An Algorithmic Lucidity

It feels immoral to even think of using techniques to motivate oneself; one should instead just use one's free will to choose the correct action. How utterly degrading it would be, how insulting to the very notion of human dignity, to stoop to the level of contemplating one's own psychology using mere cause-and-effect reasoning, as if one were some sort of animal, or a machine!

But this moralizing is itself immoral, because it doesn't work. If I'm not smart enough to do the right thing for the right reasons, then I might at least aspire to do the right thing for the wrong reasons for the right reasons.

So we have this metric space, which is this set of points along with a way of defining "distances" between them that behaves in a basically noncrazy way (points that are zero distance away from "each other" are really just the same point, the distance from one to the other is the same as the distance from the other to the one, and something about triangles).

Let's say (please, if you don't mind) that a sequence of points \((x_n)\) in our space is fundamental (or maybe Cauchy) iff (sic) for all positive ε, there's a point far enough along in the sequence so that beyond that point, the distance from one point to the next is less than ε. Let's also agree (if that's okay with you) to say that our metric space is sequentially precompact iff every sequence has a fundamental subsequence. If, furthermore, the precompact space is complete (all fundamental sequences actually converge to a point in the space, rather than leading up to an ætherial gap or missing edge), then we say it's compact. It turns out that compactness is an important property to pay attention to because it implies lots of cool stuff: like, compactness is preserved by homeomorphisms (continuously invertible continuous maps), and continuous functions with compact domains are bounded, and probably all sorts of other things that I don't know (yet). I'm saying sequentially precompact because I'm given to understand that while the convergent subsequences criterion for compactness is equivalent to this other definition (viz., "every open cover has a finite subcover") for metric spaces, the two ideas aren't the same for more general topological spaces. Just don't ask me what in the world we're going to do with a nonmetrizable space, 'cause I don't know (yet).

But anyway, as long as we're naming ideas, why not say that our metric space is totally bounded iff for every ε, there exists a finite number of open (that is, not including the boundary) balls that cover the whole space? We can call the centers of such a group of balls an ε-net. Our friend Shilov quotes his friend Liusternik as saying, "Suppose a lamp illuminating a ball of radius ε is placed at every point of a set B which is an ε-net for a set M. Then the whole set M will be illuminated." At the risk of having names for things that possibly don't actually deserve names, I'm going call each point in an ε-net a lamp. Actually Shilov, and thus likely Liusternik, is talking about closed balls of light around the lamps, not the open ones that I'm talking about. In a lot of circumstances, this could probably make all the difference in the world, but for the duration of this post, I don't think you should worry about it.

But this fear of having too many names for things is really a very serious one, because it turns out that sequential precompactness and total boundedness are the same thing: not only can you not have one without the other, but you can't even have the other without the one! Seriously, like, who even does that?!

But the reasoning is inescapable. You can't have one without the other because if every sequence has a fundamental subsequence, then finite ε-nets are a thing, which is to say (by the contraposition doctrine and De Morgan's Iron Law of Negation) that if every ε-net is infinite, then sequences that don't have fundamental subsequences are a thing. To see this, think about an infinite ε-net where no lamp lies within the lighted area of any other lamp. A sequence consisting of such lamps can't have a fundamental subsequence because the distance between successive points in that sequence is bounded below by ε.

And you can't have the other without the one because if finite ε-nets are a thing, then every sequence has a fundamental subsequence. To see this, consider a sequence. For \(k \in \mathbb{N}^+\) and for \(\varepsilon := 1/k\), we can cover any subset of our space with a finite number of ε-balls. But then by the Infinitary Corollary of the Iron Law Pertaining to the Storage of Pigeons, there must then be an ε-ball that contains infinitely many points of our sequence. Let's pick one of those points and call it \(a_k\). Then if we set \(\varepsilon := 1/(k+1)\), our ball can itself be covered by a finite number of ε-balls, one of which again contains infinitely many points of our sequence, of which we can pick one and call it \(a_{k+1}\). That triggers an induction, giving us a subsequence \((a_n)\). But then for every \(N \in \mathbb{N}^+\), if \(n\) and \(m\) are not smaller than \(N\), then \(a_n\) and \(a_m\) live in a \(1/N\)-ball, so that the distance between them is bounded above by \(2/N\), which can be made arbitrarily small by choosing a large enough \(N\), which means that the subsequence \((a_n)\) is fundamental. But this is "quod erat demonstrandum" (a Latin phrase that roughly translates as "what I've been trying to tell you this entire time").

Bibliography

Theodore W. Gamelin and Robert Everist Greene, Introduction to Topology, 2nd ed'n., §I.5.

Georgi E. Shilov, Elementary Real and Complex Analysis, revised English ed'n., §3.93.

She's a dilettante; you're a dabbler; I'm executing a breadth-first search.

Zeno knew, but did not know enough; a minute is divided
Into fragments, and each fragment sees, for points it o'er presided:
A small change, of which I take the distance
Along each fragment's lost existence:
The root of the sum of the squares
Of the length and the width and the height
Of the change in the range as the fragment is spanned
As the fragment is stricken from sight!

By philosophy I understand the discipline of discovering truths about reality by means of thinking very carefully. Contrast to science, where we try to come up with theories that predict our observations. Philosophers of number have observed that the first ten trillion nontrivial zeros of the Riemann zeta function are on the critical line, but people don't speak of the Riemann hypothesis as being almost certainly true, not necessarily because they anticipate a counterexample lurking somewhere above ½ + 10²⁶i (although "large" counterexamples are not unheard-of in the philosophy of numbers), but rather because while empirical examination is certainly helpful, it's not really what we do. Mere empiricism is usually sufficient for knowing (with high probability) what is true, but as philosophers, we want to explain why, and moreover, why it could not have been otherwise.

When we try this on topics like numbers or shapes, it works really, really well: our philosophers quickly reach ironclad consensuses about matters far removed from human intuition. When we try it on topics like justice or existence ... it doesn't work so well. I think it's sad.

I've been taking a summer course at a university which I won't name, because whenever I do, I'm always tempted to replace one of the words with an obscenity that starts with the same letter, which is probably a bad habit. The topic is contemporary sexuality, which seemed like a fine choice for knocking out one of my remaining so-called "general education" requirements, and maybe even learning something relevant to my interests.

The class ends on Thursday the ninth, and I had intended to make a lot of progress today (the sixth) filling out the workbook (worth half a letter grade) due then. I didn't get very far. The task shouldn't be difficult; my goal is only to reduce the probability of my receiving a C in the class by means of circling the appropriate letters for the multiple-choice prompts (for which the answers are conveniently provided) and scribbling responses to the short-answer questions, glancing at the reading as necessary. There was once a time when I would have regarded this behavior as sinful: of course what you're supposed to do is carefully do the corresponding reading by the assigned date before thoughtfully filling out each workbook section, only using the multiple-choice answers to check your work. But if I've abandoned my moral scruples sometime in the past five years, then I also throw far fewer crying fits, and I don't think these changes are unrelated.

But when you don't respect the work, even doing a lazy job takes a certain amount of self-command. One of the workbook questions asks, "How do you define 'virginity' and what behaviors do you believe cause one to 'lose' his or her virginity?" I wrote, "It's pointless to argue about the definitions of words; once you know what behavior someone has engaged in, then calling it 'virginity' or 'non-virginity' doesn't give you more information". Snarky passive-aggression? Maybe, but when you ask a retarded question, what do you expect?

My patience broke when I got to the article arguing that we shouldn't use baseball metaphors to talk about sex (because those are sexist and oppositional) but should instead use pizza metaphors. Except it doesn't say metaphors, it says conceptual models.

In the evening, I received an email from the University. "Be prepared for Graduate School" says the subject line, although I don't understand the motivation for capitalizing Graduate School but not prepared. Morally, I expect such an email to say something like, "Make sure you've chosen a valuable or exciting topic on which to advance the frontiers of human knowledge, as is the function and sacred duty of scholars!" But of course it's just telling me that "[t]test preparation workshops for the GRE, GMAT, LSAT, CBEST, and RICA are starting soon."