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There are in fact 3 different compositions represented in this page.
To me, the correct stance to take on the issue of complex emergence from simpler substrates has already been clearly suggested in considering Conway's Game of Life (GOL). In that "designed universe" (the GOL game board), only the rules of play were designed, while the forms of being which it might manifest were very certainly not designed and could not be planned-for. So with a very simple rule set in an appropriately designed mode of existence, that being the algorithmic and relational space of a 2-dimensional GOL grid, we can see that forms of arbitrary complexity are possible, and that simple and even moderately complex forms are readily emergent. By "forms" we mean patterns in their own identifiable right at a level of emergence obviously higher than the basic atom of that universe. Furthermore, it's also been shown that second-order emergent levels (patterns made of first-level patterns) can emerge spontaneously. And more than that, these emergences can arise quite commonly from random initial patterns. That is, ordered complexity really can naturally derive from random simplicity as long as there is enough "stuff" and time available to supply the requisite processes.
For me it is a very simple matter to understand the early cosmos we inhabit and its means of generating complex and amazing beings as much like a very large, dense, randomized GOL board. We know there was plenty of energy and then matter available, and that the rules were simple and of an appropriate style (more on that style elsewhere). And based on what we observe in GOL, we know that this is all that's required to permit the emergence of more complex forms. And we also know that complex forms can become the substance of building even higher levels of forms, on and on without bound, as long as the matter and energy remain readily available. One may watch GOL and call it a miracle or a mystery, but it's really neither. You can analyze to your heart's content just how particular patterns come to emerge and you can trace it down to the implementation of the specific laws of the game. But this won't satisfy the curiosity about "Why these forms?" or "What do these forms mean?" In fact, and this is crucial to understand, to predict the location, timing, and nature of the higher-order forms that will arise from any given nontrivial starting pattern, the most efficient and only reliable means is to run the program and observe!
Through contemplating Conway's GOL, one learns just how and why it is both OK and correct to simply allow that such forms have their mode of being, are derived from their universe naturally and appropriately, were never designed, and have real persistent existence. There's no designed outside-in explanation for the patterns that emerge. They simply emerge and then their being defines themselves. Their subsequent interactions further define their purpose. It's no more boring nor exciting than that.
Furthermore, there have been many variations on the original GOL model. Some have 6-way relationships instead of 4. Some allow for inter-atomic influence at greater distances. Some have different "probability" factors or other algorithmic differences from computing interactions among the atoms. Some have multiple states per atom, instead of only 2. It's been demonstrated that there are many ways to set up the rules which result in "interesting" emergent patterns. Even so, some rulesets are more stable and interesting than others. But in all cases the rules are amazingly simple, and yet without design and without even the ability to predict them, the most complex and interesting forms of existence in the GOL world can emerge naturally.
It should also be noted, for completeness, that within a GOL world, designed beings can be created, and they can be designed with arbitrary complexity and with arbitrarily grand purposes. (This is much like our world, in which natural derivation occurs, but human design does too.) For example, prime number generators have been designed into standard GOL boards and they function quite well. One may note that designed patterns are not at all like random patterns, and one might be tempted to assert that no random pattern could ever result in a pattern which we might have designed; but that's only partially correct.
Arbitrary degrees of complexity and unbounded levels of emergence are possible in such a world, however, its goals would never bring it to create something like a prime number generator until it had first derived universal constructors; then the universal constructors might come to value mathematics, comprehend prime numbers, and then design a way to generate an unbounded set of them. In other words, a prime number generator will always have design at its root, but it's possible for the creator of that design to be fully derived and itself without a design in its history. Here again we see that designed and derived beings, even if they had similar orders of complexity, are ontologically very disparate.
There are some additional observations one can learn studying GOL which help dispel those objections. Consider:
All that said, the one big difference between designed and naturally emergent patterns is that all the naturally emergent patterns came to persist because they and all their ancestors were functionally competent at persisting. For this reason, we should NOT expect extrinsically designed beings to look or behave much like intrinsically derived beings. It's in this way that designed and emergent forms of being are necessarily, fundamentally, ontologically distinct. While the complexity level of a prime number generator is effectively just as (un)likely to naturally pop into existence in GOL as a human being or a computer is in the real world, the human being truly is naturally derivable (some say even inevitable) over time specifically because it was necessarily loaded with derived intrinsic purpose from the beginning of its derivation history. Thus we can't measure how likely something is to appear in reality simply by gauging its order of complexity. We must also consider its path of derivation. Natural emergent derivation is capable of increasing the complexity of beings from low order to higher orders over time. Design works in the other direction, with the perfect form of the being conceived from the start, and its existence representing a continual war against the human-dreaded "forces of entropy" until it eventually and inevitably fails to meet its extrinsic purpose. This is the nature of the "death" of a designed being. The nature of the "death" of a deeply derived and naturally emergent being is of a wholly different character and significance.
Considering these kinds of truths about how complex and interesting patterns can emerge naturally from a simple and undesigned substrate; and given that our cosmos's rules, while still very simple by our standards, are notably more interesting than those in GOL (for example it's a 3D+time model instead rather than 2D+time, and the number of fundamental states would seem to be at least 6, representing the known quarks); and given that our universe appears to be practically unbounded in spatial scope and is inconceivably larger than the size of any one atom; and given that the number of atomic interaction cycles which have been available to the universe is inconceivably great compared to the most extensive GOL instance we've ever run; it seems fairly obvious that our universe could easily be expected to give rise to patterns such as those we see, without design, and with only the ongoing existence of "experience" to guide it. Furthermore we should expect that in our cosmos, the only way to reasonably predict the future with anything approaching complete accuracy will always be to merely "see what happens", and we can note that any attempt to model the cosmos within any tool which is itself made of the stuff of the cosmos will necessarily fail to measure up to the completeness of the task unless it so overwhelms the subject of its purpose in scope as to make that subject effectively irrelevant.