Snowflakes are elegant examples of both simplicity and complexity in the natural world. Surely we all remember learning as children that no two snowflakes are the same. Every flake has a unique and complex pattern. But then as we moved from elementary to high school and learned the basics of chemistry, we discovered that snowflakes are also simple. In fact, at the molecular level, they’re identical: they are all frozen water. How can this be? How do identical water molecules form billions and billions of unique flakes? The answer lies in the interplay between natural laws and chance. The regularity of natural laws makes each snowflake a beautiful six-sided crystal. The random motion of the individual molecules in the air makes each flake unique.
I’m fascinated by natural systems which become more complex over time via the interplay of law and chance. And snowflakes aren’t the only example. A small seed, over decades, can grow into a full-grown tree that is far more complex than the seed from which it began. Newly formed barren volcanic islands can be colonized by lichen and algae, then by more complex plants, then by animals that can swim or fly, gradually forming a complex ecosystem over millions of years. And a single fertilized egg cell, in just nine months, can become a living, breathing, human baby.
More complex things take more information to describe. A single water molecule can be described with a small amount of information; describing an entire snowflake requires much more information. A single seed might required a lot of information to fully describe, but a full-grown tree would require vastly more. The DNA in a single fertilized egg cell, encoded on paper or in a computer file, is several billion bits (“gigabytes”) of information. The information required to completely describe an entire cell is still greater, and the information required to describe an entire baby is far, far greater still. How is that information created?
Some advocates of Intelligent Design theory argue that the ordinary mechanisms of evolution cannot significantly increase biological information—specifically the information of DNA and protein sequences. They argue that going from relatively simple living organisms, with small genomes and small numbers of proteins, to more complex organisms required God to act in ways beyond ordinary natural laws and random processes.
Evolutionary creationists, however, believe that God created the biological information in our DNA and protein sequences through the natural laws and random processes that he designed and sustains. In other words, God created biological information through evolutionary mechanisms in ways analogous to how God creates the information needed to describe each new snowflake, each new tree, each new ecosystem, and every new human being.
The theme of this series is biological information. Dennis Venema has already written several posts about biological evolution and genetic information. Here, I want to expand the conversation further to include examples from some of my areas of expertise: mathematics, physics, computer science, and game theory.
There are many kinds of information. Natural laws and chance can create vast amounts of some kinds of information. Natural laws and chance can copy or convert some kinds of information into other kinds of information. So the natural laws and random processes of evolution which create the biological information of DNA and proteins are just part of a larger set of processes which God uses to create and govern this universe.
Some kinds of information have precise mathematical definitions. For example, Shannon information is related to the number of bits of information used for encoding and transmitting messages. Kolmogorov complexity is a measure of the size of an algorithm required to describe an object. The internet has many technical articles describing how those kinds of information are defined and used. I won’t get quite so technical here.
While there are many kinds of information, in these blogs I’ll focus on just four steps of information creation and transformation. I’ll avoid technical or mathematical definitions and stick with intuitive understandings.
- Combinatorics: A few simple pieces and a few simple rules can combine to create a vast space of possibilities.
- Chance: Random events in physical, biological, or computational system almost always increase the amount of information required to describe the system.
- Evolutionary adaptation: As an object or organism adapts to a complex environment, variation and selection can cause information about the environment to be duplicated in the object or organism.
- Co-option: Variation and selection can cause parts of an object or organism to gain new functions, often leading to greater complexity.
For the rest of this post, I'll focus just on the first step: combinatorics. Plastic toy bricks provide a nice example of combinatorics. Let’s say you have a set of 500 bricks, with about a dozen different types of bricks. A description of each type of brick, and a list of all the ways that any two types can snap together, could be written down on about one page of information. However, the number of different ways you can combine those 500 bricks—what I call the combinatoric possibility space—is vast. You could play for a lifetime and never come close to putting together every possible combination of those 500 bricks. Most of the shapes would look like abstract sculptures. But tiny subsets of that possibility space look like toy houses or trees or airplanes or sailing ships. Those potentialities were built into the combinatoric possibility space as soon as the bricks were first designed.
The game of chess provides another example. The board has 64 squares. There are only 32 pieces and 12 different types of pieces. Yet the number of unique ways you could arrange those 32 pieces is approximately 1057. (That’s a 1 followed by 57 zeros. For comparison, there are estimated to be about 1080 atoms in the visible universe.)
In addition to the possibility space of all arrangements of those 32 pieces, the rules of chess—which can be written on a few pages of paper—create another vast possibility space, namely, the space of all possible games of chess. The rules of chess specify all legal moves you could make from one arrangement of pieces to other arrangements. Sometimes the rules are completely deterministic, such as when there is only one legal move. Often there are multiple legal moves, and the player selects one. A computer might be programmed to randomly select from among all legal moves, or perhaps it might select randomly but with probability weighted by how “good” each move is. Starting from the standard opening position of chess, a common estimate is that there are about 10120 different possible games.
Snowflakes are an example of possibility spaces. When God designed the physics of water molecules, he effectively designed the space of all possible ways they could combine into snowflakes. Every time it snows, the random motion of molecules in the air explores a tiny portion of that space.
DNA provides another wonderful example of possibility spaces. DNA molecules are strings of just four kinds of nucleobase molecules (typically labeled C, G, A, or T). But they can be combined in almost any order into long strings. The DNA in mammalian cells has about a billion nucleobases strung together. This means that there are about 10500,000,000 different possible ways to put together a DNA string that long. When God designed the DNA molecule, he also created the vast possibility space of creatures which could be generated by all those combinations.
The natural laws of biochemistry describe the mutations by which a cell can move from one location in the combinatoric possibility space of DNA to another. Living organisms over the long history of life on earth have only explored an extremely tiny portion of that enormous possibility space.
Particle physics provides yet another stunning example of combinatorics. Consider just three fundamental particles: protons and neutrons, and electrons.1 The properties of these three particles, and the mathematical formulas which model how they move and exert forces on each other, can be written on roughly a single sheet of paper. These three types of particles combine into roughly 100 different types of atoms. And those 100 types of atoms combine into a tremendous variety of molecules.
Look around your room and think about every different solid, liquid, and gas in the room. Then think about the thousands of different biological molecules in your body. Each has unique properties. Those molecules combine to form almost everything we see. Yet each is made from a different arrangement of just those three kinds of particles.
I believe that when God designed electrons and quarks and the laws which govern their interactions, God had in mind all the possible things which could be made by combining just those three particles in different ways: stars and mountains, oceans and amoebae, plants and people. Our universe thus far has only explored a tiny fraction of all possible combinations of just those three types of particles.
When we humans create plastic toy bricks or games like chess, where a few simple pieces and a few simple rules can combine into possibility spaces so extensive that we couldn’t explore it all in the lifetime of the universe, we are imitating one aspect of how God chose to create our universe.