Yuri Tarnopolsky                                                                                                             ESSAYS                                                                  17. On Complexity

complexity. unified picture of the world. Ulf Grenander. pattern theory.

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Essay 17. On Complexity


Sciences on one side and humanities on the other seem to be separated by a cultural chasm that became obvious to  C. P. Snow  in 1959.  There was no such sharp divide in times of Lucretius (94?-55? BC), Aristotle (384-322 BC) , St. Isidore of Seville (560-636; he is Patron Saint of: computers, computer users, computer programmers, and Internet), Leonardo da Vinci (1452-1519), and, actually, up to the times of C. P. Snow. The scientists in the beginning of the twentieth century were men and women of general humanitarian culture, with interests in arts and humanities, and Albert Einstein is a popular example. Some familiarity with science was also a part of general culture. Science was an aspect of human curiosity and creativity and technology had just started its Cambrian Explosion: dramatic diversification of types of products.


The change around 1960 was, probably, a result of the new role of science and technology and the divergence of the life of Things from the life of humans (see Essay 4, On New Overcoats). Science and technology smoothly wriggled out of the shell of general culture as a separate second culture because of:

     ¤   increased competition for human time (see Essay 2, On the Chronophages or
Time-eaters) between both,
     ¤  decline of the monetary reward for humanitarian knowledge and expertise, and
     ¤  overwhelming complexity and specialization of science and technology.        

In my opinion, the divide between sciences and humanities is not absolute. The shared human language unites the two cultures like the language of genetic code unites all living forms.

...Conductivity, wavelength, voltage, electrophoresis......temperature, entropy, energy, order........class struggle, revolution, domination......gene, meme, DNA, selection ........virtue, happiness, suffering, duty, love, perfection...


The vocabulary of sciences and humanities spreads from narrow and highly special terms of natural sciences, like tensor, mitochondria, and quark, to the words of strictly humanitarian usage, like guilt and hubris (they might be appropriated by physics in the future).

At some historical point, a word of the common language (for example, charge, decay, resistance) was selected as a scientific term, usually, for the reason of analogy. Other scientific terms were originally invented for internal use, but later infiltrated humanities and common language (entropy, diffusion, algorithm) for the same reason.  Latin and Greek roots went both ways, retaining their general meaning. Thus, the Latin posse (have power), gave potential, power, possibility, and impotence).

I believe that if there is a substance of the unified knowledge, it is analogy and metaphor. I believe that at a certain level of abstraction, a large picture of the world can appear, which we cannot see by using narrow terms. Accordingly, if we use either wide and vague or exact but very abstract terms, we cannot see the details of the picture. It is a tradeoff. We are not trained to see the whole because our education and division of mental labor reflects the historic evolution of knowledge with its diversification and specialization into philosophy, literature, physics, biology, and thousands of narrow strips.


Human nature, living nature, and physical nature are separated only in our mind. For an observer from Mars, they both are the Nature of the Earth, but only because of a big distance.

Charge, energy, power, speed, acceleration, resistance, competition, even culture (microbiology and anthropology), strange and charmed (quarks), together with scores of other words, are in common use by both sciences and humanities. Although theoretical physics tends to break with analogy, even the color of quarks is not just a nostalgic artifact of sensual perception but a meaningful analogy with the three basic colors.


The unified picture of the world is in the state of a permanent growth, like a regenerating tissue covering the lesion. To watch pieces of this jigsaw picture join and fuse has been my major single passion. Strangely, the picture has been getting only simpler with time. But you can never make money on anything simple except aspirin, and the adepts of a unified world picture will crouch somewhere below the English Major.

There is another ambidextrous concept that overstepped the divide from the humanities to natural sciences: complexity.

It seems that the complexity of modern life is as oppressive as a humid hot July day in a big city. Simple living  becomes a dream, but a related Web site looks like a window into complexity.


Regulations, laws, rules, tax code, OSHA and EPA requirements, paperwork, documentation, bureaucracy, special interests, political correctness, politics, economy, technology, computers, programming, education, science, air transportation, parking space, ethnic fragmentation, ethnic sensitivity, world community, international relations, police activity, globalization, dealing with protesters, Arab-Israeli conflict, ethics of medical research, spread of AIDS, religious influence on secular life, and countless other issues are components of modern complexity.


Fortunately, the growth of complexity is partially offset by its loss. Thus, the relations between people seem to drift toward simplification. The loss of loyalty, for example, takes a good deal of complexity fall off our shoulders. The topic of the loss, however, better suits a separate essay (see Essay 34. On Loss).


We can certainly solve all the problems, except finding parking space downtown, by having a czar with full power for every problem that cannot be solved by a town meeting. We would simply overturn any czar who acts against  majority. Social complexity, therefore, displays between the simplicity of  absolute dictatorship and an ultimate democracy.


What is complexity? What is more complex and what is less so? How to measure it?


The subject turns out to be very complex. Complexity today means:

       ¤ a particular science about large dynamic systems, strongly impregnated by
       ¤ difficulty of understanding (i.e., amount of work needed for understanding, which is not
                    as shallow as might seem)
       ¤ the static property of being complex, often, in a very narrow aspect, like complexity
                    of calculations and computer programs, but also in a wide view, like complexity
                    of a civilization..

A host of definitions can be found and to review the subject would take a book. I will limit myself to references to Murray Gell-Mann, John Horgan, and Chris U. M. Smith.  None of the sources on the Web or otherwise seems satisfying to me as far as the big picture is concerned.


I am going to present my understanding of what complexity is by playing with a set of nine Lego-like blocks that can be connected in various ways. 


First, let us examine a block:

This is a description of the block:

This block is a square with four connection points shown as red and blue dots, according to the picture. The block has its top and bottom, as the fill shows.

Here is another block:

It differs from the first in the location of the dots. Here is yet another  block:

  This is how two blocks can be connected:

  These are rules of connection:

                The blocks connect by touching with the dots of the same color.

    They do not rotate in the plane. They do not flip.

  Here is a combination of all three blocks :

  Here are some other combinations:


The blocks and the rules of connection define a space for all possible combinations and  form a kind of a creative system for producing combinations. We will call the system of the above three blocks and the rules of connection SYSTEM 1. It has a certain complexity that we don't know how to measure. We can compare, however, two systems that do not differ much, i.e., one is produced from the other in a small step.

This is how it could be done.


Let us form a new system by adding four new blocks, without changing the rules.

Although we add more blocks than there were initially, this is a small step. If it seems big, we can add them one by one. If we changed the blocks and rules at the same time, it would be a bigger step. But we can always divide a change into minimal steps (an assumption).

The new blocks (not the system yet) are more complex than the previous three because they have more kinds of dots: three colors instead of two. Therefore, the new system, let us call it SYSTEM 2, is more complex.

SYSTEM 2 with seven blocks is more complex than the system with either three or four blocks because it uses more different types of blocks, some of them more complex.

This is just one example of what  we can make of the seven blocks.


 Naturally, the number of  combinations in SYSTEM 2 is larger than in SYSTEM 1.

 Now let us try something quite new: a mutation of the shape. We add two identical triangular blocks:

 ......................                                       ...
  There is only one way we can connect such triangle with only one of  the seven:


This SYSTEM 3 of nine blocks is more complex than SYSTEM 2 because it has more types of blocks: square, as well as triangular.

 SYSTEM 4 comes next, in which the rule of matching colors is relaxed and the following combination, for example, is possible (it does not use all nine blocks):


SYSTEM 4 is less complex than SYSTEM 3  because it has less rules, even though it generates more combinations.  If dots of all colors are equally  connectable, then they can be reduced to just one color, which makes the blocks much simpler.

We can have more combinations by allowing rotation of the blocks in plane (SYSTEM 5). We will require, however, the sides of the blocks be approximately parallel.

The next two combinations by the rules of SYSTEM 5 look rather complex:




In fact, with the relaxed rules of connection, we can eliminate not only the colors, but also the fill that distinguishes between the top and bottom of the squares, and even the dots, so that the above combinations look very trivial:

 A completely chaotic system, without any rules and with simplest blocks, is very simple. The billiard balls form such a system on the pool table.

 One can try designing various Lego-like systems and studying their complexity. The Microsoft Draw, which is a Microsoft Word function, is very convenient for this. It allows for a rather high complexity of combinations. I drew a picture....couple (120K) made of two kinds of blocks: closed lines and fills. At this point I let the reader guess what the rules for the combination were.


To summarize, instead of trying to evaluate complexity of a single object, we do it for the system that generates it. All the objects generated by the system are of the same complexity, however they look.  It seems to contradict our intuitive concept of complexity because large combinations look more complex than a single block or a couple of blocks. We should not mix up size and  complexity, however. Still, we can take the number of different blocks in combinations within the same system as a measure of a partial complexity within the same system.

The type of connection can vary, too. For example, it can have a direction and require two different types of connecting points, which would add to the complexity of the system :


I do not believe it is possible to find a universal numerical measure of complexity for everything in the world. Moreover, it is not necessary. Instead of measuring complexity of an individual object by a number, we compare any pair of systems and by transforming one into another, we can trace the number of steps that reduce or increase complexity. We may not have an exact measure in complicated cases, but we can still have a good idea about the difference between two systems.   We might have even a scale of complexity by selecting a zero point.

In this concept of complexity I use the same principle that Confucius used to quantify human virtues: by comparing two selected individuals and thus establishing a partial order in moral values (see Essay 13. On Numbers ). It is the same approach as with the beauty contest. There is no absolute numerical measure of beauty but we still can run a beauty pageant by ordering the contestants.

 This concept of complexity, which I would call pattern (or chemical) complexity, is based on fundamental ideas of Pattern Theory developed by Ulf Grenander. It is not limited to static structure and can be applied to dynamic systems where transitions from one static structure to another take place according to a separate set of dynamic rules.

I came to Pattern Theory from chemistry, which can be considered as an application of Pattern Theory.

The main idea of Pattern Theory is that most (if not all) of our knowledge about the world can be presented as blocks connected with bonds. This atomistic principle comes from deep antiquity, from Democritus, but Ulf Grenander developed the simple ancient idea into a rigorous mathematical (and by no means simple) edifice. The ideas, implications, and applications of Pattern Theory are inexhaustible. I tried to outline some in my manuscript The New and the Different.

Ulf Grenander and I attempted to apply Pattern Theory to history in our History as Points and Lines, with no prospects of publishing until history changes its course.

Pattern theory covers not just ethical systems, beauty contests, and Lego, but also social relations, biological forms, digital anatomy, molecules, genealogies, language, philosophical ideas, personal relationships, and Everything in the World.
Pattern Theory presents a universal abstract language for describing any system of any complexity. Therefore, it is also a language for describing evolving complex systems.   It does not discriminate between sciences and humanities.

One way to solve a problem is to appoint a czar. A town meeting is another one. The third one is to ask a sage.


1. Researchers involved in measuring complexity can compare the pattern complexity  with Kolmogorov’s complexity and its controversial implications concerning random sequences. In pattern complexity, anything without rules is inherently simple.

2. In authoritarian society some blocks have many copies and are interchangeable, and others have a few or one. In liberal democracy, ideally, all blocks are unique, but the function of bureaucracy is to keep the uniqueness down.

3. The following is the inventory of the individual blocks used in this Essay. They don't have either names or numbers. They are their own symbols, like pictograms in hieroglyphic script. By writing particular terms on the blocks and specifying rules, we can construct particular systems in various fields of knowledge, from poetry to molecules.


4. Pattern Theory:  Ulf Grenander, Elements of Pattern Theory, 1996. Baltimore and London: Johns Hopkins University Press.
Ulf Grenander, General Pattern Theory, 1993. Oxford: Oxford University Press.

P.S. (2016) It occurred to me while re-reading that complexity, like energy, does not have any absolute measure and can be only compared for two objects. It is, probably, an evidence of the fundamentality of the concept of complexity. But I can only raise the question whether a unit of complexity exists. I doubt it does. In Pattern Theory, it depends on a subjective knowledge and choice of an observer of images, templates, and configurations. Complexity is a bridge between sciences and humanities. It is mathematics with an anthropic principle: it makes sense only because humans exist.


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