There has been a really weird… phenomenon, I think mostly on Twitter, where some people referred to 2+2=4 as a basic, incontrovertible fact, and some other people have been invoking higher mathematics to say that sometimes 2+2=5. Since I’ve got a master’s degree in math, I thought I might as well explain what’s going on, and when 2+2=5.

To get an obvious point out of the way so as to avoid misunderstanding, in normal conversation, when people talk about 2+2=4, they are clearly referring to the natural numbers under one of the the ordinary definitions of addition (such as under the peano axioms, where addition is combined succession). Given that set and that operation, 2+2 equals 4, always, no exceptions.

Where higher level mathematics comes in is that a mathematician is free to consider any set whatsoever, and to define any operations on it that he wants. Most of these sets and operations won’t be interesting, but they will be valid. Let me give you an example of such a set with such an operation, so you can see what I mean.

Let us define the set F, such that F={2,5} (that is, F has two elements, one called “2” and the other called “5”). Let us define an operation on F×F→F (the cross product of F, mapping to F), called +. It is defined such that + of any two elements in F is the element “5” in F. Thus, 2+2=5. Under this definition, it is also the case that 2+5=5 and 5+5=5.

OK, so we now have a set F and an operation + such that in F under + as we’ve defined it, 2+2=5. Whoohoo. This is obviously a completely uninteresting set and operation. There’s nothing to say about it. There’s no reason one would ever want to actually define it, other than as a joke. (There is a certain similarity, here, to the classic set of outcomes to a coin flip, “heads I win, tails you lose”, if that is of any interest.)

But this is it. It’s not some great truth of higher mathematics, it’s a trivial side-effect of what is doable in higher mathematics which has generalized the more common mathematics.

I should note that there are very interesting sets and operations to define. For example, given a prime number m, the set of non-negative integers less than m, with addition defined to be addition modulo m, is quite interesting. (For those not familiar, modular arithmetic is basically doing arithmetic and if the result is larger than the modulus, you keep subtracting the modulus until it isn’t. Thus (2+6) % 7 = 1, since 2+6=8, but 8 is bigger than 7, so we subtract 7, and 8-7=1. You can also define modulus in terms of remainder after division.) These are quite interesting because the operation we’ve defined as + is closed, so every number has an additive inverse. It would take too long to explain why that’s interesting, but the short short version is that it’s the basis of many cryptographic algorithms, including some of the most widely used public key cryptographic algorithms like RSA which underly most encrypted web traffic.

Thus I hope it’s clear that it is not the case that higher mathematics is all uninteresting. It is only that the abstractions which are used in higher mathematics have boring side-effects. This is no different from the oddity of “Buffalo” meaning (1) an animal, (2) a city in NY and (3) to confuse someone allowing one to claim that “Buffalo buffalo buffalo buffalo buffalo” (meaning that buffalo from the city of Buffalo confuse buffalo from the city of Buffalo) is a perfectly valid English sentence. It is, but this is, at most, an amusing accident. It doesn’t mean that English is stupid, nor that people who study language are stupid. It’s just that complex things occasionally have odd quirks. What is true of the English language is also true of higher mathematics.

And none of this means that the people who say, “well, actually, sometimes 2+2=5…” aren’t intentionally missing the point.

In closing, let’s never forget the point in this XKCD comic:

Over at Amatopia, Alexander Hellene discusses nihilism, primarily in art. It’s a good post, worth reading. There’s one segment of it that I want to discuss, though, because I think that it somewhat misses the bigger picture:

But endless moping about the meaninglessness of everything isn’t a harmless amusement. It’s a dangerous idea that spreads despair and brutalizes people into looking into the sewer, metaphorical or otherwise, for meaning when they should be looking to something higher, whatever that may be. Instead of aspiring to lofty heights, those at the top would rather wallow in filth, and they want to drag you down with them.

There are two ways in which this misses the bigger picture.

The first is that, unless you believe in God, nihilism is true. That is, based upon one’s worldview, if there is no God, then nihilism is the correct conclusion. Atheists, as so much of the elites of American culture are, are either nihilists or lying to themselves.

This brings me to the second way in which it misses the bigger picture: if an atheist comes up with a meaning for his life, it will necessarily be an idol. You can see this in progressives/wokesters. They have decided that the meaning of their life is helping the downtrodden and oppressed. They would do far less harm, even to the downtrodden and oppressed, if they just wallowed in filth in the gutter. Everyone would be better off if atheists do not make up secular meaning in their life. That’s how people get to guillotines and death camps.

The biggest reason, though, that I think that it’s actually healthy to encourage atheists (who are the only people who can be nihilists) to take nihilism seriously is that it is here that they face their own beliefs directly and without disguise. It’s a commonplace observation that people rarely get out of self-destructive habits before they hit rock-bottom. Nihilism is atheists hitting rock bottom. The longer they go on pretending that everything is fine, the more entrenched in this mistake they will get.

In short, nihilism is a symptom, and it is a mistake to treat symptoms rather than the diseases that create them.

Just don’t let your kids watch the product of nihilistic artists until you’ve prepared them to see through it. But, you know, prepare them at an early age.

Over on Twitter, Benjamin Kit Sun Cheah wrote a very interesting thread on Wuxia (Chinese heroes) and the meaning of this genre. He kindly gave me permission to quote it in full here since that’s much easier to read than a Twitter thread if you’re not used to Twitter.

Among the hottest fiction trends today, and the genre I’m working on next. I’ve been looking into the genre for years, but everywhere I looked I found too many power fantasies, too few actual wuxia. It shows a lack of understanding of the genre. Wuxia should be the stuff of legends. Highly-skilled warriors in a milieu of danger and respect. Adventure in exotic realms. A world where you can earn your place with your sword. But beyond that, wuxia has one more element: Ethics.

It’s right there in the name. Wuxia is commonly translated as ‘martial hero’ into English. The meaning of ‘hero’ is well-known. ‘Martial’ has a neutral connotation. It means the ways of war. The meaning of wuxia seems obvious: a hero who uses martial arts. But this is not what wuxia means in Chinese. Chinese is a logographic language. Every ‘word’ is a written character that carries certain meanings. Every character in turn is made up of radicals. Radicals are smaller characters that convey pronunciation, and most importantly, MEANING. Keep this in mind.

Wuxia is a transliteration of 武侠. 武 carries the meanings of ‘martial, weapons, military’. 侠 means ‘chivalry, gallantry, hero’. But this is in English. In Chinese it carries a much, much deeper meaning. 武 is composed of two major radicals. 止: Stop 戈: A dagger-axe, an ancient Chinese weapon. (When used as a radical, the word loses a stroke.) Therefore, the true meaning of 武 is: to stop the dagger-axe. The English ‘martial’ has a neutral connotation. 武 has an innately noble purpose: to stop the dagger-axe, to defend and protect.

Morality is hard-coded into Chinese martial arts. The Chinese took it very, very seriously. Some Chinese martial arts masters worked as police officers, soldiers, and bodyguards. Others were civilians, but they assisted the police in arresting outlaws. Others ‘took back the art’ by defeating (and crippling or killing) bandits who had trained in martial arts.

侠 is an interesting case. This is the simplified version of an older word: 俠. 侠 has two radicals: 人: man 夹: squeeze, pinch, wedge, carry under your arm This implies that 侠 is ‘a man who wedges’ or ‘a man who carries’. 夾 from 俠 can be interpreted in two interesting ways. The ‘official’ interpretation I’ve seen is of two smaller men lifting up a larger man. My other, artistic interpretation is a large man wedging himself between two others. With this artistic interpretation, if we look at 俠 again, we see this: A man who places himself in between a tall man (on the left) and two smaller men (behind him). A big man protecting two smaller men with his body. What is the deeper meaning of 侠? A man who lifts up and supports other people. This is how the Chinese viewed chivalry and gallantry. Or, if you look at it artistically: A man who shields the innocent with his body.

Put these characters together and we get the meaning of the true meaning of 武侠: A man who stops the dagger-axe and supports others. Martial skill is thus used to protect the innocent, NOT to puff up your ego. To be a hero is to help others, NOT to merely be strong. Many stories tagged as ‘wuxia’ miss that. Without this element of ethics, of a hero willing to shield others from the dagger-axe with his own body, there is no wuxia. Today, there are ‘dark wuxia’ stories where the MC is a VILLAIN, which defeats the genre altogether!

With this in mind, let’s examine Xianxia. Xianxia is the transliteration of 仙侠. 侠 is known by now. But what about 仙? English says it means ‘immortal, fairy, sylph’. But that’s not quite what it means. 仙 has two radicals: 人: Man 山: Mountain A 仙, an immortal, is thus a man who goes up to the mountains. But why? Isolated in the mountains, free from the concerns of the mortal world, a man can tap into the abundant qi of nature. Through cultivation, a Daoist becomes an immortal. Through cultivation, a Buddhist gains enlightenment. They do not gain superpowers.

More precisely, these powers are NOT the point of cultivation. They serve as way markers. Confirmation that you are on the right path. And, if needed, skills TO HELP OTHERS. To pursue powers at the expense of your own development is to lose the Way and fall into delusion. Not only that, the archetypal xian is a hermit. He goes up the mountain AND STAYS THERE. Cultivation is a long, difficult, tedious process. He needs to be free of worldly distractions. Why would he climb down the mountain?

The answer, for this genre, is 侠. What is the deep meaning of xianxia? A man who goes up the mountain to cultivate himself and gains powers along the way, then climbs down the mountain to use his skills to help others. Modern-day xianxia show ‘heroes’ gaining power, beating up bad guys, attracting a girl, gaining more power, ad infinitum. This is a power fantasy, with optional harem elements. The purpose of power is not MOAR power. It is to be used to help others – or not at all. Without this moral element, a wuxia / xianxia story is not wuxia / xianxia. It is a mere power fantasy. But it is the dominant trope today. A mere aesthetic to dress up a hollow fantasy, no more. A shadow of the true meaning of the genre.

Only one thing left to do: Overturn the heavens and the earth. SAGA OF THE SWORDBREAKER, coming 2021 / 2022.

I have a small comment of my own to add, which is that you see the sort of perversion of a genre which Benjamin has described almost anywhere in which you see vivid world-building, no more than ten to twenty years later. It will inevitably come about when there are people who grew up with the world-building but who reject the heroism in it for the various reasons that people reject heroism. (Mostly it’s because they’re bad people and contemplating great virtue makes them feel bad about their vices, rather than encouraging them to increase their virtues, but that’s a topic onto itself.) These people, having spent so much time in such worlds in their imagination, long to tell their own stories in the same setting, though not at all to tell the same sorts of stories.

(This is a reason, by the way, that you will tend to see a golden age in which a type of new fiction has some particular excellence at the beginning, but then the genre becomes a swamp in which it is still possible to find diamonds. At first, people enter the genre to tell the sorts of stories this new fiction lets them tell especially well. Later, people who are used to the genre it want to tell all sorts of stories that have a superficial resemblance to the originals, and most of them are bad because they do not fit. The diamonds are those stories telling stories which do actually fit the genre.)

Somehow or other, my ten year old son discovered my blog post The Least Jedi, wherein I make fun of one of the worst movies of all time, point-by-point. He’s actually having me read it to him in place of a bedtime story and finds it very funny. (This is not actually the first time; he made me watch The Last Jedi with him, and now has me reading it to him again.)

This got me to wondering why he’s enjoying it so much. I think that part of it is the same reason I enjoy Mystery Science Theater 3000—I’d love to be involved in making a movie so much I’d be willing to help make a bad movie. As G.K. Chesterton once said, “anything worth doing is worth doing [even if one can only do it] badly”. But I think that there’s another aspect to this, too.

Sarcasm is not polite. Sarcasm about a work of art, however, is (except in particularly grumpy company) not impolite. It can be very mean to make fun of a person, but it is not mean to make fun of a movie. And the problem with politeness, especially from a child’s perspective, is that you never quite know what people who are being polite really think.

I think, then, that part of the pleasure of sarcasm, especially for younger people—who, after all, are the people who generally enjoy sarcasm the most—is that one can take it far more at face value than most of the things one comes across. It is a vacation from veiled meanings and subtle hints. In sarcasm, we find good and bad openly called by their names.

There is a lesson, there, for people who write sarcastic things. They have a special appeal to children for a reason; it behooves us to make sure to write them very well, since children will absorb errors when they see them far more than will adults.

Something I’ve noticed about Twitter is that I frequently come away from it feeling less emotionally balanced than when I went to it. The obvious thing to do, therefore, is to figure out why, so that I can figure out what to do better, or at least how to approach it or whether it’s possible to approach it usefully. The first thing I notice when I consider this is that Twitter—by which, of course, I mean the tweets of the people I follow on Twitter—contains a great deal of complaining.

This is a little bit odd because I’ve generally selected people to follow on the basis of having said something insightful or something funny. So the first question is: what did I do wrong in how I’ve selected people to follow?

The answer there seems to have several parts. One of which is just that it’s far easier to complain than to come up with anything insightful to say. When people run out of insight, they may simply turn to complaining because it’s better than nothing. Another explanation is that there simple are people who one needs to turn retweets off for, because they will retweet things that they will not write. That’s easily done, and not the majority of what tends to cause my day to be worse after going to Twitter. I think that another part of it is that there are people who have valuable things to say, but minimal self-control when they themselves become upset. I don’t think that there’s anything practical to do with these people other than to unfollow or mute them. A happy medium is actually to only read lists, and just leave these people off of the lists.

There are also people who have valuable things to say, but fundamentally misunderstand the medium in which they are saying them. Twitter is called “micro-blogging” for a reason, though people frequently think of it as merely conversing with friends. This dissonance can produce things that would make sense if heard only by the person to whom they were written but are highly liable to misunderstanding in public, where they actually are. Much of the above about people with little emotional self-control does apply, though on rare occasions it may be helpful to point out to these people that they are making public statements that really are meant for private audiences; since this is, fundamentally, a mistake, they may possibly be helped.

There is also a problem with Outrage Quoting. This is harder to know what to do with; one thing that helps is to block the idiots who are frequently quoted. And, of course, using lists, muting, and unfollowing are all options.

Ultimately, I suspect that the correct approach is just to narrow down yet more carefully the list of people whose tweets I see and to double-down on my rule about only reading twitter via Tweetdeck using lists.

Still, I think it’s also worthwhile to do some introspection on why I’ve broken down on that rule I imposed on myself; what am I looking for? I’ve got more than enough to do, I certainly don’t need Twitter in order to stave off boredom. I’ve got enough to read (and write) that I shouldn’t have time to be bored in the next twenty years (God willing). It could be looking for human contact, or hoping to find people to help me think through some topics that I’m thinking about. I could just be craving a certain sort of stimulation, since the current environment requires more patience than I’m used to practicing.

A wag I know one described the characteristic masculine and feminine addictions on the internet as:

Men play video games to pretend to be good at doing things. Women use social media to pretend to have friends.

I’m going to leave the second half of that alone, but the first half is interesting. Video games, at least for males, are great and dangerous for really the same reason: they have a much lower effort-to-reward ratio than real life does.

I should clarify that by “real life” I mean skills that still work when the electricity is out. For example, lifting heavy things, carving wood, playing a piano, flying a kite, boxing, riding a bicycle (fast), shooting a bow and arrow, building a miniature ship inside of a bottle, dancing, building a fire from gathered wood and starting it, etc. All of these skills, and many, many more, take a very long time to get good at, typically with a long time at the beginning which has little result besides besides frustration.

In video games, by contrast, one can typically learn the relevant skills to get some rewards within an hour, and often within a few minutes. It is true that they will sometimes have skills which are difficult to master, but even those tend to only require hundreds of hours to master, not tens of thousands, and they almost never involve enduring physical pain along the way.

All this is correct for video games for their intended purpose: relaxation. Video games, used well, are fun. They are a restorative to a weary soul who has been ground down by the trials and tribulations of doing real things, in situations and environments which were not designed to be enjoyable. The quick fun and easy rewards help one to remember the slow enjoyment and eventual rewards of good work in the real world.

The problem is something that really is more dangerous to young men—it is possible to become so used to the ease and comfort of video games that the difficulty of real life becomes insurmountable. Without the rewards of accomplishment coming on the schedule a young video game player has been trained to expect them, he may face crushing disappointment. Instead of being a restorative to a fallen creature in a fallen world, enabling him to face the world in which he lives, it may be an impediment which makes it harder for him to do real things.

As in all things, the trick is to use things in the right way and to avoid their pitfalls.

According to the tribal witch doctors we call weathermen, it’s going to hit 90 degrees today. Yesterday wasn’t too far from that, and where I live it tends to be at least fairly humid most of the time. Plus, I’ve just always dealt better with the cold than with the heat. But this put me in mind of something interesting—why people used to get up so early in the day.

There was a tree that had sprung up in the middle of a large bush and went unnoticed for long enough that its trunk was now about 4 inches around. This is close enough to the neighbor’s house that the tree was growing over the fence and getting in the way of her lawn crew. Both to be a good neighbor and for the sake of the lilac bush it was growing it, it was time for the tree to go.

I don’t own any kind of powered saw; when I remove small trees and thick branches, I do it with my bow saw. It’s not that hard, but it is work, and work creates heat. Generating a lot of heat isn’t much fun on a cold day, and it’s miserable on a hot day, so I decided to saw the weed tree down in the morning, before work, and before the heat of the day settled in. This got me to thinking about why it was that farmers—who for most of history was most of humanity—would get up at the crack of dawn. If you’re going to be doing a lot of manual labor, it’s far preferable to do it early in the morning, before the day has gotten hot. Once the day gets hot it takes a long time to cool off again, and often doesn’t cool much before the sun sets.

Whether or not people actually kept working throughout the day, there would have been basically no way for them to have continued to work at the same level of output as before it got hot. The human capacity for work is dependent on the environment. Of course where it gets really hot people have a tendency to eat a big meal then take a nap during the hottest part of the day. As the saying from tropical regions goes, only mad dogs and Englishmen go out in the midday sun. Even outside of hot regions, though, it makes sense to take advantage of the times when one can actually put in hard work, rather than just slowly suffer.

This, I think, helps to explain why waking up late is so associated with laziness. If a man who works outside waits to get up until it’s hot, he’s not going to be able to get much nearly as much done in a day. There is, also, the issue of light availability; one does not do much farming by lamplight. I don’t want to entirely discount it, but especially in summer, there is just a ton of daylight available. Outside of farming and other outdoor labors, the issue of heat, in the days before air conditioning, is far more pressing than the issue of light. Even inside of buildings, it gets quite hot on a hot day. It may be delayed by an hour or two from how hot it gets outside, but having put in work inside of an unairconditioned building in summertime, it doesn’t stay cool nearly as long as one would like. Granted, one can still write as quickly in the heat, but many of the other trades still involve moving and pushing and pulling; things that generate heat.

I’ve no way of knowing if this really was it, but it does strike me that someone who waits to get out of bed until one can no longer reasonably expect him to do work would seem very lazy, and indeed might well be very lazy. (Programmers, who always work in air conditioned environments and who move exceedingly little while they do so are notorious for preferring to work at night, rather than during the day. Having a wife and children I’ve mostly adapted to working during normal hours, but it would be very easy to slip back to a more natural way of working…)

[Update: Paul, in the comments, pointed out that I got the phrase about mad dogs and Englishmen. It comes from a 1931 Noel Coward song, Mad Dogs and Englishmen, and contains the refrain “Mad dogs and Englishmen go out in the midday sun”, as well as descriptions of how everyone else is too sensible to go out in the midday sun. You an see Mr. Coward performing the song on YouTube.]

I’m currently hosting this blog on wordpress.com, but since I rent my own server anyway, I’m thinking of doing the work to move over to hosting my blog on my own server. WordPress does make a plugin for integrating in the wordpress social features, like the wordpress.com reader. It’s about $40/year to use, so, I’m wondering whether anyone actually uses the wordpress.com reader to read this blog. If you do, could you leave a comment letting me know?

There is a story in my family about my grandfather, which comes from the 1950s or perhaps the 1960s, when he was not a grandfather but only the father of three girls. It’s not a long story; they had a television and would sometimes watch it. He had a habit of occasionally, and out of the blue, telling his children to put Tarzan on it.

They would then explain, once again, “but Daddy, you can’t just put Tarzan on. You have wait until they broadcast it.” The way the story is told, I suspect that he knew it perfectly well, and was only teasing his children. In my own experience young children never grow tired of telling their parents things that their parents already know, and a parent pretending to not know something makes for a game that’s a little bit like a baby bird learning to flap its wings.

The very curious thing about this story is how dated it was. When I was a boy, the story required no explanation, for television was still broadcast over radio waves at the time, on a schedule, and if you missed a broadcast you had no way of seeing it again. (Incidentally, when I explained that format of watching shows to my then-five-year-old son, the look of horror on his face was priceless.) This story made perfect sense to me without any explanation.

From what I understand there are still companies using the broadcast television model, though mainly over cables rather than over radio waves, though even that is, so I hear, still going on a little bit. However, the predominant way in which children watch video these days is all video-on-demand over the internet. Whether it’s Netflix or Amazon or Disney+ or YouTube or one of several others, the dominant way a child watches a cartoon is to think of what cartoon he wants to watch, go to the appropriate player, and hit play. Not everything is on streaming services, of course, which is why the DVD and Blu-Ray disks exist; they require a bit more exercise to put into the blu-ray player, but, barring one having a more important obligation, one can still watch whatever one is in the mood for, at approximately the moment one wants to watch it.

This is a story that I will no doubt pass on to my own children, but it is curious that I will have to include an explanation that what will sound perfectly natural to them was actually a joke at the time.