## October 31, 2003

### Linearity

Posted by shonk at 02:38 AM in Geek Talk | TrackBackIn my analysis class today, the professor was lecturing on the Hann-Banach Theorem, which states that if you have a bounded linear functional defined on a subspace of a normed linear space, there is an extension of the functional to the whole space that preserves the norm. Don't worry, I'm not going to go into the theory or its applications, but I did want to comment on something the professor said in class, namely: "The idea of linearity is deeply ingrained in us". Now, he was specifically commenting on our natural inclination to try to deal with linear functions, since they're so much nicer to work with. For example, the first time we encounter something like *(x + y) ^{2}* in algebra, we want it to be equal to

*x*instead of the clumsier

^{2}+ y^{2}*x*. This reasoning, aside from conforming to our aesthetic sensibilities, is also founded on sound thinking, since mathematics has far more tools to deal with linear functions than other varieties.

^{2}+ 2xy + y^{2}However, I got to thinking about his statement and I think it may be more broadly applicable than he intended. The idea of linearity is, indeed, deeply ingrained in us. The most challenging books or movies are often those that reject the traditional linear forms of narrative; they are challenging in large measure precisely because they stray from that ingrained idea of linearity. The linearity that characterizes our perception of time makes clichés like "time flies when you're having fun" both memorable and slightly ridiculous. We like to think that time ticks inexorably by, each second lasting precisely as long as the last. This linearity applies to our perception of space as well: we see the earth as flat, we like our roads straight and our buildings upright, we figure distances as the crow flies.

In a way, we might see the 20^{th} century as the century of non-linearity: the Theory of Relativity debunked the notion that time and space are linear or flat, quantum mechanics and chaos theory question the very possibility of linear determinism, writers like Joyce, Nabokov, Amis and Márquez explored non-linear narratives, as did movies like "Un chien andalou" and "Memento", networks and hyperlinks changed the ways we learned, read and interacted. Certainly, non-linearity wasn't *unique* to the 20^{th} century - strong strains of it are present in Newton's (and Leibniz') calculus, impressionism, the work of Diderot and Sterne, etc. - but I don't think it's a stretch to say that it has been most prominent in the last century or so.

I admit that this isn't an important observation, but I guess I just find it fascinating that art, culture and science are all, seemingly in tandem, redefining themselves in non-linear terms. And I think that part of the reason the adaptation to these new terms is so difficult for so many is precisely because linearity *is* so deeply ingrained in us. Heady thoughts indeed for a Thursday afternoon analysis course.

**Bookkeeping Note:** I've added a new section to my links, "Literature Online". I've collected links to some of my favorite books and poems under that heading. Book titles are linked to full-text versions, novelist names to their biographies and poet names to a collection of their poetry. Enjoy.

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