Hiding behind chaos and error in the double pendulum

If you want a visual intuition for just how unpredictable chaotic dynamics can be then the go-to toy model is the double pendulum. There are lots of great simulations (and some physical implementations) of the double pendulum online. Recently, /u/abraxasknister posted such a simulation on the /r/physics subreddit and quickly attracted a lot of attention.

In their simulation, /u/abraxasknister has a fixed center (block dot) that the first mass (red dot) is attached to (by an invisible rigid massless bar). The second mass (blue dot) is then attached to the first mass (also by an invisible rigid massless bar). They then release these two masses from rest at some initial height and watch what happens.

The resulting dynamics are at right.

It is certainly unpredictable and complicated. Chaotic? Most importantly, it is obviously wrong.

But because the double pendulum is a famous chaotic system, some people did not want to acknowledge that there is an obvious mistake. They wanted to hide behind chaos: they claimed that for a complex system, we cannot possibly have intuitions about how the system should behave.

In this post, I want to discuss the error of hiding behind chaos, and how the distinction between microdynamics and global properties lets us catch /u/abraxasknister’s mistake.

Eukaryotes without Mitochondria and Aristotle’s Ladder of Life

In 348/7 BC, fearing anti-Macedonian sentiment or disappointed with the control of Plato’s Academy passing to Speusippus, Aristotle left Athens for Asian Minor across the Aegean sea. Based on his five years^[1] studying of the natural history of Lesbos, he wrote the pioneering work of zoology: The History of Animals. In it, he set out to catalog the what of biology before searching for the answers of why. He initiated a tradition of naturalists that continues to this day.

Aristotle classified his observations of the natural world into a hierarchical ladder of life: humans on top, above the other blooded animals, bloodless animals, and plants. Although we’ve excised Aristotle’s insistence on static species, this ladder remains for many. They consider species as more complex than their ancestors, and between the species a presence of a hierarchy of complexity with humans — as always — on top. A common example of this is the rationality fetish that views Bayesian learning as a fixed point of evolution, or ranks species based on intelligence or levels-of-consciousness. This is then coupled with an insistence on progress, and gives them the what to be explained: the arc of evolution is long, but it bends towards complexity.

In the early months of TheEGG, Julian Xue turned to explaining the why behind the evolution of complexity with ideas like irreversible evolution as the steps up the ladder of life.^[2] One of Julian’s strongest examples of such an irreversible step up has been the transition from prokaryotes to eukaryotes through the acquisition of membrane-bound organelles like mitochondria. But as an honest and dedicated scholar, Julian is always on the lookout for falsifications of his theories. This morning — with an optimistic “there goes my theory” — he shared the new Kamkowska et al. (2016) paper showing a surprising what to add to our natural history: a eukaryote without mitochondria. An apparent example of a eukaryote stepping down a rung in complexity by losing its membrane-bound ATP powerhouse.

Computer science on prediction and the edge of chaos

With the development of statistical mechanics, physicists became the first agent-based modellers. Since the scientists of the 19th century didn’t have super-computers, they couldn’t succumb to the curse of computing and had to come up with analytic treatments of their “agent-based models”. These analytic treatments were often not rigorous, and only a heuristic correspondence was established between the dynamics of macro-variables and the underlying microdynamical implementation. Right before lunch on the second day of the Natural Algorithms and the Sciences workshop, Joel Lebowitz sketched how — for some models — mathematical physicists still continue their quest to rigorously show that macrodynamics fatefully reproduce the aggregate behavior of the microstates. In this way, they continue to ask the question: “when can we trust our analytic theory and when do we have to simulate the agents?”

Evolution of complexity

My cats are jittery. The end of the world must be near.

In this post I will outline the questions I wish to answer, the existing approaches to these questions and the intuitions I wish to formalize. Thus I’m going to break all the promises I made in the last post — but really, this should have been the last post, somehow I got sidetracked…

There is an intuition that life has gotten more complex over geological time. More hierarchies have been built, more specialization of parts, possibly even more intelligent. This intuition is not only true for life, but for societies. Governments, bureaucracy, culture, social divisions — all seem to become more complex with time. The modern division of labor is the pinnacle of this. Why?

Orthodox evolutionary theory does not have an answer to this. And well it does not, because the answer is both nasty — and, I will argue, deathly wrong. Orthodox evolutionary theory proposes that natural selection is the major driver of evolutionary change, that organisms change their forms and types because of differences in fitness. Ergo, things must have become more complex, specialized, etc. — because they are more fit. Humans, as such recent players in evolutionary history, are very complex, very specialized, and very smart — hence we are the evolutionary climax, the last of the rungs of the evolutionary ladder to perfection…

This answer is the disowned natural bastard of orthodox evolutionary theory. It is never given out in modern polite scientific discourse, and most biologists would argue militantly that it is not true. Like all disowned bastards of royalty, however, its head constantly pokes up with alarming frequency, each time with new insurgent allies looking for a coup d’etat: eugenics, racism, social darwinism, all the bad movies using evolution as deus -ex-machina (X-men!) etc. Hear hear, it seems to say, Evolution is making us better, and we should hasten her along.

I’m fully convinced that this bastard child is incredibly wrong and we ought to have its head on a pike. The trouble is, parts of its parents might have to go also. The parts that lead to this bastard child, at any rate — I will argue that the long trends of macroevolution have little to do with natural selection.

For simplicity, let’s focus on complexity alone for the moment. Current (polite, modern, scientific) discussion about the evolution of complexity hinge around if it is increasing at all — it turns out “complexity” is an awfully slippery thing to measure. McShea did a great deal of work (1991, 1996, 2005) here, although him and colleagues aren’t the only group (e.g. Heylighen 1999). In any case, there’s a vast literature on this. I would carefully submit, however, that complexity has in fact increased, since we don’t find any pre-cellular living forms (the earliest living form cannot have been a fully formed bacteria!) and even our oldest Archeabacteria are well removed from the earliest living form (say, a self-replicating RNA). Nor did anything as complex as mammals show up right at the beginning of life. Although this work on the definition and measurement of complexity is incredibly valuable, we don’t need a thermometer to tell us that boiling water is hotter than ice.

The question is therefore why. To use the answer of orthodox evolutionary theory, that natural selection drove the extinction of simple organisms and made organisms more and more complex, is intensely unsavory. It’s more than just the political and cultural distastefulness of the answer and the capacity for people to abuse this fact, I — and many others — do think it’s actually wrong. But then there must be another force outside of natural selection that can drive evolutionary trends on the geological scale. What is this other force? I’ll summarize the existing arguments — and my opinions (why else write a blog?) — below:

Gould (Gould 1997) has famously argued that complexity, on average, has nothing to do with natural selection. The increase in complexity over time, he says, is simply the result of a random walk. Sometimes complexity is good for the organism and it grows more complex. Sometimes complexity is less good and organisms grow less complex. However, for biochemical reasons, the first life forms had to be very simple. On the other hand, complexity cannot go below zero. Thus, the evolution of complexity is that of a random walk with an absorbing state of zero. The average complexity then naturally increases, but complexity itself, per se, doesn’t do anything for fitness at all — what is good for fitness is entirely environmentally dependent.

Gould thus posits no positive force for the increase in complexity. McShea, who concluded that complexity was increasing after all, considers drift to be a positive, but weak, force for the increase of evolution. With Robert Brandon, they coauthored a book arguing that this is the case. A positive review is here. The idea is that mutation is a natural driver for diversity, which is synonomous with complexity. After all, one does not mutate into the same thing that one was. Thus, in the complete absence of selection, evolution progresses into greater complexity. McShea and Brandon called this the Zero Force Evolutionary Law (hmmm… I pick up a hint of Asimov here).

I think Gould’s idea is very clever, but it contradicts empirical evidence. There are traits that seem to make its carrier fitter over a wide swath of environments; the eusociality of ants must have contributed to their dominance in the world’s ecology. Might complexity be such a trait? Probably not, considering the dominance of bacteria… but we cannot reject that complexity has an effect on fitness overall, as Gould does. Besides, according to Gould, if all of complexity is driven by environmental conditions, then all lineages should show theoretically unlimited movement in complexity. Thus, according to this theory, whenever environments favor the loss of mitochondria in eukaryotes, eukaryotes should lose them and good riddance. Unfortunately, for all strains of eukaryotes, losing mitochondria is death — regardless of the environment, so there is no strain of prokaryotes with eukaryotes as their ancestor. This is the intuition I hope to tighten later. Similarly, of the many billions of cases of cancer that has occurred throughout evolutionary history, there is no species of single-celled organism that had a amphibian, reptile, or mammalian ancestor (I’m not sure about sponges). Once the organism dies, the cancer dies (unless it’s kept alive in a lab). None of the cancer cells could revert to unicellularism, although it’s undoubtedly advantageous for them to do so. Although multicellular to unicellular evolution has certainly occurred, they don’t seem to have ever occurred in species where the viability of the organism depends on an intense and obligate integration evolved over billions of years. Thus, unlike Gould’s claim, there seem to be some plateaus of complexity that, once stepped onto, cannot be descended from.

McShea’s idea, on the other hand, I’m skeptical about. It reminds me of a flavor of mutationism and orthogenesis that has been soundly routed in the course of the history of evolutionary thought, with good reason. Natural selection is an awfully powerful force, strong enough to beat the second law of thermodynamics every time. Most mutations increase diversity, yes, but most mutations also make us closer to a ball of gas, and yet we aren’t balls of gas. The authors seem to believe that it is a gentle breeze of a force blowing in the background, such that if the selection for or against complexity averaged out to nearly zero over time, then this force is sufficient to provide a long term trend. With no mathematical model behind their reasoning, I cannot formed an informed opinion of whether this might be true — but I highly doubt it. What I think is much more likely to happen is that the evolution of complexity would be precisely as dictated by natural selection, whether it is a random walk or not, and only increased slightly at all points in time by the gentle breeze. Consider the following 2 minute drawing in GIMP:

Different prediction for the evolution of complexity

It looks terrible, I know — and the axes are unlabeled. Okay okay, x axis is time and y axis is some measure of complexity. Let’s say that natural selection is the black line, so sometimes complexity is selected up and sometimes down, but there’s no overall trend (well, there should be, since it’s a random walk with an absorbing condition — but bear with me here). Red is what I think McShea and Brandon is proposing, that there’s a gentle background force moving complexity up. But I think that the gray line is what would happen — the evolution of complexity, according to McShea and Brandon’s force, would exactly reflect natural selection, with no trend. The force would nudge complexity to be a bit higher than what it would otherwise be, but that’s it.

Wow this post has gotten long already — you can’t say very much in a post! Next post — and I won’t break any promises this time — will deal with my own thoughts on increasing complexity and its links to the holey landscapes model.