# Pandämonium: Fernweh, Wandertrieb Und Zugunruhe

Our experiences do not define us, yet we are nothing more than our past and our future. Compelled to make new mistakes and relive old ones.

To be content is to be unhappy. To idle is to die young. We desire change. Our instinct is to wander. I grow restless.

Time slips away as moments become shorter. Darkness deafens the fragile senses. The Silence is blinding.

Who knocks? Is it death? …Is it reality?

amn’t a linguist by training, but every so often I come across a peculiarity of the English language that makes me wish I were. Whether it’s the differences in zero-marking between British and American English or the fact that we can ask “Aren’t I?” but not say “I aren’t,” the language is riddled with curiosities that developed for one reason or another and, when I’m lucky, they carry funny names like the “expletive it.” I want to seize this opportunity to write down some of my thoughts on two phenomena I recently encountered for which there does not seem to be suitable references online. These notes are rough, so take it with a vaguely predicated amount of salt.

Now, I’m certainly not the first to say this, but not all of “the rules” of English make sense, nor do they always seem to help clear up meaning. Irregardless, as a former student of The Greek, The Latin, and The Arabic, I enjoy learning about these international laws governing our languages. And you should too, if for no other reason than to enforce arbitrary syntactic structures upon your peers’ utterances at the most opportune (inopportune for them, of course) times. Perhaps follow it up with a “Speak English, please” to add insult to insult. [Whenever I’m caught in this situation with my proverbial pants down, I plead ignonce.] In all seriousness though, I still find it to be a constant struggle to adhere to MLA, APA, CMS, and other TLAs and my writing is often riddled with confusing punctuation and even more perplexing quasi-verbiage.

Caveat lector: The following has some math(s), but I’ll try to keep it self-contained.

1) Perhaps no distinction annoys the Descriptivists more than the fewer vs. less divide of 1770. Purists treat this as a matter of life and death, as if it were an eleventh commandment decreed by God herself that for all instances in which objects may be counted one must use “fewer” and for all other instances one must use “less.” I very much doubt any person truly follows this to a T, and even the Merriam-Webster Dictionary of English Usage prefers the common usage of less in many instances. Notwithstanding, I believe I have found proof that this rule is not from God, but in fact man-made and impossible to satisfy. Consider the rational numbers and the real numbers, which consists of the rational numbers and the irrational numbers. There are infinitely many rational numbers and thus there are infinitely many real numbers, and certainly more real numbers than rational numbers. However, there are only countably many rational numbers while there are uncountably many real numbers. That is to say, we can count off the rational numbers in a systematic way (1/1, 1/2, 2/1, 3/1, 1/3, 1/4, 2/3, 3/2, …) whereas we cannot do so with the real numbers. Therein lies the problem. Are there fewer rational numbers than real numbers, or are there less rational numbers than real numbers? [A more symmetric phrasing is: which are there fewer/less of: rational numbers or real numbers?] In just one sentence we are talking about a single fundamental type of thing: numbers. Yet, numbers, when gathered in big enough groups, go from being countable to not. Thus it becomes ambiguous whether we ought to employ fewer or less. I do not know of other things that can be both countable and uncountable, but it seems almost ironic that “number” is the very word that leads to a contradiction of the fewer/less rule.

2) This second example has to do with adjectives, or modifiers, in a broad sense. There are lots of adjectives out there. Rumor has it, they’re in the top five most used parts of speech. As a refresher, here are some adjectives: blue, round, tall, fast, fake, honest, upcoming, fuzzy, melted, et cetera. How do adjectives work? You can learn more than you probably ever thought was possible here. There’s a surfeit of neat stuff, but let me break down the relevant bits. Functionally, what does an adjective do? It modifies a noun. How it does that depends on the adjective-noun combination.

Say you start with a noun, which naturally has some definition. The definition defines a set of properties that the noun satisfies. Then you modify the noun with an adjective. This modification usually has the impact of introducing further properties that the noun phrase (adjective + noun) satisfies. For example, suppose you have the noun “paper” which clearly has some definition. We know that papers can come in many colors though, so we modify it with an adjective to “blue paper.” We’ve now added the property of blueness to the set of properties, which causes a restriction. Simply put, the more properties there are that need to be satisfied, the fewer things there are that can satisfy all of them. Notice though that “blue paper” is both “blue” and “paper;” it satisfies two sets of properties, the first being the singleton set of blueness and the other being the set of properties of being paper. Phrased differently, “blue paper” is in the intersection of things that are blue and things that are paper. Linguists call this kind of adjective “intersective.” For the mathematically inclined: $\{\mbox{blue paper}\} = \{\mbox{blue stuff}\} \cap \{\mbox{paper}\}$.

Another kind of adjective is the subsective adjective, and as the name suggests it has to do with subsets. Take the noun “programmer.” Again it has a set of properties that define it. Now suppose we modify it with “clever” to get a “clever programmer.” We certainly still have that a “clever programmer” is a “programmer, i.e. $\{\mbox{clever programmer}\} \subset\{\mbox{programmer}\}$. However, just because someone is a “clever programmer” does not mean they are “clever”. It seems that rather than being another separate property that the “programmer” satisfies, the modification from “clever” affects a property. So instead of the property set expanding to include an additional property, “clever” alters a property within the set of “programmer” properties. The property adaptation in this case roughly is: “knows how to write computer code” becomes “knows how to write efficient computer code.”

Other types of adjectives may or may not exist depending on the school of thought you’re working with. A common example of this is the privative adjective with words like “fake.” A fake gun is not a gun and a gun is not a fake gun. Thus we have that the intersection of the modified and unmodified noun phrases is empty: $\{\mbox{fake gun}\} \cap \{\mbox{gun}\} =\O$. What we see is that property set for “gun” is not expanded by the adjective”fake,” but rather a crucial property (a gun discharges projectiles such as bullets) is negated (a fake gun cannot discharge projectiles). We can see something somewhat similar with temporally shifting modifiers like in the phrase “past president.”

In math(s) though, there seems to be some examples of adjectives which are distinctly different in behavior from the ones above. Rather than adding to, narrowing, or negating the properties, some adjectives widen. A clear example of this is in the noun phrase “general eigenvector.” One definition of an “eigenvector” of a matrix $A$ is a vector $x$ that satisfies the three properties that 0) $x\neq 0$, 1) $\exists \lambda \in \mathbf{C}, m\in\mathbf{N}, (A-\lambda I)^m x= 0$, and 2) $m=1$. A “generalized eigenvector” is only required to satisfy the first two properties, i.e. it is not required that $m=1$. So we have that $\{\mbox{generalized eigenvector}\} \supset\{\mbox{eigenvector}\}$. Other examples include skew fields, gaussian/eulerian/algebraic/etc. integers, and non-associative rings. By symmetry to the term subsective adjectives, I think (and a handful of other people on the internet agree) these adjectives should be called supersective. They have the ability to remove a property, and therefore loosen the noun phrase. Whether or not real examples exists outside of math(s), I am not yet sure. The closest I’ve been able to get to one is “dog food” vs “food” but please be careful.