The Arc of Complexity

“Everyone knows” that evolution has become more complex over the past 3.8 billion years. One species has spawned 30-100 million species. Organelles, multicellularity, tissues, and social systems have all appeared in life forms over this span. And everyone knows that technology has increased that complexity further. It is obvious that the mechanical complexity of the technium has increased in our lifetimes, if not in the last hour. The only problem with this conventional wisdom: we have no idea what complexity is, or how to define it.

What’s more complex, a cucumber or a Boeing 747? The answer is unknown. We have no way to measure the difference in order and organization between the two and don’t have good working definition of complexity to even frame the question. Seth Lloyd, a quantum physicist at MIT, has counted 42 different mathematical definitions of complexity. Most of them are not universal (they only work in small domains not across broad fields like life and technology), and most are theoretical (you can’t actually use them to measure anything in real life.) They are more like thought experiments.

Yet we have an intuitive sense that “complexity” exists, and is increasing. The organization of a dog is much more complicated that a bacteria, but is it ten times more complicated, or a million times? Most attempts to distinguish complexity aim at the degree of order in the system. A crystal is highly ordered to the point that a large hunk of diamond could be described mathematically with a small set of data points: you take carbon and repeat it X times in configuration Y over three dimensions. A bigger piece of diamond (or any crystal) has the same repeating “order” just extended further in space. Its structure is very predictable, and therefore simple, or low in complexity. What could be simpler than a homogenous bit of stuff?

On the other hand, a heterogeneous, mixed-up piece of granite, or a piece of plant tissue, would offer less predictable order (you could not determine what element the adjacent atom would be), and therefore more complexity. The least ordered, least predictable things we know of are random numbers. They contain no expected pattern, and therefore by this logic (less order = more complexity) they would be the most complex things we know of. In other words a messy, randomly ordered, chaotic house would be more complex than a tidy, well-ordered house. For that matter, a chopped up piece of hamburger would be more “complex” that the same hunk of meat in the intact living cow. This does not ring true for our intuitive sense of complexity. Surely complexity must reflect a certain type, a special kind of order/disorder?

Crystals just repeat the same pattern over and over again. So an extremely highly ordered sequence, like the repeating pattern in a crystal, can be reduced to a small description. A trillion digits of this sequence, 0101010101010…, can be perfectly compressed, without any loss of information, into one short sentence with three commands: print zero; then one, repeat a trillion times. On the other hand a highly disordered sequence like a random number cannot be reduced. The smallest description of a random number is the random number itself; there is no compression without loss, no way to unpack a particular randomness from a smaller package than itself.

But the problem with defining randomness as the peak of complexity is that randomness doesn’t take you anywhere. The “pattern” has no depth. It takes no time to “run” it because nothing happens while it runs. A highly ordered sequence, such as 0101010101010 doesn’t go far either, but it goes further than randomness. You at least get a regular beat. A meaningful measure of complexity then would reckon the depth of pattern in the system. Not just its order, but its order in time. You could measure not only how small the system could be compressed (more compression = less complexity), but how long the compression would take to unpack (longer = more complexity). So while all the complicated variations, and unpredictable arrangements of atoms that make up a blue whale can be compressed into a very tiny sliver of DNA code (high compression = low complexity), it takes a lot of time and effort to “run” out this code (high complexity). A whale therefore is said to have great “logical depth.” The higher complexity ranking of a random number is shallow compared to the deeper logical complexity of a complicated structure in between crystalline order and messy chaos.

“Logical depth” is a good measure for strings of code, but most structures we care about, such as living organisms or technological systems, are embodied in materials. For instance, both an acorn and an immense 100-year old oak tree contain the same DNA. The code held by tree and its seed can both be compressed to the same minimal string of symbols (since they have exactly the same DNA), therefore both structures have the same logical depth of complexity. But we sense the tree – all those unique crenulated leaves and crooked branches — to be more complex than the acorn. So researchers added the concept of “thermodynamic depth” in quantifying complexity.  This metric considers the number of quantum bits that are flipped, or used, in constructing the fully embedded string of code. It measures the total quantum energy and entropy spent in making the code physical, either in a small acorn or, with more energy and entropy, in a majestic spreading tree. So a tree with more thermodynamic depth has more complexity than its acorn.

But when it comes to quantifying the complexity of a cucumber versus a jet these theoretical measures don’t help a lot. Most artifacts and organisms carry large hunks of useless, insignificant, random-like parts that raise the formal complexity quotient, but don’t really add complexity in the way we intuitively sense it. The DNA string of the cucumber (and all organisms) appears to be overrun with non-coding “junk DNA” while most of the atoms of a 747 – in its aluminum – are arranged purely at random, or at best in mini-crystals. Real objects are a grand mixture of chaos and order, and tend to hover in a sweet spot between the two.

It is precisely that goldilocks state between predictable repeating crystalline order and messy chaotic randomness that we feel captures real complexity. But this  “neither-order-nor-non-order” state is so elusive to measurement that Seth Lloyd once quipped that “things are complex exactly when they defy quantification.”  However Lloyd, together with Murray Gell-Mann, another quantum physicist, devised the 42nd definition of complexity in the latest attempt to quantify what we sense. Since randomness is so distracting, producing “shallow” complexity, they decided to simply ignore it.  Their measurement, called effective complexity, formally separates the random component of a structure’s minimal code and then measures the amount of regularities that remain. In effect it measures the logical depth after randomness is subtracted from the whole. This metric is able to identify those rare systems (out of all possible systems) that cannot be compressed, yet are not random. An example in real life might be a meadow.  Nothing much smaller than a meadow itself could contain all the information, subtle order and complexity of myriad interacting organisms making up a meadow. Because it is incompressible, a meadow shares the high complexity quotient of shallow randomness; but because its irreducibility is not due to randomness, it owns a deep complexity that we appreciate. That difference is captured by the metric of effective complexity.

We might think of effective complexity as a mathematical way to quantify non-predictable regularities. DNA itself is an example of non-predictable regularity. It is often described as a non-periodic crystal. In its stacking and packing abilities it shares many predictable regularities of a crystal, but rare among crystals, it is non-repeating (non-periodic) because each strand can vary. Therefore it has a high effective complexity.

But the increase in effective complexity in life’s history is not ubiquitous, and not typical. While a rise in complexity can be seen retrospectively in the broad lineage of life, it usually can’t be seen up close in a typical taxonomic family. There’s no factor associated with complexity that will consistently gain across all branches of life all the time. Scientists intuitively expected larger and later organisms to acquire more genes. In broad strokes that can be true, but on the other hand a lily plant or ancient lungfish both have 40 times the DNA base pairs than humans. Many “laws of acceleration” have been proposed for evolution and all of them disappear when inspected quantitatively in a specific range. One of the earliest laws proposed, Cope’s Rule, posited in the 1880s, says that over time life evolved larger bodies. While a trend toward large body size is true within some lines (notably dinosaurs and horses), it is not true of life as a whole (we are much smaller than Tyrannosaurus Rex), and it is not true for all dinosaur lineages, or all branches of ancestral horses. More importantly, in most orders of life, the largest organisms may tend to grow larger, but the mean size of the organisms remains constant. The smallest stay small. Stephen Gould interprets this as “random evolution away from small size, not directed evolution toward large size.”

A second proposed universal law says that over evolutionary time longevity increases for the lifespan of a species. Over time, species are more stable. And in fact at certain epochs in the past, the longest-lived species did live longer. But the mean longevity did not increase. This pattern is repeated throughout the tree of life. Maximum diversity increases, but the mean diversity does not. Maximum brain size increases in many animals over time, but the mean brain size does not. When we apply the feeble measurements we have for complexity, we find that maximum complexity increases over evolutionary time, but mean complexity does not.

Typical pattern of increasing size among foraminifera (similar to plankton)

Nowhere is this more evident that in the makeup of the bulk of primeval life on earth. Life began 3.8 billion years ago, and for the first half of that time — its first two billion years — life was bacterial. This first half of life’s show was the Age of Bacteria, because bacteria were the only life.

Today, in an expanded biosphere crammed with 30-100 million species, bacteria still make up the bulk of life on earth. [ck]  Maverick scientist Thomas Gold estimates that there is more bacterial mass living in the crevices of solid rock in the earth’s crust than there is in all the fauna and flora on earth’s surface. Some estimates put the total bacterial biomass (in terms of sheer weight) within soil, oceans, rocks, and in the guts of animals to be 50% of all life on this planet. Bacteria are also significantly more diverse then visible life. The bacterial world is where gene hunters go to find unusual genes for drugs and other innovations. There is nowhere bacteria have not colonized. Bacteria thrive in more extreme environments – cold, dry, pounding deep pressure, scorching heat, total darkness, toxic elements, radiation intense – than any other kind of life. They produce most of the oxygen for the planet. They underpin most ecosystems. They dwarf the rest of life in genemonic variety. The bacteria breeding between the interstitials of loamy soil and in the depths of the ocean and in the warm tub of your own intestines are all as highly evolved as you are. Each bacteria is the result of an unbroken succession of 100% successful ancestors, trillions of generations long, and each is the product of constant, hourly, adaptive pressure to maximize its fitness to its environment. Each bacteria is the best that evolution can do after several billion years.

In every biological way we can measure, bacteria are the main event of life on earth. If an exploring probe from another galaxy landed on earth and began a life census, they would quickly and correctly deduce that after 3.8 billion years of evolution this planet was still in the Age of Bacteria.

As far as we can tell most bacteria are the same as they were a billion years ago. That means for the bulk of life on earth, the main event has not been a steady increase in complexity, but a remarkable conservation of simplicity. After billions of years of steady work, evolution produces mostly more of the same.

It would entirely fair, then, to represent the long arc of evolution much as Stephen Gould has in this diagram above. As he renders it, the main event in both epochs – early and late — is the reigning pyramid of simple bacteria. In the middle of the sloping bulge are the mid-size organisms – the grasses, plants, fungi, algae, coral of the world. Not as simple or collectively massive as bacteria, these semi-simple organisms nonetheless constitute the bulk of life that interacts with us directly. They don’t form the infrastructure; this middle life forms the architecture. These simpler organisms introduce the change and variety in our lives. At the bottom edge, extending to the right, is the thinning, minor “long tail” of the more complex organisms These are the scarce charismatic organisms that star in nature shows. The message of this sketch, Gould concludes, is that “the outstanding feature of life’s history has been the stability of its bacterial mode over billions of years!”  Not just bacteria, he reminds. Horseshoe crabs, crocodiles, and the coelacanth are famously stable in geological time.

But it takes a peculiar kind of blindness to see stability as the chief event in this movie. In scene one there is a nice hill of simple bacteria. In scene two, the hill has enlarged and grown a long tail of weird, complicated, improbable beings. They seem to come out of nowhere, and because of their complexity, must be more surprising and unexpected that the arrival of life itself. Sure, in a quantitative, almost autistic kind of reckoning, nothing much happened because the hill of bacteria is basically the same. This is true! The veracity of this observation is the foundation of the orthodox arguments against a trend for complexity in evolution. The drive in evolution is to keep things simple and thus the same.

Everything is the same, except for, well,,,, the addition a few little meaningful details. This blindness reminds me of a quip by Mark Twain on the consequential difference “between lightning and a lightning bug.” Apes, proto-humans, and humans are all basically the same. Nothing really happened between scene one, Homo erectus and scene two, Homo sapiens. The genes between the two are likely to be 99.99% conserved. When the long thin tail of language appeared in one ape, it was a minor alteration compared to the bulk of everything else about apes that remained stable. Yet, how that additional ‘bug” changes the meaning of everything else! So by the reckoning of the imagination, everything has changed over time.

To be fair to the orthodoxy, I don’t think Gould and others would deny the power of very small changes to have profound effects. The contention is whether these small changes (like occasional lines of increasing complexity) are the main event or simply a side effect of the main event. To rephrase Gould, are we really just witnessing random evolution away from simplicity, rather than random evolution directed towards complexity? If this kind of evolutionary complexity is a minority thread, then how can we claim it is being driven, pushed by evolution?

The evidence lies in deep history. The two scenes in the cartoon diagram above are not beginning and end, but the middle. Their action takes place in Act Two of a long movie, the great story of the cosmos. The first Act begins long before this sequence appears, and the third Act follows it. The long arc of complexity beings before evolution, then flows through the 3.8 billion years of life, and then continues into the technium.

Seth Lloyd, among others, suggests that effective complexity did not begin with biology, but began at the big bang. (I argue the same in different language in The Cosmic Origins of Extropy and The Cosmic Genesis of Technology.) In Lloyd’s informational perspective, fluctuations of quantum energy (or gravity) within the first fempto seconds of the cosmos caused matter and energy to clump. Amplified over time, with gravity, these clumps are responsible for the large-scale structure of galaxies – which in their organization display effective complexity. Recently three researchers (Ay, Muller, and Szkalo) determined that effective complexity is primed to generate phase changes. A phase change is the weird transformation, or restructuring, that the molecules of an element like water undergoes as it assumes three very different forms – solid ice, liquid, or steam. Systems (like a galaxy) can also exhibit phase changes, producing new informational organization with the same components.

In Lloyd’s scheme, “gravitational clumping supplies the raw material necessary for generating complexity,” which in turns generates new levels of effective complexity in the form of self-regulating atmospheric planets, life, mind and technology. “In terms of complexity, each successive revolution inherits virtually all the logical and thermodynamic depth of the previous revolution.”  This ratcheting process keeps upping the effective complexity over deep time.

This slow ratchet of complexity preceded life. Effective complexity was imported from antecedent structures, such as galaxies and stars, that teetered on the edge of persistent disequilibrium. And as in the organizations before them, effective complexity accrues in an irreversible stack. Lloyd observers, “This initial revolution in information processing [in galaxies and clusters of galaxies] was followed by a sequence of further revolutions: life, sexual reproduction, brains, language…and whatever comes next.”

In 1995, two biologists, John Maynard Smith and Eors Szathmary, envisioned the major transitions in organic evolution as a set of ratcheting organizations of information flow. Their series of eight revolutionary steps in evolution began with “self-replicating molecules” transitioning to the more complex self-sustaining structure of “chromosomes.” Then evolution passed through the further complexifying change “from prokaryotes to eukaryotes” cell type and after a few more phase changes, the last transition moved it from language-less societies to those with language.

Each transition shifted the unit that replicated (and upon which natural selection worked). A first, molecules of nucleic acid duplicated themselves, but once they self-organized into a set of linked molecules, they replicated together as a chromosome. Now evolution worked on both nucleic acid and chromosomes. Later, these chromosomes, housed in primitive prokaryote cells like bacteria, joined together to form a larger cell (the component cells became organelles of the new), and now their information was structured and replicated via the complex eukaryote host cell (like an amoeba).  Evolution began to work on three levels or organization; genes, chromosomes, cell. These first eukaryote cells reproduced by division on their own, but eventually some (like the protozoan Giardia) began to replicate sexually, and so now life required a diverse sexual population of similar cells to evolve. A new level of effective complexity was added: Natural selection began to operate on populations as well. Populations of early single-cell eukaryotes could survive on their own, but many lines self-assembled into multicellular organisms, and so replicated as an organism, like a mushroom, or seaweed. Now natural selection operated on multicelled creatures, in addition to all the lower levels. Some of these multicellular organisms (such as ants, bees, termites) gathered into superorganisms, and could only reproduce within a colony or society, and evolution emerged at the society level as well. Later language in human societies gathered individual ideas and culture into a global technium, and so humans and their technology could only prosper and replicate together, presenting another level for evolution and effective complexity.

At each escalating step, the logical and thermodynamical depth of the resulting organization increased. It became more difficult to compress the structure, and at the same time, it contained less randomness and less predictable order. Each upcreation was also irreversible. In general, multicellular lineages do not re-evolve into single cell organisms, sexual reproducers rarely evolve into parthenogens, social insects rarely unsocialize, and to the best of our knowledge, no replicator with DNA has ever given up genes. Nature will simplify, but it rarely devolves down a level. However nature is nothing but a collection of exceptions. There is no rule in biology that is not broken or hacked by some creature, somewhere. Yet here the trend is mainstream and representative of the mean. When life does complexify in levels, it does not retract.

Just to clarify: within a level of organization, trends are uneven. A movement toward larger size, or longer longevity, or higher metabolism, or even general complexity may be found only in a minority of species within a family, and the trend may be subject to reversal on average. So when biologists search for a measured increase in some body characteristic over evolutionary time they typically find patchy distribution as soon as their survey widens beyond a narrow taxonomic branch. Consistent directions of evolution are absent across unrelated subjects in similar epochs.  The trend toward greater effective complexity is visible only in the accumulation of large-scale organizations over large-scale time. Complexification may not be visible with ferns, say, but it appears between ferns and flowering plants (recombining information via sexual fertilization).

We can take the second panel of Gould’s diagram as a wonderful and ideal illustration of this escalation. The long thin tail of increasing complexity is actually the trace of the major transitions in evolution. The shift to the right is a shift in the number of hierarchical levels that evolutionary information must flow through. Not every evolutionary line will proceed up the escalator (and why should they?), but those that do advance will unintentionally gain new powers of influence that can alter the environment far beyond them. And, as in a ratchet, once a branch of life moves up a level, it does not move back. In this way there is an irreversible drift towards greater effective complexity.

The arc of complexity flows from dawn of the cosmos and into life. Complexity theorist James Gardner calls this “the cosmological origins of biology.” But the arc continues through biology and now extends itself forward through technology. The very same dynamics that shape complexity in the natural world shape complexity in the technium.

Just as in nature, the number of simple manufactured objects continues to increase. Brick, stone and concrete are some of the earliest and simplest technologies, yet by mass they are the most common technologies on earth. And they compose some of the largest artifacts we make: cities and skyscrapers. Simple technologies fill the technium in the way bacteria fill the biosphere. There are more hammers made today than at any time in the past. Most of the visible technium is, at its core, non-complex technology.

But as in natural evolution, a long tail of ever complexifying arrangements of information and materials fills our attention, even if they are small in mass. Indeed, demassification is one avenue of complexification. Complex inventions stack up information rather than atoms. The most complex technologies we make are also the lightest, least material. For instance, software in principle is weightless and disembodied. It has been complexifying at a rapid rate. The number of lines of code in a basic tool such as Microsoft’s Windows has increased ten fold in thirteen years. In 1993, Windows entailed 4-5 million lines of code. In 2006, Windows Vista contained 50 million lines of code. Each of those lines of code is the equivalent of a gear in a clock. The Windows OS is a machine with 50 million moving pieces.

Increasing complexity of software

Throughout the technium, lineages of technology are restructured with additional layers of information to yield more complex artifacts. For the past two hundred years (at least) the number of parts in the most complex machines has been increasing. The diagram below is a logarithmic chart of the trends in complexity for mechanical apparatus. The first prototype turbo jet had several hundred parts, while a modern turbo jet, 30-50 times more powerful, has over 22,000 parts. The space shuttle has tens of millions of physical parts yet it contains most of its complexity in its software, which is not included in this assessment.

Complexity of manufactured machines, 1800 – 1980

We can watch our culture complexify right before our eyes. As an almost trivial yet telling example, author Steven Johnson has noted how the plot lines of movies and TV have become more complex within his own lifetime. The number of characters involved in a story has doubled, the number of twists increased, the frequent insertion of complicated literary devices such as flashbacks has increased the levels of engagement. If a movie is a program (as in a computer program), then over time these narratives accumulate the equivalent of subroutines, parallel processing, and recursive loops, elevating the story into a more adaptive, living thing.

Our refrigerators, cars, even doors and windows are more complex than two decades ago. The strong trend for complexification in the technium provokes the question, how complex can it get? Where does the long arc of complexity take us?

Three scenarios of complexity for the next 1,000 years:

Scenario #1. As in nature, the bulk of technology remains simple, basic, and primeval because it works. And it works well as a foundation for the thin long tail of complex technology built upon it. If the technium is an ecosystem of technologies, then most of it will remain highly evolved as microscopic and plant equivalents: brick, wood, hammers, copper wires, electric motors and so on.

Scenario #2. Complexity, like all other factors in growing systems, plateaus out at some point, and some other quality we had not noticed earlier takes its place as the prime observable trend. In other words, complexity may simply be the rosy-colored lens we see the world through at this moment, the metaphor of the era, when in reality it is a reflection of us rather than evolution.

Scenario #2A. Complexity plateaus because we can’t handle it. While we could make technology run faster, smaller, denser, more complicated forever, we don’t want to beyond some point because it no longer matches our human scale. We could make living nano-scale keyboards, but they won’t fit our fingers.

Scenario #3. There is no limit to how complex things can get. Everything is complexifying over time, headed toward that omega point of ultimate complexity.

If I had to, I would bet, perhaps surprisingly, on scenario #1. The bulk of technology will remain simple or semi-simple, while a smaller portion will continue to complexify greatly. I expect our cities and homes a thousand years hence to be recognizable, rather than unrecognizable. As long as we inhabit bodies approximately our size – a few meters and 50 kilos  — the bulk of the technology that will surround us need not be crazily more complex. And there is good reason to expect we’ll remain the same size, despite intense genetic engineering and downloading to robots. Our body size is weirdly almost exactly in the middle of the size of the universe. The smallest things we know about are approximately 30 orders of magnitude smaller than ourselves, and the largest structures in the universe are about 30 orders of magnitude bigger.  We inhabit a middle scale that is sympathetic to sustainable flexibility in the universe’s current physics.  Bigger bodies encourage rigidity, smaller ones encourage empheralization. As long as we own bodies – and what sane being does not want to be embodied? – the infrastructure technology we already have will continue (in general) to work. Roads of stone, buildings of modified plant material and earth, not that different from our cities and homes 2,000 years ago. Some visionaries might imagine complex living buildings in the future, for instance, but most average structures are unlikely to be more complex than the formerly living plants we already use. They don’t need to. I think there is a “complex enough” restraint. Technologies need not complexify to be useful in the future. Danny Hillis, computer inventor, once confided to me that he believed that there’s a good chance that 1,000 years from now computers might still be running programming code from today, say a unix kernel and TCP/IP. They almost certainly will be binary digital. Like bacteria, or cockroaches, these simpler technologies remain simple, and remain viable, because they work. They don’t have to get more complex.

At the same time, there is no bound for the most complex things we will make. We’ll boggle ourselves with new complexity in many directions. This will complexify our lives further, but we’ll adapt to it. In fact, ongoing complexification – even in the thin bleeding edge – suggests a fourth scenario.

Scenario #4. Complexity gets more complex. We make or discover technological systems that require new, more complex definitions of complexity. Not merely, as I quoted Seth Lloyd above, because our definitions of complexity indicate ignorance, but because in fact we are finding/making things more complex and need new definitions. “Logical depth” won’t be enough for a definition as we keep making software more complex. As an example in the financial world, the invention of stock ownership added complexity to a marketplace. Then we invented making bets on those shares (stocks), and then we invented making bets on those bets (a derivative) and then bets on the bet on the bet (second order derivative), each layer of relation between bits adding complexity, and requiring new ideas about complexity. We keep adding new levels and ways to complexify our economy, till the complexity exceeds our ability to measure it (or understand it). Over time we are increasing the complexity of complexity itself, by inventing/finding new ways for bits of information to relate to other bits. It is those intangible relations that form complexity. As far as we can see, there is no limit to the new ways bits can relate. Keep in the mind the “spooky at a distance” entanglement between quantum bits to understand how complicated complexity could get.  In a thousand years, the concept of “complexity” will probably be as dead as the old notion of “metaphysics” because it will have turned out to be too blunt for the dozens if not hundreds of concepts it probably contains.

A movement towards complexity is what I call the “least objectionable theory” of the universe. Not everyone agrees with the trend, but not everyone agrees on anything at this scale. However fewer people in both science and faith disagree about the large scale movement toward greater complexity (whatever that is) over time. Defined as an irreversible escalation of increasing relations among the information flowing through structures, then complexity is rising.

The great “least objectionable” story so far: The arc of complexity begins in the ultimate simplicity of the big bang creation of something/nothing, and steadily sails through the universe, pushing most of it slowly towards complexity in the news levels of life, while at a few edges it accelerates through technology, making complexity itself ever more complex.

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• Andy Felix

Amazing article. Leaves my mind boggling at how complex I am myself?

• Devonavar

I find your urge to measure complexity paradoxical and the definitions you use of complexity incomplete. You are turning to math for a precise definition of something that is not inherently mathematical, and then complaining that the mathematical definitions you can find do not cover your intuitive sense of what complexity is.

I think you nod towards the solution in Scenario #4, but you still don’t quite get it right because you are treating complexity as a metric of things rather than a metric of ideas. You are confused because you are asking what the complexity is of something that does not have it. Systems do not have complexity; explanations do. Complexity is a measure of how difficult it is to explain something; it applies to concepts and ideas, not mass or systems. I do not mean to imply that systems cannot be more or less structured or that it can be easier or harder to describe those structures. I am also aware the “complexity” is a mathematical concept with a long history, but I think you are confusing this limited concept with our more general intuitions about complexity that are much broader and are not captured mathematically.

What I really want to drive home is that complexity is a measure of explanation, and explanations are relative to a specific conceptual framework. Attempting to measure increases or decreases in complexity over time only works if the conceptual framework itself does not change â€” a circumstance that does not apply to the real world.

I would argue that complexity is proportionate to the length of an explanation, and that complexity is a measure of the quality of that explanation (as per Occam’s Razor), not of … well, I’m not clear on what you think complexity is a measure of, since explaining this seems to be the entire purpose of your post. I would stick to applying complexity to information rather than applying it to the things that you have information *about*.

I would also hesitate to conflate the amount of information (measured in bits) with the amount of explanation. Your example of an ordered sequence versus a random sequence is a case in point. Let’s take Beethoven’s Fifth Symphony (approximately half an hour of music) versus half an hour of white noise (i.e. a sequence of random bits) as an example. There is far more information in the white noise because it cannot be compressed (though, if it’s truly random, it’s hard to see how it constitutes information, which is inherently structured and has meaning). But, if you want to explain the two, most explanations for Beethoven’s Fifth will be far longer, since there is much more to explain, be it historically, musically, or demonstratively. You can explain white noise by listening for five seconds, saying repeat for half an hour, and leaving it at that. White noise is uniform and unstructured, no matter how many bits it takes to record it. Explaining Beethoven’s Fifth by listening requires a full half hour.

On a side note, you have done this as well in your explanation. Your description of a random number could be encapsulated in a sentence with *two* commands: “Print any digit; repeat 1,000,000,000,000 times.” An added precondition is that “any digit” must be any digit in the base-two numerical system â€” i.e. the commands are sensical in your chosen system of representation / explanation. In a base 2^trillion numerical system (a different conceptual framework for explanation), both of your examples would have equal complexity because both could be represented in a single digit.

Is there a sense in which the binary system is “less complex” than my base 2^trillion system? I suppose, but only because we (the explainers) find it easier to explain using binary than in base 2^trillion. I would argue that this is arbitrary. Most of us find it even easier to use decimal than binary, and that, on these terms the random sequence “100” and the ordered sequence “111” have equal complexity when looked at in decimal (5 and 8 are both single digits and thus have equal complexity). Or, if the explanation is in terms of number of “parts” 5 is less complex because it is prime.

Moving on. I believe that a lot of the historical increase in “complexity” (and *especially* the increase in the “complexity” of the technium) that you see is reflective of advancements in our systems of explanation and the amount of information our society holds as a whole. Certainly, the amount of information our society tracks has increased as our population and understanding of the world has increased. In parallel, the vocabulary that we use to explain the world has expanded, as have the number of “basic” concepts that we use in explanation.

I think you are talking about the right things in the wrong way when you write “We keep adding new levels and ways to complexify our economy, till the complexity exceeds our ability to measure it (or understand it).” Complexity doesn’t exceed our ability to measure or understand; it *is* our ability to measure or understand. An explanation is complex to the degree that we do not understand it. As our understanding of the world grows (i.e. we aquire and internalize more and better information about it), the things that we understand become simpler, and we discover new complex things that we never even had the framework to grasp before. Adding new levels to our economy does not mean we are making it more complex. It means that our understanding of the economy is becoming better, and we are thus discovering new layers which are new and therefore complex. We didn’t invent second order derivitives. We (meaning certain bankers) discovered them; specifically we discovered there was money to be made in making bets on bets on bets. We (meaning everyone else) also discovered that we didn’t fully understand the consequences of allowing such bets to be made, because this new discovery wasn’t fully understood (and was therefore still complex). We now understand a lot more about second order derivitives in the financial markets, and can see new complexities raising their heads in the future.

I would like to end by addressing your “least objectionable” story. I think your story is just Occam’s Razor revisited: The simplest explanation is the best explanation (and, yes, I know that’s a loose paraphrase). But, I think this boils down to nothing more than the fact that explanation is grounded in simplicity; an explanation is nothing more than describing the unknown in terms of the known or the complex in terms of the simple. Or, put another way, an explanation is a way of finding unity or order within multiplicity or chaos. Once something is “fully” explained, it shifts from complex to simple, and further explanations can be built on top of it.

The Big Bang is fundamentally an explanation, not an event. Scientifically, it is the bedrock that supports every other explanation within the conceptual framework of physics (and thus, many other disciplines). In this sense, I think you are right, there is an “arc” to complexity that “begins in the ultimate simplicity of the big bang creation of something/nothing”. However, I think it is more accurate to say that the arc traces the explanations that science offers, not complexity itself. As science grows, each new theory builds on the last, and is this, in one sense “more complex”. More generally, our stockpile of knowledge of the universe in general grows, and there is therefore a larger “fringe” of complexity at the edges of our knowledge that we do not understand. However, I think it is likely that at some point our scientific paradigm of explanation will shift, and the big bang will no longer be the most likely explanation for the creation of the universe. Occam’s Razor will do its work, and a new framework will give us a new sense of what is complex and what is simple. Likely as not, the Big Bang theory will still be contained within that framework, but will no longer be considered primal or original (Or, perhaps it will just be written off as a mildly diverting TV show from CBS).

• @Devonavar: Thanks for your very long post. You raise several points, but they mostly connect back to your contention that the “rising complexity”, is only our rising understanding, or our way to explain the world which is not getting more complex. I think you mean that a rainforest is not getting more complex, rather we come to understand more levels about it.

This is true. Our understanding, as in science, does increase over time. And complexity is surely an informational measurement — because as I explain, complexity is itself a informational relationship — but in addition to the way our understanding is getting more complex (an assumption I did not bother to address in this piece), the world itself is getting more complex.

To say that a worm and an elephant are equally complex in reality, but that our explanation of one them is more complex, seems something you’d have to prove because on the surface this idea seems like nonsense.

• Vasu Srinivasan

The source code of Solaris open-sourced last year was 2 million lines of code. Does that mean Vista is functionally 25 times better than Solaris? Or does that mean Solaris is more efficient than Vista? (Note: I don’t belong to either camp).

Lines of Code is a very useful technique during project estimation phase, but loses its value with time and does not measure complexity well.

Refactoring the fat, without scraping the vital organs, is a hard job in software engineering. So, when functionality increases, in some organizations, they refactor the code very well and hence the code can do additional things without incurring too much,by way of lines of code. In some organizations, they might not do this task of refactoring the code while adding functionality, very well.

-Vasu Srinivasan
http://blog.amusecorp.com

• Alex Tolley

Very interesting essay.

What if the complexity (however measured) is more like the 2nd law of thermodynamics – order : disorder = simplicity : complexity.

The complexity results from an unlikelihood of returning to simplicity. However in this case, the driver is Darwinian selection mechanism.

• Arthur Smith

Kevin, nice essay here. On the scenarios, there is a fifth, I think, suggested by your analogy with evolutionary eras and also I believe something Thomas Homer-Dixon talks about in some of his books (The Upside of Down is probably the one I’m thinking of):

Scenario 5: Things get too complex for us to handle (like 2a) and everything collapses, and starts off simpler again (but not as simply as the previous time).

By the way, Steven Gould and Thomas Gold have both passed away (Gould in 2002, Gold in 2004) – not sure your use of present tense in their cases was intentional, but it seemed odd.

• @Arthur Smith: Books give an author an immortal presence. Does Darwin say in his books, or as Darwin said in his books. I think both work.

• Alex Tolley

“Scenario #3. There is no limit to how complex things can get. Everything is complexifying over time, headed toward that omega point of ultimate complexity.”

One thing you seem to forget is that complexity implies costs – costs of building and maintaining the complexity in DNA and software. If the utility of the complexity does not exceed the costs, evolution via reproductive success will favor the simpler over the complex.

This would imply scenario #1 is more likely. BUT, simplicity in our materials has a cost too – maintenance. If structures could self repair, then maintenance costs decline substantially. This suggests that our simplest materials, concrete, steel and glass might not be optimal for the future and that more complex materials might be more successful and replace them.

Another piece of the evolution of complexity seems to be the number of species increases. This may be an artifact of the preservation of fossils, but the data shows that biodiversity has been increasing. We also see this in our technologies, and I think we are seeing a rapid burst in speciation of computer languages, both general purpose and domain specific.

• Noel Cody

The most intuitive definition of complexity for me, Kevin, is one based on what the organism or system can “do” â€“ what it’s capable of, whether it’s conscious or unconscious of its actions. However that can be quantified.

• Devonavar

@KK*

I’m sorry I can’t be more clear … the idea that I have just bubbled up; I haven’t refined it, and even writing my response I knew there was lots missing.

I’m not really sure how you *could* prove my assertion though. At the very least, you would need a working definiton of complexity, and if we had that, I wouldn’t need to say anything. What I’m trying to do is not so much make a statement about reality but to define complexity in a way that fits the intuitive meaning. You can’t “prove” definitions; you create them. I was hoping my rambling might help bring a workable defintion into focus.

In any case, I wouldn’t say a worm and an elephant are equally complex in reality becuase I don’t think things are fundemantally simple or complex. Things exist. Complexity is something we apply to those things when we try and grasp their meaning. In our current understanding of biology, the biological complexity of the elephant is greater â€” but a different understanding of biology could change or reverse their relative complexity.

As you say, complexity is an informational relationship, and, thus, what we consider to be our “simple” building blocks of information matters. If the only vocabulary we have to descibe elephants and worms is horses then the elephant is less complex because it is arguably more similar to the horse than the worm. It is only because our understanding of biology is based around collections of cells that the elephant seems more complex to us. There’s nothing fundamental about it.

This is why I think it’s a mistake to talk about the generalized “complexity” of the universe, evolution or anything else. Yes, you can attempt to measure the information it takes to descibe the universe in bits and characterize those bits as more or less complex. But really what you are measuring is the efficiency of how you use those bits to represent meaning, not the things themselves.

Take pi. Represented numerically in bits, pi has infinite complexity because it is a random, infinite sequence. That doesn’t mean pi is fundamentally “infinitely complex”; nor is it beyond our ability to understand. It does mean that it can’t be adequately represented in numerical bits, but it is quite easy to represent it in other ways. In words it is succinctly captured as the ratio between the circumference and the diameter of a circle. In UTF-8 unicode, it is 11001111:10000000 (a decidedly finite representation â€” but still using bits). Most of the time, math takes this even further, and represents pi as a single greek character (π).

When asking “What’s more complex, a cucumber or a Boeing 747?” the answer is not unknown. There is no answer; the question is incomplete. You need to ask which is more complex in terms of ____. What representation are you using to explain them? Unless you answer this question (and you give several possible, equally feasible answers), you have nothing to measure.

My rambling here is provoked by these words: “…we have no idea what complexity is, or how to define it.” I responded because, arrogantly enough, I *do* think I have some idea how to define it. Define it in terms of explanation. Complexity is a measure of how well we represent information, not a fundmental property of the universe.

• Mark Dow

By the characterization of an “almost autistic kind of reckoning, nothing much happened because the hill of bacteria is basically the same”, I assume you mean something like “self-centered” or “narrow” — surely not autistic. But finding meaning, attaching meaning, to our emergence on the tail of a grand ecosystem seems more narrow, the definition of anthropocentric. Yes, we are the topic of the nature channel.
The length of description required to get a grasp of a human or any species is longer than the genome, epigenome and local environment. We are not independent of the hump, and are meaningless and hopelessly lost outside the context of the hump. Description of the range of possibilities for large animals requires a great deal of information about the details of the hump, its history and structure. It is a system that is not separable as a simple sum of parts. Cucumbers and 747’s have an intimate and non-trivial relationship that should be included in a tally of their complexity.
I don’t see any purpose or inevitability to this drift away from (growth along with?) the dominant more simple systems, but I do appreciate the value of descriptions of individuals, species, technologies and the range of interactions. Your points about the increasing variety and layering of levels of complexity are compelling.
(femo->fempto seconds)

• John B

Determining metrics to quantify complexity is a very interesting problem. I won’t presume to have anything even approaching an answer, but I did want to contribute this notion:

Complexity of a thing might be related in an interesting way to the amount of information stored in or by the thing.

• Tom Crowl

Article and comments interesting but I don’t think these questions will be resolved until there’s a clearer agreement on a definition of “complexity.”

A clear distinction needs to be made between complicated and complex since I think the real questions discussed involve chaos/order, energy/entropy, etc.

For instance, I would suggest that a pocket watch with ten-thousand parts that tells the time, day, month and even the next eclipse is complicatedcomplex.

As are detailed explanations and TV scripts.

Each has lots of pieces but don’t change each time they’re run.

Complex systems MUST have chaotic elements which arise either from feedback mechanisms between multiple interacting “non-chaotic” elements of the system or from system elements that are in themselves chaotic.

So to move the argument you have to address the question…

Is the world getting more complex? Or more complicated? Or both?

I would contend, in technical terms that increasing complication sucks. And that we need some well defined complexity to clean it up!

As for measuring complexity…

Not sure how to do that but suppose it would relate to how small of a butterfly’s flutter in Chile it takes to produce how big of a tornado in Kansas.

• Tom Crowl

I seem to have dropped a couple of words in my comment…

Its supposed to say complicated but NOT complex

Ooops!

• Sam

I expect our cities and homes a thousand years hence to be recognizable, rather than unrecognizable.

Uh, really?

Our body size is weirdly almost exactly in the middle of the size of the universe. The smallest things we know about are approximately 30 orders of magnitude smaller than ourselves, and the largest structures in the universe are about 30 orders of magnitude bigger.

So what? The main issue to me seems to be the transfer of information – human technology is just way better at rapidly sending signals across vast distances than anything in the biological world. There doesn’t seem to be much from stopping the entire earth from conglomerating into one gigantic brain. The future Earth may have humanoid robots of some kind, but I have a lot of trouble imagining them (or any other subsystems on the planet for that matter) as having any significant degree of autonomy.

• Aaron Davies

Regarding computers of the distant future, have you read Vernor Vinge’s A Deepness in the Sky? He has a space-faring civilization some unspecified number of millennia from now still using Unix systems, indicated by their 1970-epoch date system (presumably with 64-bit time_t!).

• Aaron Davies

@Devonavar: I think perhaps the answer you want to give to “Whatâ€™s more complex, a cucumber or a Boeing 747?” is “mu“.

• Devonavar

@Aaron Davies

Thanks, mu fits very well. Mu is the answer to most paradoxes; in most cases, recognizing how you’ve misconceptualized the question you are asking makes teh paradox go away.

• E. Rietman

Kevin, this is really good stuff. Thanks for having the guts to post it and the guts to put it in a book. Itâ€™s risky to suggest that complexity is increasing, but it is obvious to anyone who thinks about it for only a moment. I want to point out a few things you are apparently thinking about but, from my read of your Blog, havenâ€™t yet put together.

747s and algae cells are both dynamic systems. They both operate in specific environments. They both use energy, process matter, process information, have some degree of complexity, and perhaps some degree of awareness. No one will doubt the first four of these, but the fifth may be a little questionable. But consider a thermostat. One could say it is aware of the room temperature. It processes information. It doesnâ€™t process matter. It does process energy. And it has a degree of complexity, albeit, low. The 747 also contain subsystems that have some degree of awareness. The algae cell is aware of light and chemo-toxins. When the 747 is parked and engines turned off, it is in a stable attractor point, except for some subsystems in standby mode that are in a limit cycle.

The point I am getting at is that complexity, however you want to measure it, is perhaps not enough for describing 747s, algae cells, human brains, cities, and ecosystems. All these systems also can be placed on a scale of information processing, matter processing, energy processing, awareness, and complexity. The challenge is to devise not only metrics for complexity, but also metrics for these other attributes.

• Finn

Robert — Kolmogorov complexity DOES establish existence of a universal complexity without reference to an external system. That’s exactly the purpose and wonder of it’s discovery.

There are many different methods of describing something. Kolmogorov showed that amongst those methods, there are a set of optimal methods, which are seperated by a constant value associated with the method of description. Thus you can pinpoint this ‘complexity’ to within a constant.

So yes — you *can* have a universal complexity.

One catch — the proof is linked to the proof that establishes the existence of universal computation. Which means that there is a part that relies on the halting of a computer’s programs. Which, as Turing demonstrated, is unprovable to know whether a given program will ever halt or not. Kolmogorov complexity is similar — one can establish an upper bound for an objects complexity, but there might be a long running, shorter program that has not yet halted, which describes an object in a simpler (read shorter program length) way.

Translated: we can never prove that something is not simpler than it looks!

Finn

• Naveen Bachwani

What was equally interesting and informing was the discussion that Devonavar started, in response to your post.

It is this high quality exchange of ideas that makes me return each time to The Technium…

• RobertJ

It’s great to see you attacking this problem. It is a difficult problem, as most of us try to frame our “intuitive” notion of complexity into a mathematical form, without asking if our intuitive notion could in fact be wrong.

Scientific rigor reminds us that we are not measuring the complexity of an organism, but only the complexity of our perception or “model” of the organism. This I think, was also Devonavar’s point.

Not only our knowledge but also how we frame it into a model affect the measure of complexity. Kolmogorov complexity examplifies this, as it cannot put a bound on complexity without reference to an “operating system” or language within which the string is programmed.

I take issue with a third hidden assumption: That complexity can be measured for a single isolated object. This ignores the mutual dependency between organisms in the biosphere. In figure 2 above, the long tail of large organisms is a new habitat for microorganisms. The introduction of large organisms increased the diversity among bacteria as well, both as hosts and as creators of new biotopes. (forests, marshland, coral reefs e.t.c) and the large organisms depend crucially on micro-organisms for their survival.

If you then look at the genome, it is not new letters, but the network of interdependencies between genes that increases complexity and let us state that a mammal genome “is more complex” than bacterial genomes of the same size.

Finally, you find the same thing in your technium. The aluminium in a can is perhaps less “complex” than the rock in a stone axe, but the “intuitive” complexity is regained when you consider then web of production methods that brought forth the can and the web of technolgies underlying these production methods.

I think that taking this network-centric view is crucial in the study of complexity, which differ from statistics in that the latter is more concerned with averages and typical values.