But there are various variables that influences the number it rolls, which we simply are unable to see. We assume symmetry, but how precise is that symmetry? — Jeremiah
Nature doesn't produce dice. Only humans do. However they still illustrate the essential principle of how to understand randomness or spontaneity in nature.
So you are making the standard Laplacian complaint that, in principle, complete knowledge of nature is possible, and so all future events can be calculated from determinate microphysical laws.
Well firstly, we now know that Newtonianism in fact fails at the limits. Quantum mechanics says existence is irreducibly indeterministic - and that ontic claim can even be phrased epistemically in terms of this being due to the fact we can't ask two different (non-commutating) questions of reality simultaneously. Like where are you exactly/what is your momentum exactly?
And complexity theory shows that the very idea of calculation is also self-limiting in this fashion. Because calculation is a digital way of describing an analog world, there is always round-up error in any attempt to model real world events.
No computer could ever specify the initial conditions of a calculation to an infinite number of decimal places. And if error compounds exponentially while the calculation proceeds in linear time (polynomially), then error must swamp any claims to accuracy in a few steps if it is describing a non-linear or chaotic event (one with less constraints than the kind of regular dynamics that Newtonian mechanics was designed to describe).
So we know that this idea of a mechanically deterministic universe is itself an idealisation. It is not the "natural state" of nature. Newtonian physics describes the world after it has reached the limit of a process of symmetry breaking and thus spent its many degrees of freedom. It is the world in as determinate state as it can get - yet not actually determinate, as quantum physics and complexity theory reveal.
Anyway, back to dice and how they illustrate this.
We make dice as perfect and symmetrical as we need them to be. Which in turn means we are matchingly indifferent to imperfections that are beyond what might affect our purposes in having a die.
What we want is a die that a thrower can throw in a fashion which leaves them with no way of telling what number will roll. So it must spin easily (bevelled edges) and yet fall flat on one face (break the symmetry of spinning) without favouring any one outcome. So if you are really concerned about dice being fair, you buy machined dice. You pay extra for the engineering and certification.
You insist on certainty that the die will break symmetry in a way that is entirely spontaneous to you.
But if you wanted to insist on that level of spontaneity in terms of nature itself, then you would have to get down to harnessing some kind of quantum noise or quantum emission process. Even nature doesn't know when an atom will decay - just that it has a completely exact and predictable poisson distribution. (The propensity to decay remains constant in time - which tells us something deep about the constraints that form nature, that is, our particular Universe.)
For all we know the cycle of the moon, or the time of the day could affect the number it lands on. So is it our inability to see all the variables and how it plays out that makes it unpredictable? — Jeremiah
When it comes to dice, we could in theory measure these further variables. But until gamblers start troubling casinos with such high tech approaches to beating the house odds, no one has reason to care.
So the human situation shows directly that randomness is about how much we have practical reasons to care about constraining the physics of events. We don't let gamblers drop dice. They must roll them properly.
The difficult mental leap - the one I've argued for - is to see that this principle is true of nature also. And quantum physics is the best argument. Nature can only ask questions of itself (hey little particle, what's your exact location/momentum?) to a limited degree of precision. And yet this doesn't really matter on the general scale of things.
Quantum fluctuations only disrupt nature on the tiniest or hottest possible scale of being. The Universe itself is now so cold and large that it is pretty much entirely classical in practice. There is infinitesimal chance of it doing something "quantum" like winking right out existence, or fluctuating into some other bizzare arrangement.
So indeterminism is basic to existence. And yet existence has become a place where everything is more or less as good as determined.
The question then becomes, why do humans still find randomness useful? Why do we invent ways of introducing chance back into the world of dull mechanical routine?
Obviously it is because we enjoy creating zones of freedom in which we can pit our wits. Games of chance are a way to practice our skills at strategy and prediction against "unpredictable nature". And so the kind of randomness we are really modelling there is the unpredicability, or non-computationality, of complexity.
We can try to calculate the future. But also such calculation is impossible. Which is where the pleasure and pain of being lucky/unlucky comes in.
But what if we select something at random out of say 10 possible choices? Then we know what we are gonna get; we are gonna get one of the 10 possible choices, but it was still a random selection. Is that saying we have simply removed the decision form our hands, and allowed variables we can't see to make the selection? — Jeremiah
Yes, you are describing epistemic uncertainty - something we have got mathematically and mechanically good at "manufacturing".
And then the deeper issue you want to address is ontic uncertainty - the randomness of nature itself.
And as I say, we can either rely on our own actions to result in our desired level of uncertainty (as i insisting gamblers roll dice properly, and don't bring moon gravity measuring devices with them into the casino). Or we could try to harness uncertainty by tapping into nature's own level of physical indifference. We could get down to quantum level processes. Or step up to uncomputable non-linear or chaotic processes.
Of course, people will still insist that at the bounding extremes of nature - the micro-physical and the macro-complex - Newtonian determinism must still reign.
But that is simply old-hat physics. We know that at the limit, things are actually different. The physics of the classical middle ground - the computationally simplest possible physics - no longer applies.