So it is still merely a conjecture on Wolfram's part.
Granted.
On the other hand why does it make a difference whether or not there are truly random events just so long as there is effective randomness from complexity. As I pointed out earlier you couldn't tell the difference between tombola randomness and QM randomness.
One reason for bringing up the possibility of true randomness from deterministic rules is to (possibly) object to the assertion that "if you allow true randomness in QM, then determinism is already false" as has been claimed above.
I say "possibly" because if "determinism" means the clockwork universe, then I do believe that's false. If "determinism" means that everything arises from fundamental processes that can be described by deterministic rules, then I have no reason to believe that's false.
I do think it makes a difference whether we're talking about true randomness or seeming randomness (if we define randomness as deviation from deterministic reaction, simply b/c if only the latter exists then free will becomes impossible, even if the illusion of free will remains possible).
In other words, for our purposes here, it doesn't seem that human inability to discern determinism is an adequate test. So if tombola randomness is actually just determinism that's so complex we can't tease it out, that's significant relative to this discussion. But perhaps that's not what you meant -- correct me, please, if I'm misinterpreting.
Back to your equation, I see a potential problem with the condition "Let's call the precise state of a human's mind at the point of the decision
X". If "at the point of the decison" is defined as the point at which the decision is made (in itself a possibly impossible "point" to define, but let's ignore that for now) then there is only one resulting value.
So how about this.... let's call the precise state of a human's mind at the point when s/he begins to consider the decision
X. Then we have some room for the path from
f(X) to
a or to
b, which I think more accurately describes the situation.
You ask, "Would there be anything at all that accounted for the difference between decision outcomes
a and
b or would this difference be purely arbitrary?"
I'd be a liar if I said I knew for certain.
My brother calls to say that my estranged father is dying. It's a medical certainty that he will die within 2 days, although he will likely live at least another 12 hours. He's conscious and will probably remain so for a day. It takes 4 hours for me get to the hospital.
Do I try to see him before he dies? Simple
a/
b choice. I leave or I don't.
Is my ultimate decision determined by random firings in my brain, even if I can't perceive them? Is it determined by non-random firings in my brain mapped out by the rigid laws of physics, even if I can't perceive that either? Is it determined by an act of will? Is there an interplay of these?
No matter which decision I reach, it certainly will seem, in hindsight, as though I could have chosen the other. It will seem that the other was not impossible.
And because I have no basis for declaring that either choice is, in fact, impossible, I therefore concede free will. Perhaps I'm wrong, but until a mechanism is shown, I side with free will (somewhere Neil Peart is smiling).
Is it arbitrary, purely random? I don't think so. Then what accounts for it? I believe that the emergent phenomenon of individual conscious processing accounts for it. I believe that this "weighing of things" that we do, this process of decision-making, is very real, and in cases when we're not simply reacting, when we don't feel overwhelmingly compelled, it could go either way.
Of course, I could be wrong.
But either way, it's very clear that the Modern Synthesis accounts very well for how beings with such capacity of choice could arise. And more than that, it accounts very well for why we should feel the need to choose in situations such as this... why this situation should cause me more distress than, say, a choice between the piece of pie closest to me, and an identical one farther away on the cafeteria shelf.