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Does one have to be aware that one is exercising one's free will, in order to have free will?

Hard to see why, in my opinion. If (say) a free action is one that you undertake such that, at the moment of acting, it was at least logically, and perhaps even physically, possible that you either perform that action or not perform that action, those facts themselves at least seem to be independent of what your awareness is. What would be interesting is an argument that shows that only IF one is aware of the facts just described could those facts obtain ... but at the moment I don't see how to generate such an argument. Perhaps in the mix here is the thought in the other direction, a kind of old-fashioned argument for free will, that states that if one believes one is acting freely then one IS acting freely--or using our conscious experience of (or as of) acting in a way in which it seems to us that multiple options are logically and perhaps physically available as a sufficient condition for acting freely. (Your question concerned whether such awareness was a necessary condition, but here it is offered...

Is there a role of mathematics in the development of human consciousness?

interesting question. not sure I understand it exactly. but I can refer you to some fascinating work that touches on it -- mostly anything by Douglas Hofstadter, but you might start with Strange Loops and/or Godel Escher Bach ... both spectacular works that trace the essence of consciousness to self-referential recursive (mathematical) processes ... he'd be a good place to start. best, Andrew Pessin

Why is it so difficult to accurately discuss consciousness? People have been fumbling around with strange thought experiments and neologisms like "qualia" for a while now, yet there still doesn't seem to be any clear language to use while discussing the "hard problem of consciousness". The closest I can get is to frame the question using a computer analogy. A computer can compute, and then it can provide output to show a human user what it's computing. Our minds seem to be providing "output" that we might call consciousness or experience; why isn't it just computing in the dark? Yet even this analogy seems clumsy and inaccurate. So what makes consciousness so uniquely impossible to discuss in a clear fashion? I've never come across any other topic where language itself failed to grasp the subject of discussion.

That's why it's called the "hard problem".... :-) And perhaps that's why some philosophers take an 'eliminationist' or materialist approach to the subject (eg Dan Dennett, Paul Churchland): the 'problem' itself is so ill-defined because there really is no such phenomena to be analyzed ... Words like "consciousness" are so vague and unclear that they cannot constitute a subject of scientific investigation, which will eventually dispense with them altogether ... That's not very helpful, of course, but then there is no real answer to your question ....! best, ap