[Piers]

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1) prove that determinism exists.

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If you want to know what you will do in the future, just go to the appropriate point in space-time and see. Whatever you do, is what you were always destined to do. There is nothing you could possibly have done to change this.

The universe is its own model. A list of everything that has is and will happen. To see the future just go look it’s as simple as that.

However I should add that its my belief that its impossible to improve on this. To be able to predict the future with 100% accurately before hand is not possible, at least not without in effect reconstructing the whole universe, but I do not that believe that a being can achieve this either directly or indirectly.

LaPlace's Demon cannot be shown to exist, so his argument disappears in a puff of smoke.

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2) illustrate free will without determinism.

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Free will exists at the level of human experience. If we construct a model of the universe to examine the details of interpersonal behaviour of humans, it makes sense to include the assumption of free will into the model. This does not mean we believe that the assumption of free will is a true and accurate reflection of reality. It just means that the assumption of free will is useful for our current purposes.

When we create models of the universe, we do it to achieve certain ends and it is those ends that are important. It is perfectly acceptable to make assumptions known to be false if they make the ends easier to achieve.

Consider the large amount of our current technology base that has been developed using principles of Newtonian mechanics. People, who firmly believe that the assumptions of Newtonian Mechanics are wrong, still use the theory successfully.

We can only reason within the models we create. It is impossible for us to accurately model the universe. It makes sense to construct the models we reason within to reflect practical needs rather than impossible fantasies.

It is very useful to assume humans have free will, irrespective of whether they do or not.