I know this has been discussed before, but I have read Sam Harris' book Free Will and Michael Shermer's book The Believing Brain, and I must say that I agree with both authors. Studies show that our brains make a decision on an unconscious level three tenths of a second and sometimes more before we even consciously know we're going to act. To take a short quote from Shermer's book: "The neural activity that precedes the intention to act is inaccessible to our conscious mind, so we experience a sense of free will. But it is an illusion, caused by the fact that we cannot identify the cause of the awareness of our intention to act".

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A few facts may help clear up some of your questions.

The first one is the uncertainty principle. The uncertainty principle is given by the equation:

ΔxΔp ≧ ℏ/2

where Δx is the uncertainty in the position x of a particle and Δp is the uncertainty in its momentum. The right half of the equation is a small constant. For ordinary everyday objects momentum is relatively small. (Momentum is the product of mass and velocity.)
So that, while the inequality holds, it is not a practical limitation on measurement. However elementary particles move at extremely high velocities and have considerable momentum on the experimental scale even though their mass is small. Consequently the uncertainty becomes a significant factor in the physics of elementary particles. This is precisely what your quote from Hawking says, in different words.

The second fact you need is a more accurate description of the double slit experiment. When a single electron is fired, a single dot appears on the detection screen behind the double slit barrier. Each time one is fired another dot is added to the picture. After a large number of particles have been fired and the screen has collected a large number of dots, a pattern in their distribution emerges. This is the kind of pattern that the interference of waves would produce. The conclusion is that the particles also have a wave character which is what is behind this interference pattern:

The above shows the progression towards this pattern as the dots accumulate. In the first three—a,b, and c—no pattern is apparent, but in d it shows clearly. (The whole picture is not visible, an artifact of the narrow width of this post.)

The pattern is created by the interference of the particle waves, not by any force. If only one slit is open, no interference pattern occurs, just the expected distribution of dots around the slit.

Maybe that is enough physics for now. There's more of course, but I think these two things can help to clarify what is going on.

Here is the whole picture showing the interference pattern in the last frame.

I'm familiar with these images. I find them very interesting, but what about the wave-particle duality? Do not the electrons behave like a particle when observed, but create the interference pattern when unobserved?

I find it strange that you can't many videos on this experiment. I've searched Vimeo, YouTube, Google, etc. YouTube will just give you a lecture by the computer animated "Dr. Quantum" or a Yale discourse on such experiments. I know "Through the Wormhole" filmed this process, but there's not that many.

You said, "The pattern is created by the interference of the particle waves, not by any force." Isn't the momentum of the electron shooting from from the apparatus a "force"? I'm not quite sure what you meant here by "not by any force."

I want you to leave you with two quotes that maybe you've come across:

"It will remain remarkable,  in whatever way our future concepts may develop, that the very study of the external world led to the conclusion that the content of the consciousness is an ultimate reality." -Eugene Wigner

"Science cannot solve the ultimate mystery of nature because, in the last analysis, we ourselves are part of the mystery that we are trying to solve."

- Max Planck

It's almost as though the physicists have merely acknowledged that when matter or light are unobserved, they behave as waves, but when observed, they behave as particles. Is this truly an explanation?

Do not the electrons behave like a particle when observed, but create the interference pattern when unobserved?

Hitting the recording device counts as being observed. If the double slits are monitored so that it's known which slit the electron went through, then the electrons don't interfere.

It's almost as though the physicists have merely acknowledged that when matter or light are unobserved, they behave as waves, but when observed, they behave as particles.

When something is "observed" - interacts with a big environment - its state becomes entangled with the state of the environment.  This leads to the something being in a definite state - at least, as we observe it.  Since we ourselves are part of that environment. 

There are different theoretical interpretations of this "quantum state reduction". 

Do not the electrons behave like a particle when observed, but create the interference pattern when unobserved?

No, that is not the correct interpretation of the experimental result in these pictures.

Isn't the momentum of the electron shooting from from the apparatus a "force"? I'm not quite sure what you meant here by "not by any force."

No, momentum is not a force. You need some elementary mechanics. There are four known forces in physics—weak, strong, electromagnetic, and gravitational.

It's almost as though the physicists have merely acknowledged that when matter or light are unobserved, they behave as waves, but when observed, they behave as particles. Is this truly an explanation?

This is not what any physicists are claiming to my knowledge. It seems a distortion of the so-called Copenhagen interpretation of wave-particle duality.  There are several different explanations of wave-particle duality and it is unquestionably one of the most philosophically interesting issues in physics, but much too large a topic for here. Duality is not present in the theory of David Bohm, which you mentioned earlier.

Perhaps you've mentioned it before, Allan, but do you have a position in regards to determinism/free will/incompatibilism, etc.?

I did mention it earlier in this thread. I believe it's on the second page of responses, but I don't claim it is a really well thought out position.

I don't believe it is possible to establish either free will or determinism rigorously, but I am inclined toward the possibility of a measure of free will based on our direct experience of it— with the caveat that strong and perhaps unconscious impulses often direct choices and that learned behavior repeated many times can become unconscious. 

I like Dennett's book Freedom Evolves.  It's an excellent analysis of free will.

I'd suggest that free will is logically incompatible in both a deterministic universe (entirely causal) as well as an indeterministic universe (one where acausal events happen). In this way the hard incompatibilist position is stronger than the hard determinist position in that it addresses both possible worlds. ;-)

Daniel Dennett takes the compatibilist position. Do you disagree with this point-of-view?

Yes, I disagree with Dennett.

I think Dennett (and other compatibilists) defines "free will" in a way that most people don't "feel" they possess when they hear the words "free will". He basically moves the semantics away from the definition that holds all philosophical import. In other words, people think that they and others could have, of their own accord, done otherwise...and it is this that is tied to so many other important topics (economics, ethics, religion, politics, criminality, social, the way people feel about others, etc.).

I think the compatibilist usage of "free will" is no different than defining "god" as "nature" or "the universe". Sure, anyone can do so...and that particular definition of "god" surely would exist. But it's not helpful, not needed (in fact it takes away from the productivity of the topic), and it's just a way to avoid the more common types that most people believe (and a way of avoiding the issues around such a belief).

It also allows people to hold on to thinking they possess an ability that people like Dennett would, when pressed only, say they wouldn't.

I also think Dennett contrives various points surrounding the free will topic. For example, his usage of the word "evitible" is a play on words (I go over that in my book). 

Let me explain your "free will." Your brain records your experiences and events in your life. Somewhere within your brain are all of these experiences. When you have a choice to make you chose from the most likely of the stored events. This is almost a subconcious effort. It gives the illusion of "free will" and your choice might even be from something similar that worked for you in the past.

Now consider that the above is about one person only. Every other person would have went through this also, but they do not have the same experiences that you had. To do so they would have had to of been you. This opens up a whole set of other experiences giving some people more choices than others.

Not only does this define "free will" in terms we can identify with, but it makes true free will next to impossible!

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