me momentarily shelve writing poetry, something which I had not done even after I realized making poetry can be as stupid and vexing a process as making soufflé. I couldn’t put a sock in the mouth of my inner George Costanza: neuroticisms would bust out like hives whenever I considered the infinite combinations/lineations/spellings/what-have-yous of language—or in other words, my pen tip was overweight. In the world’s heaviest dictionary exist all possible subsets of language, subsets which, though mostly untapped, express in every way every thing. Why write when out in some abstract nether is a Plato-esque and perfect “whole?” To make a poem is to “select.” But to select is to chop most of the universe. Even successful poems start to vaguely feel like strange amputations of a “total context.”

I did resume writing poetry, but only after finding a heuristic metaphor for poetry’s infinity in my advanced logic class. Nearly halfway through the course we developed a proof for “compactness,” a property of some logics including the staple first-order logic. In a “compact” logic, we know that if we are presented with an infinite set of sentences that contain a contradiction, we do NOT require a likewise infinite set in our proof to detect the paradox, i.e., if some set gamma is inconsistent, then a finite subset of gamma is inconsistent, i.e., all proofs in a compact logic must be finite! This, I’ll aver, is such an intuition irritant it may be universally befuddling. Still, the metaphor for “wholeness” it provides is a help. Certainly an infinite set is whole/complete, but finite partitions of that infinity can be in a sense whole/complete as well, much like all the constellations in our celestial sphere are docketed as a set meaningful to us, even if they are only a pinch in the enormity of stellar bodies. So I am satisfied that there is a point where a finite subset—whether it contains tokens of research or poetic filaments—is sufficient, or enough.

This is not an exposition on infinity so much as it is a call for a quelling of nerves. Infinity is forever not-quite-at-our-tips, so why fuss? Especially considering a piece of work in the humanities may be considered “whole” even in its status as a piece of the “whole.” I think that, say, a set of facts in conjunction with an interpretation on those facts can be such that their scholar (no doubt a gumptious egghead) can construct a complete or whole or satisfying picture, even if there is some unknown fact or interpretation out there that renders the scholarship “wrong.” Or that, say, even if I bring a limited mastery of language to the craft, I can still construct a poem that is a complete or whole or satisfying catalogue of my experience without the total context of “the infinite poem.” Those listless bookworms inching around the library could act just a tad perkier.  

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