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I was thinking about the notion of sample spaces and was wondering whether they can be “objectively” analyzed for their fruitfulness.

For example, in the case of dice rolls, it seems obvious and natural to consider the sample space as representing each one of the outcomes ranging from 1-6. But what about for other events in life?

For example, I can imagine a world where I wasn’t typing this sentence out and instead was talking to my girlfriend or an enormous number of outcomes that I can think of. If I conceived of anything that is coherent as part of the sample space, it would include an infinite number of things, some of which may break physical laws! I could also conceive of all possible things that don’t break physical laws but yet aren’t typical of things that I normally do (for example, I haven’t ever went skydiving, so that doesn’t seem like a reasonable counterfactual).

So, are these sample spaces of outcomes mind dependent or mind independent? If they are mind dependent, does that mean even “obvious” sample spaces like the ones in dice rolls and coins are not as objective as we think they are?

My final comment on this would be to involve quantum mechanics since it seems to be the only theory that postulates “true” randomness and actual possibilities. In a sense, the sample space of measurements would simply be all possible measurements that we have observed so far.

However, let’s say that a deterministic theory underpinned quantum mechanics. Then, in a very “real” sense, each measurement would technically have only one possible outcome. And yet still, one can still imagine other outcomes from an epistemic perspective that may have resulted from different possible deterministic theories. This would be similar to how one can imagine a coin landing on heads even if it actually lands on tails and we live in a deterministic universe where the “heads event” at that point wasn’t technically possible given the initial conditions that led up to the coin toss.

So if we have no determinism, we can imagine other outcomes. But if we have determinism, we can also imagine other outcomes. In the first, other outcomes would not violate any laws yet they would still be unrealized. In the second, other outcomes would violate the deterministic law and also be unrealized. Is the sense in which things are possible under determinism and a lack of determinism the same? How should the sample spaces of each be treated?

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The relevant principle here is that sample sets represent uncertainty. A simple example should illustrate the point...

Suppose I roll a dice, which shows a four. Since the outcome has happened, it is entirely determined and certain. However, if I ask you to guess what number was shown, you have to consider the other five options. For you the sample set of possible outcomes contains six numbers, even though I know five of them to be ruled out. Importantly, your sample set does not include the number seven, for example, because you know that to be an impossible outcome.

Sample sets, therefore, should contain outcomes that you are unable to rule out, regardless of whether the outcomes may be the result of a determined or undetermined process. As the simple example I gave shows, they are most certainly mind-dependent, in that one person might have knowledge to rule out possibilities that another person cannot.

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  • Wow that’s an interesting way to think about things, next! Do you think there are mind independent sample spaces at all then or are they always dependent on knowledge in your opinion? Commented Apr 6 at 9:08
  • @MarcoOcram in the dice example, the random experiment is "the guessed value will match the actual value", right?
    – ac15
    Commented Apr 7 at 5:45
  • @ac15 yes, that's right. Commented Apr 7 at 5:47
  • @Mikhail I always quiver with fear when asked a question like that which requires a definitive answer I haven't properly thought about- it's an invitation to make an even bigger ass of myself! Let me think about it and nag me to death if I don't get back to you. Commented Apr 7 at 5:50
  • @MarcoOcram I guess Mikhail didn't understand it this way, but rather that it was literally about assigning probabilities to past events
    – ac15
    Commented Apr 7 at 5:51
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In a deterministic universe there would be no concept of alternative possibility nor any concept of imagination. An imagined thing itself is an alternative possibility.

True randomness is not a possibility. True randomness is necessary for any kind of evolution. A deterministic system does not evolve, nothing changes, nothing new emerges, complexity and entropy of the system remain constant.

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  • This sounds more like your personal ramblings than anything else to be quite honest Commented Apr 6 at 9:09
  • When every event is completely determined by the previous event, then everything happens with absolute certainty and accuracy. This is the very idea of determinism. None of this is my personal opinion. Commented Apr 6 at 11:28
  • the game of life [ en.wikipedia.org/wiki/Conway%27s_Game_of_Life ] is a counter-example to "A deterministic system does not evolve, nothing changes, nothing new emerges, complexity and entropy of the system remain constant", right?
    – ac15
    Commented Apr 6 at 15:04
  • @ac15 On the contrary. GoL demonstrates a deterministic system beautifully. No runtime input is accepted, complexity (pixel count) remains constant, everything is determined by the initial setup. Commented Apr 6 at 15:20

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