Simplicity and complexity

Some chaotic items (not a systematic outline):

• Occam's razor; Popper; empirical content of theories; Einstein's adage (theories should be as simple as possible but no more simple)
• number of items; number of tokens; number of types 
• Kolmogorov complexity: algorithmic compressibility
• apparent complexity joined with underlying simplicity: Mandelbrot set
• contrast: random-color pixel bitmap does not look complex, but has in general max Kol. complexity
• caveat: any bitmap can result from a stochastic/genuinely random plotting process; monkey at the typewriter writing a novel, given enough time
• complexity as a bareer to understanding
• simplicity and complexity in visual art
• (moderate?) complexity as source of pleasure 
• ornament as an addition to complexity; boring rectangles in modern architecture
• network; interrelatedness; tangledness
• simplicity and complexity of biological organisms
• apparent tendency toward increased complexity (multicellulars; ever more complex brains?)
• does Gould have a reservation? 
• simple organisms do not disappear; they are in general viable

This article could quite possibly be broken down into separate subjects. It implements the idea that doing semantic lexicography on names of concepts is not good enough; one should have a more discursive treatment. And that discursive treatment turns out to be rather encyclopedic. That is fine.

Kolmogorov complexity

Since in what follows, I repeatedly single out Kolmogorov complexity as one of the most plausible measures matching the intuitive concept of complexity (but beware of Manderlbrot!), let me introduce the concept. Approximately, the Kolmogorov complexity of a sequence of bytes is the minimum length of a program outputting that sequence, or the length of the sequence if all programs are longer. For instance, we could take Python to be the environment, prohibit all imports and then define Kolmogorov complexity as the length of the shortest Python program so constrained. This is usually not done in a mathematical treatment of Kolmogorov complexity. If we do not prohibit imports from Python standard library, it will feel more like cheating since the complexity will be hidden in the library being imported.

Let us have a look at some consenqunces of the definition. A string 010101...0101 has a low Kolmogorov complexity since a program outputting the sequence is very short no matter how long the sequence is. Similary, a sequence of squares (1, 4, 9, 16, 25, 36, ...), no matter how long, has a low Kolmogorov complexity. On a more counter-intuitive note, images of the Manderlbrot set have low Kolmogorov complexity, since the program generating them is very short (we can hardwire the parameters into it). That is perhaps a bit counter-intuitive since the images appear rather complex, when made with the right parameters.

Simplicity and complexity of theories

The concept of simplicity is sometimes related to Occam's razor. From what I recall, it states that a theory should not posit more entities than necessary. Whether this should be called "simplicity" is unclear. There is an Einstein's adage that scientific theories should be as simple as possible but not more simple. (Should software user interface be as simple as possible but no more simple? Is the single-button mouse of Steve Jobs better? Does Dijkstra say something about untamed complexity?) Popper criticizes the criterion of simplicity and indicates that what really matters is the empirical content of a theory. The empirical content is the set of all potential refuting/falsifying observations. An empirical theory says the more, the more it forbids. But what is the complexity (lack of simplicity) of a scientific theory? One measure that comes to mind to Kolmogorov complexity. The theory is more simple, the easier it is to algorithmically compress. Ohm's law is an example of a very simple law, a linear dependence. Perhaps linear dependence is in some sense more simple than a quadratic dependence. And then, Ohm's law would be more simple than the Newton's gravitational formula relating masses and distance to force.

Simplicity and complexity of geometric shapes

One could perhaphs again use something like Kolmogorov complexity to identify the complexity of geometric shapes. Let's have a look.

What is more simple, a circle or a square? To my mind, intuitively, a circle is more simple. It shows rotational symmetry. Rotation is a key operation in the space in which we are living. Human motor system depends on rotation; translation is achieved by combining rotary motions. Our space seems to show no fundamental preference to orthogonal predefined axis, although on the Earth, the vertical direction seems well singled out. However, using something like Cartesian coordinates in the context of pixels on the computer screen (pixels organized orthogonally rather than, say, hexagonally as used to be the case on CRTs), plotting a square is much easier than plotting a circle. And thus, from that perspective, a square would be more simple than a circle. What is more simple, a circle or an (general) ellipse? Circle is more simple, since it has fewer parameters. Moreover, in the context of human artifact making (e.g. of a cup or a vase), the rotational symmetry of a circle is of great importance. It seems much more simple to make material artifacts showing rotational symmetry. Indeed, cups, plates and vases use the shape of circle, not an ellipse. The simplicity of the circle is what lead Aristotle to speculate that there are only two kinds of elementry motion, circular and translational. It was Kepler who discovered that the shapes of planetary orbits are approximately elliptical (he would have thought they were exactly elliptical, I guess). Any closed curve can be broken down into a composition of circular motions, per Weitz. I suspect that an ellipse needs an infinite sum of such circular motions. And then, what is more complex, a cardioid (a cycloid) or an ellipse? A cardioid would be more simple, perhaps, being composed of only two circular motions? But plotting an ellipse on an orthogonal pixel grid is rather simple: sin, cos and multiplication (plotting a circle is similar when one uses a similar algorithm, but is in a sense more simple using the Bresenham algorithm, which does not require float multiplication). What I suggest is that a technical examination of simplicity and complexity of shape may yield results a bit different from intuitive human responses.

The Mandelbrot set and other fractals are examples of shapes that appear rather complex to human inutition, but are supremely algorithmically compressible (the program plotting the Mandelbrot set is very short), and thus, they have low Kolmogorov complexity. In relation to the Mandelbrot set, each approximation of it seems to be a polynomic curve, in any case a smooth curve (assuming we use the criterion of distance from 0 greater than 2 as the escape detection criterion; we could also use a square criterion). Ever better approximation is achieved by increasing the number of iterations after which the examination of escaping is given up. The first approximation is a circle, the second is perhaps an ellipse, and further approximations increase the number of "folds" (or whatever I should like to call it). From a PNG (raster bitmap) coding perspective, the Mandelbrot set contains infinite information/structure: we can zoom ever deeper and see ever more detail; from the generability perspective, it only contains finite amout of information captured in the generation program/code. The point: what to a human can appear infinitely complex may hide great underlying simplicity.

The above may have a bearing on biological morphogenesis. Perhaps the underlying shape-generating process is more simple than the resulting shape appears to be. I don't know.

Simplicity and complexity in visual art

The right kind of complexity in visual art adds to its beauty, or to the pleasure from looking at it, I think. As regards paintings, the minimum complexity is in the empty canvas. The complexity has to be of the right kind: a random-color-pixel bitmap has great Kolmogorov complexity, but is in general no more interesting than Duchamp's pseudo-art (pseudo- by my assessment).

Modern functionalist or even brutalist architecture shows little complexity. It is rectangles all the way, quite often, with no ornament (since ornament is crime?) When in Brno in Freedom Square, when I look at the modernist Omega building, I see nothing of interest or worth noting; it could as well be a commie apartment block. Next to Omega, there is an interesting building full of ornament, with statues of men holding something (I am not good at describing these kind of things; at school, they taught me to spell properly, inflect properly and to analyze the grammatical structure of sentences, not to describe). I guess the purpose of this Omega thingy is to make the beauty of the building next to it stand out. (Okay. I dislike this modernist nonsense, at least in architecture.)

Simplicity and complexity in games

Chess is a supremely simple game, and still very popular. So is Japanese go. To my mind, go appears to be simpler; it has fewer kinds of pieces and the rules are simpler. But go seems harder to play by computers, and in that sense, one might consider it more "complex", although that may be a misnomer (see also my remarks on "computational complexity").

8-bit computer games were often rather simple (usually more complex than chess and go), and as the computer power increased, the games seemed to get more complex in general. And thus, Team17 Worms on the 16-bit/32-bit Amiga seems much more complex than Draconus on the 8-bit Atari. It seems players appreciate the increased complexity; as much as I like Draconus, I find Worms so much more appealing. This seems to relate to my previous remark that the right kind of increase of complexity in visual art seems to increase the pleasure response, although here, we are dealing with an increase of complexity in the game mechanics. (Worms also has more complex visuals, but that is not my point here.)

As something of an aside, even the simple Draconus on the 8-bit Atari understood that the right kind of visual and auditory complexity creates an aesthetic appeal. From the game mechanics perspective, the introductory graphics and animation is beside the point, and so is the celebrated introductory music. In general, computer game makers have not sided with the functionalist(?) doctrine that ornament is crime. Even in the context of the simple game of chess, makers of chess sets often make the point of making the pieces look complex in the right kind of way and thereby aesthetically pleasing. In a software implementation, one can surely implement a Unix-style console-based chess, where pieces are represented by letters, but a great deal of appeal is missing. Beauty is a great seducer of men, a saying goes. And thus, eye candy is a thing.

Complexity in font faces

Modern antiqua font faces (is that the right term?) are relatively simple, compared to fraktur. Sans-serif seems to be more simple than a serif font; and thus, Arial seems to be more simple than Times New Roman. When these faces are specified in Metafont or other font specification format (and sure enough, one installs fonts as computer files), we can again apply Kolmogorov complexity to determine one kind of complexity.

Gothic faces seem to be "cool" and more complex. These are not display faces.

Complexity of a writing script

Arguably, letter-based scripts (Latin, Cyrrilic) are more simple than east-Asian scripts. This relates well to Kolmogorov complexity. 

Computational complexity

There is what is known as time and space computational complexity (linear, n times log n, polynomial, exponential, etc.). I find that to be a misnomer. Since, Bubble sort is a supremely simple algorithm, but it has higher time computational complexity than quicksort. One could argue that the differentiating adjective "computational" saves the matter. I don't think so. I think it should be "demandingness" or the like. Perhaps "demanding" is a Anglo word, not Romance, and one would like to find a Romance (stemming from Latin) word, to have scientific vocabulary Latin-based or Greek-based.

Complexity of words

The number of syllables could be taken as one measure of the complexity of a word. And a word that is "foreign" could be taken to be more complex. Moreover, a word that is combined transparently from native morphemes could be more simple: it could be stored in the mind as indices to the morphemes (it perhaps is not so stored, but it could).

Complexity and information

Since Kolmogorov complexity can be taken to be one measure of information content, it suggests that a more complex object contains more information than a more simple object. Here, the idea is that an object contains "information" even if there is no mind trying to describe the object and even if the object is not about another object. The information amount in an object could be taken to be the information amount in a relatively complete description of the object. And thus, while e.g. a (material) vase is not an information object, it could be taken to contain information, e.g. in its shape. The matter is complicated by microscopic/nanoscopic chaotic irregularities present in material objects, including vases. A human describing or representing a vase would not usually care to represent these, in part since these are not easily accessible to inspection. If one would take the locations and other properties of the atoms in the vase to be part of complete information about the vase, the information amount would be huge.

Here, it is perhaps worth recalling the irony that a random-pixel-bitmap contains a lot of information in terms of Kolmogorov complexity (and perhaps also Shannon information measure), but to human mind, upon inspection, it contains no information at all. At any rate, it seems to contain no signal at all, only noise. This point has been made many times, including in Kevin Kelly's article on extropy, I think. At the same time, if we take a meaningful text stored in 7-bit ASCII and put these bytes into a video buffer, we also get something that looks like noise. But this apparent noise, when properly decoded, yields meaningful English text.

See also my Wikiversity userspace article on notes on a theory of interpretation, with related keywords decoding and deciphering.

Complexity of software

It was perhaps Dijkstra who said that untamed complexity is a key problem in software making, or the like. Modularization is a technique to make complexity more manageable. The fundamental module is a function/procedure, not a class, by my assessment. A larger module is the compilation unit, which can be reinterpreted as an analogue of a static class; in Python, that is a "module". I noted that the right kind of complexity adds to aesthetic appeal of visual artifacts. More complex software can be more interesting. But the purpose of most software is not to be interesting; it is to address practical problems. When the problem to be addressed is how to make the programmer harder to replace, this can naturally result in otherwise unnecessary complexity.

Use of simple language

From what I recall, Popper advocates for use of simple language in science and especially philosophy. The objective is to make refutation/criticism as easy and simple as possible. Additional layers of obfuscation make it harder to determine what exactly is being said, and then harder to discover problems. Simple language may be less aesthetically pleasing, but as far as accuracy/validity is concerned, that is beside the point. A pseudo-sophisticated person may produce superficially impressive complex sentences and challenging vocabulary that, upon closer examination, are saying little. That said, thinking back on the Einstein quote, science has to be sufficiently complex to capture the domain it is trying to capture. Arguably, Newton's mechanics is more simple than Einstein's mechanics, but that does not justify avoiding Einstein.

Simple persons

Some persons are said to be simple, perhaps to be called simpletons. Others are more sophisticated. One could talk of a complex personality, I think. The Czech humoristic novel Saturnin makes fun of someone speaking of complexity of human soul, I think; here being more of a simple person is not seen as necessarily bad. Let's also think of the pseudo-profundities and apparent sophistication of Hegel.

Complex as a noun

There is the military-industrial complex, or so they say. And some people seem to suffer from having a Freudian complex, or something of the sort. It is not clear to me what to say to elucidate these kinds of entities. A complex seems to be some compound object, perhaps not really an object but rather a quasi-object, depending on what concept of an object one has in mind.

Etymology

One can sometimes learn something from the etymology. Both simple and complex seem to be from Latin plectere (I did not bother to check a dictionary, to prevent going into yet another rabbit hole). More is to be added later.

Questions

• Does Stephen Wolfram have something like (new) science of complexity?
• Does Stewart Kaufman have some kind of science of complexity? (See also Kevin Kelly's Out of Control.)
• What does Kevin Kelly say about complexity, in his Out of Control? (Is that book online?)
• Is there Steven Jay Gould online, about complexity in biological organisms as evolutionary tendency?

Sources and inspiration 

Part of the literature I have read is listed in Wikiversity, on my user page. Popper is an obvious source here. Perhaps Hofstadter. Perhaps Kevin Kelly. I have a book by Gould, but I am not sure it treats of complexity. I wonder whether Dennett Darwin's Dangerous Idea could be relevant (which I have). Surely anthropology would have a lot to say about complexity, ornament and aesthetic response of humans to visual and other stimuli.

Further reading

• https://www.britannica.com/science/complexity-scientific-theory