Free will emerges

Free will and notes on The Beginning of Infinity, Chapter 5
Arguments against free will tend to make the reductionist assumption that, because there isn’t a physical explanation for human freedom, it must be an illusion. Broadly, reductionism is the idea that the world can only be explained as the sum of interactions of its fundamental parts. For example, the high-level properties of a substance (such as boiling point or state at room temperature) can be predicted from low-level atomic interactions. Lots of these sorts of ‘reductive’ theories are true and useful; they have reach. Newtonian mechanics was a ‘reduction’ of Kepler’s laws of planetary motion and Galileo’s theories of motion. But reductionism as a methodology is mistaken. Science doesn't progress by analysing high-level phenomena into low-level phenomena. There are lots of high-level phenomena that, while they ultimately consist of low-level phenomena, have patterns and laws that are present on the higher level but not, so far as we know, on the lower level.

Douglas Hofstadter gave the following as an example of the inadequacy of reductionist explanations for emergent phenomena. Imagine a set-up of millions of dominoes, placed closely together (so that they can knock each other over in the usual way) in a complex pattern of rows. The dominoes are spring-loaded and, if knocked over, will pop back up after a set time. If a row is knocked over it can be interpreted as a binary ‘1’, and if not knocked over, a binary ‘0’. The set-up is sufficiently rich and complex that it can perform computations, and in this instance it is set up to tell you whether the number keyed in (by placing a row of that number of dominoes at a specified position) is a prime. One domino in the set-up is the output domino: If it is knocked over at the end of the computation, it means a divisor was found and hence the input number is not a prime. If it stays standing, the input number is a prime.

Now, if the output domino stays standing after one of these operations, an observer may single it out and ask: “Why did that domino never fall?” The reductionist explanation would be: “Because the domino behind it never fell, because the domino behind that domino never fell, because…” – and so on, or: “Because none of its neighbours ever fell, because none of their neighbours ever fell,” and so on.  This answer is true, but it is not an explanation: It merely states the already obvious fact that no domino will fall unless one of its neighbours falls. To explain why the domino didn’t fall, we have to make reference to the non-physical concept of primality and to the dominoes’ emergent capacity to say whether a given number is a prime.
So reductionist 'explanations' are inadequate for some emergent phenomena. Even if the series of physical events described by the reductionist account did actually happen, such emergent patterns have to be explained on their own terms. If human freedom is emergent, it doesn’t conflict with physical determinism -- it just has nothing to do with it.

There is a reductionist idea that the mind cannot affect the physical world, on the grounds that only physical events can cause other physical events. But the idea of a ‘cause’ is abstract; at the purely physical level, cause and effect are interchangeable. The laws of motion can retrodict as well as predict. So a cause is just an explanation we infer for why something happened, and there is no reason to think that physical explanations are the only explanations we have.

For more, see David Deutsch, The Beginning of Infinity, Chapter 5.

Popper’s three Worlds
To describe the emergence of human knowledge, Popper posits three worlds -- World 1, the world of physical events; World 2, that of mental states or subjective experiences; World 3, that of products of the human mind: theories, scientific (or otherwise intellectual) problems, the information (if not the physical stuff) in books and libraries. It used to be popular to deny the existence of World 1 and to claim that only subjective experience exists. Now it’s more fashionable to claim that only the physical world exists and to deny the existence of experience. The once-popular denial of the reality of World 1 was refuted by Dr Johnson on the grounds that it ‘kicks back’. If the physical world is an illusion, and that illusion behaves exactly as though it were real, then it needs to be understood and explained in exactly the same way as would a real physical world. It kicks back, and so it might as well be considered real.

To explain the physical (World 1) presence of man-made objects -- say, skyscrapers, computers, nuclear reactors -- we have to refer to the theories people formed about how to produce them. These World 3  theories could also be mistaken. A miscalculation might lead to the World 1 event of a bridge collapsing. World 3 has a logical structure that exists independently of humans. For example, though natural numbers are a human invention, facts about them can be discovered, such as the existence of primes.

The ‘human’ dimension of World 3 also kicks back. We make judgements about people’s beliefs, values, habits, etc., and explain their actions as responses, based on those things, to a given set of circumstances. I might be convinced that my boss is going to sack an incompetent colleague, but then discover that, because my boss is the forgiving type, the colleague has instead been put on a training programme. Using the laws of physics to explain all of these things reductively would be absurd. It would be as much as to say that they don’t exist, even though they kick back. If someone thinks that 4+4=9, he will be kicked back by the laws of mathematics. Similarly, (and quite rightly), people who deny the existence of free will are kicked back by the contradiction inherent in proposing such arguments as things that people ought to favour (to choose) over other theories.

For more on the three Worlds, see Karl Popper, The Open Universe, Addendum 1: Indeterminism is Not Enough.