Emergence and Design
When looking at a painting, you can be sure that every discrete aspect of it was intentional: the artist placed every single stroke onto the canvas as a deliberate act. Everything in a painting is by design.
On the other hand, on areas other than the Arts, there are observations that were not explicitly designed, but instead arise as a consequence from a simpler core set of rules. This is what can be called emergence.
Take, for example, the prime numbers. The definition of what a prime number is, by design, “a natural number greater than 1 that is only divisible by 1 and by itself.”
As an emergent property, there are infinite prime numbers. Nowhere on the definition said that this would have to be the case. And how this follows from the definition is very understandable:
1. Assume there is a finite number of prime numbers. Those could be enumerated as { p1, p2, ..., pn }.
2. By construction, take q = p1 p2 ... pn + 1.
3. q is not evenly divisible by p1 (because the remainder of the division of q by p1 is 1). The same argument is made for each i from 2 to n, following that q is not divisible by any prime number other than 1 and itself.
4. From (3) and the definition of prime numbers, q is a prime number.
5. Finally, q, a prime number (from 4), is not present in the list of all prime numbers on (1), reaching a contradiction. Therefore, (1) must be false, meaning there is not a finite number of prime numbers. There are infinite prime numbers.
The proof above is a classic and the result is first recorded on Euclid's Elements in 300 BC.
The proof is more simply that, the only way to not introduce a contradiction into the mathematical system of rules is to accept the fact there are infinitely many prime numbers.
This was not designed for, but, I argue, emerged as a natural consequence of the very definition.
There are plenty of examples in mathematics of observed complexity arising from seemingly trivial definitions. Another very popular example is Conway's Game of Life.
Ever since Darwin published “On the Origin of Species” in 1859, there is public discourse about the Evolutionist and the Creationist theories of how Nature came to be.
In his seminal work, Darwin postulated a simple set of core rules: (1) genetic hereditability; (2) random variation; and (3) natural selection; from which all biological complexity could emerge.
These rules are consistent with fossil records and current-day observations of geographically isolated species. To this day, Evolution is the most successful scientific theory which explains the diversity and complexity of life on Earth.
By and large, the precise line of reasoning I hope to have conveyed in this piece is that complexity is no evidence for design.