Consider a system composed of many interacting parts. I take the defining characteristic of an emergent property is novelty. That is, the whole has a property not possessed by the parts alone. I argue that there are five other characteristics of emergent properties. These characteristics are common but they are neither necessary nor sufficient for novelty.
1. Discontinuities
2. Unpredictability
3. Universality
4. Irreducibility
5. Modification of parts and their relations
I now add another characteristic.
6. Diversity
Although a system may be composed of only a small number of different components and interactions, the large number of possible emergent states that the system can take is amazing. Every snowflake is different. Water is found in 18 distinct solid states. All proteins are composed of linear chains of 20 different amino acids. Yet in the human body there are more than 100,000 different proteins and all perform specific biochemical functions. We encounter an incredible diversity of human personalities, cultures, and languages.
A related idea is that "simple models can describe complex behaviour". Here "complex" is often taken to mean diverse. Examples, how simple Ising models with a few competing interactions can describe a devil's staircase of states or the multitude of atomic orderings found in binary alloys.
Perhaps the most stunning case of diversity is life on earth. Billions of different plant and animal species are all an expression of different linear combinations of the four base pairs of DNA: A, G, T, and C.
One might argue that this diversity is just a result of combinatorics. For example, if one considers a chain of just ten amino acids there are 10^13 different possible linear sequences. But this does not mean that all these sequences will produce a functional protein, i.e., one that will fold rapidly (one the timescale of milliseconds) into a stable tertiary structure, and one that can perform a useful biochemical function.
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