I'm not exactly sure why people try to use mathematics and physics to try to prove free will...
Human beings obey different laws than math and science. One of the very first posts on this thread talked about humans not having free will because we can not choose to disobey the laws of gravity.
We live in a physical world and we have to obey the laws of that world. Free will has to do strictly with making decisions. While you can't decided to disobey the law of gravity, you can decide to test the laws of gravity...
When standing on the edge of a cliff, there are an infinite number of possible options(as there are with any decision that has to be made). The law of gravity is a law only because every time something is dropped, it falls to the ground.
One of the first things you learn in any philosophy class is that the scientific laws are based on precedence only. Just because the pencil fell to the floor when you dropped it the first thousand times you let it fall out of your hand, doesn't mean that it won't fly towards the ceiling the thousand and first time you "drop" it.
When you decide to "disobey" the law of gravity, you are putting it to the test. Every time a 3rd grader does an experiment dropping a sheet of paper, that same law is put to a test.
In fact, every time an experiment is done to determine whether a hypothesis is a law, you are trying to "disobey" that law.
Also, there was mention of computers being able to make decisions...I'm an electrical engineering student and I spend most of my time writing code for computers to follow.
Whenever you see a computer doing something "clever", it's because a clever programmer told it to do that.
Computers are becoming increasingly "independent", but that's only because human beings are able to make them so.
When a computer has to make a decision, it has a conditional structure programmed into it for it to follow...if x = 3, then do this; if x <0, then do this, etc.
When a human makes a decision, there's an infinite amount influences to take into consideration. A computer would have to be obeying an infinite number of conditional structures, which is impossible because it would have to actively generate it's own programming.
Also, "Decision-making is an algorithmic process, a process of deterministically evaluating the values of alternative possible courses of action, and then deterministically selecting that course of action with the highest value." was mentioned on page three by moving finger...
You can't describe a decision making process as deterministic because if it's deterministic, there can't be any decision...your course of action has already been "determined" for you.
One good analogy that helped me understand free will was that we are all characters in a story...
The book's already been written, but we are within the story and we have to live it in a linear fashion.
This can go hand in hand with the idea that the only way to make objective judgements about a system is to be completely removed from the system.
Because of the flow of time, we have to make decisions to move on. Even the band Rush understood the concept. Even if you choose not to decide, you've still made a choice...
In the second Matrix movie, Neo knows he has to make a certain decision. The oracle tells him that he's already made that decision, he just has to understand it.
As a character in the book, you're locked in a certain position in time, just riding the flow, and your character has to make a decision at the end of chapter 2.
The only way for you to get to chapter 3 is to make a choice, and because time doesn't stop to wait for you, if you don't make an active decision(choose not to decide), then you move on to chapter three anyway.
That's the strongest argument for free will...that you can choose to influence what's going to happen in the future.
Sort of abstract, but it makes sense to me...
Also, say there is a God-like character...by definition He would be outside the system. He is the only objective observer. He might know what's going to happen, what you're going to do, but you still DON'T know what you're going to do, so it doesn't matter.
If there is a God that is all knowing, and He does know what's going to happen...so what? That doesn't affect your position at all...because you still don't know what you're going do. God's not making the decision, you're still the one who does that.
If Bob knows that you're going to choose blueberry pie for dessert, but you still haven't made up your mind yet, what Bob knows is irrelevant because Bob's not making the decision.
Back to my opening line...what electrons and rocks do is irrelevant because you aren't an electron or a rock. To prove that humans have free will or whether their actions are determined, look are what humans do.