If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. - John D. Barrow
This book brings up the butterfly effect, so I’m taking it as an excuse to talk a little bit about mathematics of chaos.
Before high school I managed to accelerate ahead of my class in mathematics. In high school, I got the highest score possible on the calculus AB, calculus BC, and Physics C exams. Then in college, I ended up falling in love with mathematics on account of a great teacher as well as a surrounding of peers appreciative of the beauty of mathematics. It’s difficult to express exactly why I find beauty in mathematics. Viewing mathematics as beautiful isn’t a new idea but it also isn’t exactly common. Trying to explain why mathematics is beautiful is almost like trying to explain why music or paintings are beautiful or why memes are funny.
And yet, sometimes mathematical themes and ideas will seep into the culture. The idea of chaos, sometimes described as the butterfly effect is one such idea. I’m not going to go into the mathematics or academic study of chaos. And the only review of the artistic reflection of it I will be looking at is the invocation of the butterfly effect in The Stranger.
The reason I’m not going to review chaos theory is because it would take far too much time for me to make an approachable introduction to it that would be worth sharing. Plus, there are several explorations of chaos already publicly available to introduce the concepts better than I could. Most of them are terrible, some of them are good, and I watched (or rewatched in one case) three to remind myself of some of the world of math explainers, starting with the most approachable and moving toward the most challenging.
The first is terrible. Basically, this video by Derek Muller of Veritasium will get you no better understanding of what chaos theory is all about than if you had just read the Wikipedia page. In fact, it might give you some misconceptions that you’ll have to unlearn later if you actually end up studying dynamical systems or if you try to use some of the analogies about chaos when evaluating science in news or policy. Maybe you’ll feel a bit smarter after having watched it, like you’ve learned some science about the universe, but you won’t be challenged, and it might make you more confident when you shouldn’t be. From the video:
In pop-culture, the butterfly effect has come to mean that even tiny, seemingly insignificant choices you make can have huge consequences later on in your life. And I think the reason people are so fascinated by the butterfly effect is because it gets at a fundamental question which is How well can we predict the future? Now the goal of this video is to answer that question by examining the science behind the butterfly effect.
Throughout the video, Derek keeps repeating that he is talking about the science of the butterfly effect - but it’s not exactly science in the way I understand that word. The butterfly effect is not a physical science that can be studied through empirical research, it’s a formal study of pure rules and abstractions. The very idea of chaotic systems that are extremely sensitive to initial conditions is that they can’t really be empirically verified1. The center of the video is mostly fine, but again not really much is there that you wouldn’t come away with after skimming through the Wikipedia article.
He then ends with some musings on chaotic systems being everywhere. He says the future is hard to predict especially when you go further into the future or trying to identify causes going into the past. What? What are you even talking about? This is the sort of thing that is fine if you’re writing a fictional novel or speaking by analogy but your specific examples don’t even back up these claims, they are almost completely unrelated. It’s the sort of video that is nice to get a gentle introduction to the topic but I don’t feel challenged at all to explore questions. It leaves me with the impression that I know the gist and basic ideas about chaos theory and can confidently apply them by analogy. There is a sort of online person who will invoke chaos theory to claim that climate science is bunk or we can’t impose any policies on the real world because it is far too chaotic.
Whatever. The second example of an introduction to chaos is fantastic2, by Grant Sanderson of 3Blue1Brown. This is a higher-level video that someone just starting their college mathematics journey who has a little experience with calculus should be able to mostly follow. At least, they should be able to follow if they start with the previous video of his or have been introduced to Newton’s Method of finding roots of polynomials.
One of my main goals today is to show you how this iconic shape, the poster child of math pops up in a more general way than the initial definition might suggest. Now this field is also intimately tied to what we talked about in the last video, with Newtons Fractal. And another goal of ours, towards the end of this video, will be to help tie up some of the loose ends that we had there.
You’ll notice that this video doesn’t really start with chaos itself. In fact, chaotic behavior is only found as a consequence of looking at where, in a parametric space, simple holomorphic functions shift from being bounded to unbounded. 3Blue1Brown is systematic and doesn’t spare details. You’ll notice if you watch the video, he also several times skips over some detail, trusting that the viewer can pause and verify for themselves some fact or conclusion. This channel is a great showcase in where the beauty of mathematics lies. Grant’s style is to follow his nose through to the interesting results, but also to leave crumbs of questions that can lead to more exploration and divergence into separate areas. Near the end of the video, we get a cool result.
If you draw a small disk around a point on the Julia set, it tends to expand over time, as the points from within that circle kind of go off and do their own things. In other words, points of [around] the Julia set tend to behave chaotically. Their nearby neighbors, even very nearby, will fall into qualitatively different behaviors. But it’s not merely that this disk expands…[eventually hits every single point on the complex plane].
You might notice something about this takeaway that is far different than the Veritasium takeaway of “GUESS WHAT GUYS CHAOS IS EVERYWHERE.” Here, we get the result that for this very specific holomorphic function, the liminal spaces of Julia sets behave in a chaotic way even for very slight changes in initial conditions on a measure of space that covers literally zero area. Is this useful for understanding the real world around you? Probably no! You and I probably aren’t going to encounter a Mandelbrot set out on the street in our everyday lives. But this video is challenging and opens the viewer up to exploration. It forces clear, specific thinking about the complex plane. Again, I also think Grant does a good job of showcasing the beauty of mathematics.
Finally, I watched Chaos and Noise: A Look at Stochastic Difference Equations by a former professor of mine that has under a thousand views. This video is a step up from Grant’s on the topic in terms of difficulty, is less graphically focused, and more algebraic in approach. Someone graduating with a degree in mathematics should be able to, with some effort, work their way through the details of this video and be able to understand most of it. That is to say, this video is a stretch for me at this point, so I can’t really give too much insight on it.
What Dr. Ami Radunskaya adds to the topic of chaos is the idea of stochasticity. I didn’t really mention in this blog, but one of the things about chaos is that it has basically nothing to do with randomness. Everything is deterministic in the processes being talked about so that if you knew precisely the initial conditions you would be able to know precisely how the process would end up, whether that involves settling on a fixed point, bouncing around a bunch of different values, or diverging to infinity.
I think a lot of people believe, implicitly or not, that all uncertainty is epistemic uncertainty. That the whole world could be explained and even the future could be predicted if we had every piece of information along with how that information impacted everything else in precise detail. Even wikipedia's definition of epistemic uncertainty's opposite, aleatoric uncertainty, suggests some causal mechanism for unmeasurable uncertainty. This belief is by definition unfalsifiable. There is an old saying of3 uncertain origin that "prediction is hard, especially the future" and yet no matter how difficult, the belief is pervasive that it is possible - that even people’s choices could somehow be known ahead of time. Yet stochastic models take uncertainty almost as given4, mapping out outcomes as distributional.
Does it matter which regime is true of the metaphysical world we live in? Not really. No one does or can know all matter of the universe in excruciating detail one way or another5. So let’s talk about the book from Animorphs that introduces a creature who might know about all matter of the universe in excruciating detail6.
The book begins with Cassie and Rachel7 at the circus. Rachel is a child of divorce. She mostly lives with her mom and sisters while her father takes her places every-other-week. This is also another animal heist for the opening vignette of the book. Rachel turns into an elephant to threaten the circus performer who uses cattle prods to control the elephants. She then tosses him onto a tent after threatening him so that he never uses cattle prods again. Animal solidarity baby!
This book involves a trip back into the Yeerk pool. Marco, Ax, and Tobias have been tracking assistant principal Chapman and found an entrance to the Yeerk pool in through The Gap and out through the movie theater. Rachel suggests heading to spy the Yeerk pool out to try and find where the Kadrona rays into the Yeerk pool are being beamed from. Kadrona rays mimic their own sun and Yeerks swimming in the rays are how they feed.
For Rachel’s story, the personal plot of this book, her divorced parents Dan and Naomi sit the family down for dinner. They discuss that her dad has a new job, in another state a thousand miles away. Rachel is upset. She storms off from the dinner. Her dad runs after her to offer for Rachel to come live with him. She would be able to work on gymnastics with a professional gymnastics coach and fly down frequently to see her mom and sisters. This is the choice he offers.
“Dad, are you asking me to go with you when you move?”
“Yes. I know it would be hard on you and your mom and your sisters, but we could make it work. I mean, this job pays a lot of money. You could fly back here any time you wanted. Every week if you wanted.
Was he serious? It sounded ridiculous. Was he actually serious? I sat down on the edge of my bed. My thoughts were everywhere all at once. Leave? Leave my mom and my sisters?
This was just because my dad felt guilty. He felt bad about leaving. This was about pity. He felt sorry for me or something.
“And I know it would mean changing schools,” he said, “but gee, Rachel, I think it could be okay, you know? I mean, for one thing, they have serious mountains there. We could do some rock climbing together on weekends. Go hiking. And it’s a huge sports town. I need someone to go with me to games. It would be like in the old days.” Then he winked. “And hey, it’s a much bigger city, so think of all the shopping.”
This is the first major decision in the book for Rachel. She literally flies out of her mom’s house to contemplate it. She terrifies a sleeping Tobias8, acquires a new bear morph, and considers how she will make her decision. The next major decision comes quickly on its heels as they go down into the Yeerk pool as cockroaches and are all getting swallowed by a Taxxon. Which is to say, all of them are about to die in roach morph, when all of the sudden the universe stops. They meet The Stranger9, what amounts to a godlike creature called The Ellimist.
He shows them the beauty of the earth and humanity. He tells them that the Yeerks will win and offers them a choice. The choice is to preserve the human race by being instantly transported away with family to live the rest of their life in peace or to be placed back where they were.
“We have an offer for you,” The Ellimist said. “You see, we can save a small sample of the human race. We have a planet where we would relocate you. You…some members of your family. A few others, chosen to get a good genetic sampling. As well as a few non-human Earth species that are of special interest to us.”
I was surprised to hear Cassie actually laugh. “He’s some kind of environmentalist.” she said. “That’s what he is. We’re the spotted owls. We’re the rhinos. We’re the whales. We’re the endangered species, and he’s the environmentalist trying to save us”
Initially, the group discusses and settles on no, they won’t join the Ellimist. As soon as the decision is made they are instantly back in cockroach morph.
-swer”
Instantly, we were back in our roach bodies.
IF YOU LIVE, I WILL ASK ONCE MORE.
IF YOU LIVE.
They do get out alive. The Ellimist shows them the future. Rachel confesses to the team the decision her dad is asking her. Eventually this all culminates in the team deciding to accept the Ellimist’s offer. When they do, nothing happens.
One of the great things about these books is how they weave together normal teenage life decisions with these more plot focused decisions concerning all life on earth. These two decisions are related to each other. Both will remove Rachel from the life she currently is living if she takes them. Both are an enticing escape from problems with opportunity to reclaim a part of her life that is gone. In particular, she loves her Dad and sees an opportunity to reclaim time and a relationship with him. Rachel, and the rest of them, are scared and the opportunity to leave the fight on earth and live once again in peace with their families is strong. And yet, when they’ve made the decision to end the war, nothing happens.
Then at school the next day.
Ms. Paloma, talked about what led up to the Second World War.
…
“Because we were so devoted to peace, we may have actually made the war worse,” Ms. Paloma droned on. “We’ll never know for sure, of course. You can’t really second guess history.”
You can if you’re an Ellimist, I thought. If you’re an Ellimist, you can look ahead and see it all.“Why not?”
It was Cassie’s voice…
Ms. Paloma sat on the edge of her desk.
“Because events are intertwined in ways we cannot always see, Cassie. Sometimes small things can make a huge differences. You know, they say that a single butterfly, beating its wings in China, may affect the way the wind blows here in our country. A single butterfly beating its wings may make a tiny change that becomes a bigger change that becomes a tornado. The world isn’t like math. It isn’t just one plus one equals two. It’s more complicated than that.”
Ok, so we get the impression here that Ellimist is speaking to Rachel directly through this teacher. There is more to the Ellimist’s game than meets the eye, and Rachel realizes that The Ellimist may not want them to escape the war at all. He is subtly guiding them to be able to win this battle in the war, and at the end of it they almost all die10 but they do manage to achieve their objective of eliminating the Kadrona.
One thing more that I haven’t mentioned about this series, it was initially conceived as possibly only about a dozen books. That means the last book would mark the halfway point where Jake confronts the antagonist animating his participation in the war - an emotional climax of sorts. So, this book ends with a turning point in the Animorph war against the Yeerks. The introduction of the Ellimist character signals how important the battle in this book is even if most of the previous victories have felt like they might not be moving the story forward - a practical climax at this point in the story.
But you may also notice, we aren’t halfway through. Not even close. There is a universe where this war is a one two punch with the ending of this one causing a crippling blow and one more major advantage could turn the tide completely. This could be viewed as a coming-of-age story where these children confront the great evil of the Yeerks and come out the other side victorious and mature. But that is not the story. In fact, without spoiling too much, the rough halfway point is a trilogy of books that are a pretty horrifying. I’m looking forward to them, but I’m not sure our heroes can hold their heads up as having acted as heroic.
But the end of this book is victorious. Bloody. But victorious, and hopeful. I don’t know if there is much metaphorically resonant that can be gleaned from this. I guess I’ll just have to end this particular blog post in chaos.
Next time I’m looking forward to reviewing the first book narrated by fan favorite Ax.
Oh, wait? Oh what’s that? We’re not doing Animorph #8 next time? We’re doing them all, baby! Get ready for the first Megamorph book!
Or can they?
I urge caution when clicking on this link. Math videos can get pretty graphic.
Ironically enough
Actually, even this is somewhat debatable. I think they do! But for most statistician’s professional purposes this is a philosophical distinction that isn’t much thought about. And again, unfalsifiable.
Unless of course that one is God.
GOD?!?
Narrator for this one
Poor dude can’t catch a break
Hey, that’s the name of the book!
It wouldn’t be an Animorphs book if they didn’t all almost die at some point.