On Twitter I recently saw Misha Burnett say that The Empire Strikes Back was probably the movie which ruined sequels for him.

In isolation, Star Wars had a triumphant ending. The kid from the boondocks rose to the occasion and became a hero. He showed himself worthy in the eyes of the tough kid and the pretty girl and the wise old man, and he did something that would have made his father proud. Empire took all that away from us. As the first film in a trilogy (or film number whatever in a series) the story of Luke Skywalker goes from being a boy becoming a hero to a naive recruit becoming just another soldier. The final scene of Star Wars is a recognition of a hero who has saved the galaxy and won the admiration of the beautiful space princess. The final scene of A New Hope is just a bureaucrat entering a commendation into a personnel file. And that is why I don’t like series fiction. In a series, there are no happy endings because in a series nothing ever ends.

This is a very interesting perspective, and in one sense I think that he’s right. What’s really curious is that the reason that he is correct is also why anyone remembers Star Wars well enough to make this observation.

If you just consider Star Wars (later retitled to Star Wars Episode IV: A New Hope) on its own, it’s a fun popcorn flick and possibly the second summer blockbuster after Jaws. It’s the basic Campbellian “hero’s journey,” which is fun but not very interesting. It’s not very interesting because it’s only a very narrow slice of life.

What happens in The Empire Strikes Back and Return of the Jedi is that Luke has to deal with the fact that life isn’t over until you die, even if you accomplish something big. At the end of A New Hope, they defeated the Empire’s super-weapon, but didn’t defeat the Empire. And it’s true that there is always more work to be done. It’s in The Empire Strikes Back where we find out whether the seed fell on rich soil and will yield one hundred fold or whether it fell on rocky soil and sprung up quickly but then withered quickly when the sun came out because it had no root.

In A New Hope, there is a sense in which Luke got the girl. In Empire Strikes Back, we see the much more interesting action of Luke giving up the girl for a higher calling. Then in Return of the Jedi we see, after Luke’s rigorous monastic training, his self-sacrificing love saves his father’s soul. Luke went from a kid who wanted to grow up to an actual adult who had adult adventures.

Another way of looking at it is that look progressed from natural goods to supernatural goods. At the end of A New Hope, we see Luke bathed in glory. At the end of Return of the Jedi, we see Luke alone, burying his father who he alone knew the fate of, then joining the others but even there apart, having a religious vision which showed him the deeper reality that makes the celebration good, but only a small good.

If Star Wars had remained Star Wars instead of becoming A New Hope, it would have been forgotten as one among many stories of someone getting started in life. This is the adventure movie equivalent of the romantic comedy which ends with the couple getting married. (Or, if anyone told such stories, of a man becoming a priest ending with him being ordained.) The Star Wars trilogy is the far more difficult and thus far more unique story of describing adults. We see a working marriage in Han and Leia—fighting together through suffering and pain for the sake of something besides themselves—and we see a living priesthood in Luke.

Ultimately, in life, it’s cool to see the seed germinate, but all sorts of seeds germinate. It’s the seeds which turn into fruitful plants which stay with us.

(This, by the way, is why most sequels suck—instead of telling the different story of a seedling turning into a plant, they just reset everything and try to tell the story of a seed germinating again. That’s really when all of the progress of the original story is lost.)

Oh, and even considered in the trilogy, the end of A New Hope is a whole lot of people celebrating the two men and a wookie who just saved their lives. Princess Leia doesn’t become a bureaucrat just because she eventually stops clapping and gets back to the rebellion she was leading at the beginning of the movie and which clearly wasn’t over at the end of it.

I recently heard a song from the very late 1970s in which plastic was used as a synonym for “phony”. This made me remember how in my early youth, plastic was generally (though not exclusively) used as a cheap replacement for better quality materials. While at the time of this writing (2020) it is possible to find cheap things made of plastic, it is by no means the case that plastic is merely a cheap alternative. Often, these days, plastics are a superior alternative. PeX pipes are better and longer-lasting than copper pipes. Plastic pipes are strongly preferred to metal pipes for gas lines buried beneath the ground. I’ve got more than a few glass-reinforced nylon gardening tools and prefer them to metal ones. What is commonly called “carbon fiber” is commonly an epoxy plastic reinforced with carbon fibers, and it may be the pre-eminent high-tech material of our day.

It’s interesting to trace the factors which went into plastics becoming frequently superior materials, because it wasn’t just one thing. The introduction of new plastics, such as more advanced epoxies, clear polycarbonates (possibly better know as Lexan), and PeX had a significant impact. They enabled all sorts of new uses for plastics that hadn’t been possible before, and in some cases enabled new sorts of things. Bulletproof glass, for example, is frequently made as a laminate of glass and polycarbonate, the glass giving hardness and the plastic shatter-resistance. That’s very hard to do, in an optically clear way, without plastics.

Another advance in plastics was the development of economical composite plastics. The most famous composite is, of course, fiberglass, though not far behind it is carbon fiber. Another great one that’s been in common use for 10-20 years now is glass-reinforced nylon. Almost as strong as low-grade steel, it is far more rigid and doesn’t rust—it’s great for gardening tools. More generally, composite materials have made for all sorts of things both strong and light. Ladders, camera tripods, bicycles, shovels—anything where one wants weight and strength, it is usually the case that the cheap one uses metal and the good one uses composite plastics. (To be fair, there are some very advanced aluminum alloys, these days, though.)

Another improvement to plastics was simple experience with the making of things out of plastic. The making of things out of plastic is as much an art as a science. The exact temperatures used in injection-molding plastics has an enormous impact on the quality of the resulting part. In the 1970s and 1980s, widespread injection molding of plastics was near its infancy. Worse, since all it was fit for was making the cheapest stuff possible, there was not much money or incentive to make the stuff better. Eventually, however, people learned how to do it. How to design stuff made of injection-molded plastics was another area of improvement, with the right thicknesses, reinforcements, etc. being learned through experience as well.

To my mind the most interesting advancement in plastics, however, has been learning how to make them in ways that go beyond the chemical formula. A good example of this is how polyester went from being an awful but cheap replacement for silk in the 1970s that felt akin to wearing a garbage bag to a vastly superior fabric for athletic clothes. These days, if you’re going to do something that will make you sweat, you will be far more comfortable in a wicking fabric, which are mostly made of polyester. The trick is that when making the polyester strands, instead of making them thick, so that one is almost making the cloth out of fishing line, one makes the plastic strands incredibly thin. These are then spun together much like natural fibers, and produce a fabric which is light weight, breathes well, and tends to pull moisture away from the body and allow it to evaporate on the surface. On the flip side, making polyester in very thick sheets has created its use for things like unbreakable drinking glasses. Admittedly more prone to scratching than glass is, they never shatter when you drop them and they are much better insulators, helping to keep the drink at whatever temperature it started at.

Probably the best example, however, is Ultra High Molecular Weight Polyethylene, sometimes sold under the brands Dyneema or Spectra. As a super-quick background, if you don’t know it, plastics are polymers, which means that they are a chain of simpler molecules known as monomers. These monomers are frequently liquids. The way plastics are synthesized is from a monomer stock, with a chemical reaction catalyzed by a catalyst that combines the monomers into chains, which form solids. Polyethylene comes in varieties, based on how many monomers are in the (typical) polymer chain. Low density polyethylene has a few hundred, and they tend to be in branched chains that don’t stick together well. This is the sort of plastic one finds commonly in grocery store bags. High density polyethylene has (typically) 700-1800 monomers in a polymer molecule, and arranged in much straighter lines, which stick together better. This is the plastic one finds in things like soda bottles. Ultra high molecular weight polyethylene has anywhere from 100,000 to 250,000 monomers in a polymer molecule. It has greater tensile strength than steel (by weight), similar abrasion resistance, similar friction to Teflon, and is highly chemically resistant to corrosion from acids and alkalies. It’s truly amazing stuff. And what’s really interesting is that (one of the more common forms) is made with a metal spinneret, in a matter not entirely unlike the way a spider spins its silk from its spinnerets.

In the early and mid 1900s, chemistry got most of the glory when it came to advances in technology, bringing us wonderful new materials. In the later half of that century, it turned out that chemical formulae were only a small part of the story. How you put things together is at least as important as what you put together. It’s an interesting lesson, not in the least because metals were often seen as so superior to biological materials, but it turns out that our best materials are often made by imitating biological materials.

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: