I can compress all real numbers into a single symbol: ℝ
There are many future states, but I think it’s a finite number. That’s what it comes down to, I think.
Relevant to the free will or not interests: I swear I just read about (very similar thing) in this short story by Ted Chiang, “What’s Expected Of Us”.
http://www.nature.com/nature/journal/v436/n7047/full/436150a.html
The main problem with this question for me is that it relies on the false dilemma fallacy. People made a choice, but clearly sometimes thought they had made a choice when in fact they had been prompted by the circle turning red on the monitor. So our memories are often susceptible to being rewritten - we already knew this. Our choices are susceptible to being manipulated subconsciously. Again, this isn’t news. We are not free agents who are not influenced by our bodies and the world around us, including targeted attempts to make us choose a certain way. But it doesn’t mean that when we are faced with a particular decision, we are powerless and destined to go a certain way. We are able to look at our minds and work against cognitive errors, analyse choices and identify hidden prompts pushing us in certain directions, inform ourselves and look for less biased sources of information, educate ourselves to achieve better judgement, improve our environments and empower ourselves so that we can implement those better choices etc.
When we look at a cult member, person stuck in a bad relationship, person ‘voting against their best interests’, addicts, people with psychological conditions or many other kinds of people, they are often uninformed, manipulated, disempowered, threatened or something else - their free will is being internally or externally impaired. Sometimes these people will strongly support oppressive structures, make very poor decisions or make active choices that they cannot achieve. As you pointed out, we often underestimate our subconscious and imagine that we could make free conscious decisions based on the facts, and that any right-thinking person would do the same. There are just too many other factors to entertain that idea, but a freer/less constrained/more informed will may also be possible.
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A better example is: there’s the same (countably infinite/aleph-null) number of whole numbers as there are rational numbers (even though the set of rational numbers is the set of whole numbers plus the set of all non-whole-number fractions). You can compress one infinity into another infinity as long as they’re the same kind of infinity (the same cardinality).
I still don’t think you can extract Shakespeare from the initial state of the universe.
Representation is not compression. R doesn’t “compress” anything. It is a symbol that is used to represent a set that literally contains all real numbers. I can easily represent all future states of the universe with a symbol. Here I go:
Let U be the set of all future states of the universe.
Okay, done. That doesn’t let me determine what any future state of the universe will actually be, it just lets me refer to them collectively as “U”. R is exactly the same. We can’t even represent the vast majority of numbers in R in any meaningful way (by which I mean if you chose a random real number the probability that we can specify the selected number without reference to the fact that it was selected randomly for this purpose is 0).
But again, that’s not compression. That’s mapping 1, 2, 3, 4 onto A, B, C, D. The rationals aren’t bigger than the naturals, they are the same size.
I’m taking the hypothesis that if we call the set of things at time t Ut (and ignoring the fact that as I understand it Heisenberg means this set doesn’t even exist), there is a set of rules that can be applied to create Ut+1 and Ut+2. That means that:
- The rules exist somewhere not in the universe (God? Plato’s Heaven? The fairy world?); or
- For any given t all information in Un>t is contained within Ut (information in the universe is rapidly decreasing? there is infinite information? [the latter kicks in Godel and proves that there are undetermined parts of the universe]); or
- There is no difference between Ut and Un (I’m calling this nihilism since that makes all states equivalent to maximum entropy); or
- ???
Maybe I’m characterizing this wrong, maybe the point is that the universe in its entirely exists and what we call time is just a particular trajectory along it. But that’s not the universe being mathematics or the present determining the future. That is the universe being a stone and nothing “determining” anything in any sense - the present and future would not determine one another but merely be adjacent to one another.
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Not to misapprehend your devils advocacy but…
… maybe the rules emerge as a kind of aggregate pattern, accumulating consistency over time. Locality is violated all over the place constantly, but we still treat it like a rule to be broken.
In a sense, the rules in such a case could be said to exist ‘outside’ of of the vacuum, of the set of matter and energy, space and time. Or at least the rules might not be an intrinsic part of it, they are just mostly likely patterns that systems of self-interacting (conscious) energy can model because of their extremely likely recurrence. (I’m not saying this is definitely happening for fundamental laws, but it’s difficult to prove that it is not, especially given stuff like black hole complementarity, ‘zero point energy’ and other locality violations.)
Also, yeah, there is no Ut - only the rate of diffusion of information about energy states throughout other states at C. No absolute simultaneity, just the propagation of information in light cones. Which is what interests me about what happens when those light cones are at right angles, and specifically in well formed spacetimes, like those around / in black holes.
I’ve done it before and I’ll do it again, recommend pretty much anything ever written or lectured by Sean Carroll.
Well, like I said, I was arguing about Un>t being determined from Ut. I think I can make a pretty strong case this would require some rules sitting in Plato’s Heaven to pull off if all states aren’t actually the same thing and information in the universe isn’t drastically decreasing. Try to add natural numbers without the Peano axioms and you’ll find yourself out of luck. If the rules exist within the system then the rules must apply to themselves, which means gravity is exerting a gravitational force - what happens when something bumps into it?
I think the truth is more how you describe it. Without simultaneity, and will all information being limited in the speed it can travel, there just isn’t a present at all, let alone one that could determine the future. We build things like science and mathematics so we can do a better job of getting results we want out of the world, but the universe just is, and we’ll never even really know what that means.
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I’m a little more optimistic. You and I might not, or at least, might not as our current instantiations, but the sheer fact of existence implies so much. I’m inclined to believe that any existence of mind whatsoever implies future existence of mind, and then of super-intelligent mind, and then all bets are off. We might already be living in a quantum fluctuation, 1010010000000 years after the heat death of the original universe. In any of an infinite number of such fluctuations. There may be infinitely many exact copies of ourselves doing exactly the same things somewhere out there in our own universe, and infinitely many copies of ourselves doing almost the exact same thing with one minor difference.
The implications of reality are just so weird, and allow for so much to happen, I’ve got to believe that one of those things that happens is that some mind somewhere, sometime comes to some kind of understanding that approaches unity. Or at least, a toy model of it. 
As glib as it is dubious.
They have. They just don’t ever remember. Until now. Scary huh?
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If the outcome of events are deterministic, but the mechanics are such that the outcomes are still fundamentally unknowable, then how can we ever prove or disprove that very determinism?
However, I find the common dichotomy between free will and determinism to be false. Specifically, while free will may preclude determinism, a lack of determinism doesn’t necessarily imply free will. The presence of true randomness would actually suggest it cannot be manipulated.
As for free will itself, I think it’s conceivable that we have a sort of limited free will, insofar as our brains involve choices that might be fundamentally or at least practically impossible to mathematically predict due to rapid chaotic divergence between the model and its subject. In short, though free will is indeed a poorly defined term, we may be free from ever being able to know the outcome, in which case the question is purely academic since we’d have no way to directly test the thesis.
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This is unfounded… How does the researcher determine between the effects of time pressure, which can vary in individual experience, and the action of choosing?
As well, I submit that there is a difference between making small unimportant choices and making a decision based on reflection and analysis… The former uses short term memory… The latter is well into long term memory and complex thought.
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But stuff that happens in your brain is still you. How do you separate what you do inside you head with what you do exterior-ly. Practices of yogis lowering their vital signs –
Fire at will. This one weird trick.
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If you’re asking “how?” then how do you know it is unfounded? Either you understand the experiment or you don’t…
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