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Hmmmm.  Let me try and dissect this and the above a bit, drawing on How Babies Think, a recent article by Alison Gopnik, a developmental psychologist.    

"children's brains ... must be unconsciously processing information in a way that parallels the methods of scientific discovery."

What contemporary research is suggesting is that babies are born with unconscious presumptions about the world and an unconscious mechanism for revising/updating those presumptions.  The latter involves somewhat randomly generated outputs ("play" or "experiments") that result in new inputs.  The latter are compared with expectations based on the existing presumptions and lead to revision of those presumptions when the input doesn't match expectations. To put it differently, babies are born with a pared down version of loopy scientific method and it is this that accounts for their quite remarkable learning capabilities. 

So where is the Turing machine, or formal systems, or "properties and rules," or consciousness, or free will?  Or the brain/neurons for that matter?  Its actually fairly easy to construct neural networks that will implement all of the features of the loopy scientific method, so that's not a big problem.  And one could, in principle, mimic all this on a universal Turing machine if one knew in advance all the outputs a baby will generate and all the inputs it will get as a result.  But that may well be missing the point (see Can a computer play Jeapordy as well as a human?).  Arguably the human brain is not designed by evolution to achieve a particular outcome but rather to respond adaptively to a somewhat unpredictable world.  And that in turn suggests that the brain is designed to be open to the world (see Doug) rather than, like a Turing machine, to be self-contained.  And it suggests that the brain is designed to work in terms of probability, to both detect and make internal use of stochasticity, rather than, like a Turing machine, to operate deterministically in terms of true and false.  And it suggests that the brain is designed to process information simultaneously in multiple modules, without worrying to much about consistency among them. 

If all this is so, where do formal systems, or properties and rules, come from?  My guess is that these, rather than being the essence of how the brain works are add-ons, things that appear when on begins to make sense of what the unconscious is doing, when one becomes conscious, when one starts to reflect on what one is oneself doing.  And they clearly add substantial exploratory capability: by constructing formal systems, by trying to reduce one's experiences to properties and rules, one opens up new possible worlds that would not have been reached by one's starting presumptions and experiences alone.  Conscious thought is useful.  But thinking of things in terms of formal systems, of properties and rules, can also get in the way.  As Gopnick notes

"There is a trade-off between the ability to explore creatively and learn flexibly, like a child, and the ability to plan and act effectively, like an adult."  

Perhaps the most important idea that comes out of this is that we don't have to choose between formal systems and something else, between properties and rules and chaos, between the conscious and the unconscious.  Its not a choice but a tradeoff, one we can make and remake depending on the what's going on at any given time.  And perhaps that's the opening to free will?  We're not bound by any of multiple ways of doing things that we have inside ourselves but instead have the capability to switch among them.  If we learn about them all and about each of their strengths and weaknesses.  Maybe that should be conceived as the objective of education?  To enhance the capability of adaptive change over time?