Okay, thanks for clarifying the question. If I gave you the following answer, would you say that it counts as a connection to real-world outcomes?
The real world outcome is that I run a double slit experiment with a billion photons, and plot the hit locations in a histogram. The heights of the bars of the graph closely match the probability distribution I previously calculated.
What about 1-time events, each corresponding to a totally unique physical situation? Simple. For each 1 time event, I bet a small amount of money on the result, at odds at least as good as the odds my theory gives for that result. The real world outcome is that after betting on many such events, I’ve ended up making a profit.
It’s true that both of these outcomes have a small chance of not-happening. But with enough samples, the outcome can be treated for all intents and purposes as a certainty. I explained above why the “continuous distribution” objection to this doesn’t hold.
It’s true that both of these outcomes have a small chance of not-happening. But with enough samples, the outcome can be treated for all intents and purposes as a certainty.
I agree with this in practice, but the question is philosophical in nature and this move doesn’t really help you get past the “firewall” between probabilistic and non-probabilistic claims at all. If you don’t already have a prior reason to care about probabilities, results like the law of large numbers or the central limit theorem can’t convince you to care about it because they are also probabilistic in nature.
For example, all LLN can give you is “almost sure convergence”, i.e. convergence with probability 1, and if I don’t have a prior reason to disregard events of probability 0 there’s no reason for me to care about this result.
I think davidad gave the best answer out of everyone so far, so you can also read his answers along with my conversation with him in the comment threads if you want to better understand where I’m coming from.
Okay, thanks for clarifying the question. If I gave you the following answer, would you say that it counts as a connection to real-world outcomes?
The real world outcome is that I run a double slit experiment with a billion photons, and plot the hit locations in a histogram. The heights of the bars of the graph closely match the probability distribution I previously calculated.
What about 1-time events, each corresponding to a totally unique physical situation? Simple. For each 1 time event, I bet a small amount of money on the result, at odds at least as good as the odds my theory gives for that result. The real world outcome is that after betting on many such events, I’ve ended up making a profit.
It’s true that both of these outcomes have a small chance of not-happening. But with enough samples, the outcome can be treated for all intents and purposes as a certainty. I explained above why the “continuous distribution” objection to this doesn’t hold.
I agree with this in practice, but the question is philosophical in nature and this move doesn’t really help you get past the “firewall” between probabilistic and non-probabilistic claims at all. If you don’t already have a prior reason to care about probabilities, results like the law of large numbers or the central limit theorem can’t convince you to care about it because they are also probabilistic in nature.
For example, all LLN can give you is “almost sure convergence”, i.e. convergence with probability 1, and if I don’t have a prior reason to disregard events of probability 0 there’s no reason for me to care about this result.
I think davidad gave the best answer out of everyone so far, so you can also read his answers along with my conversation with him in the comment threads if you want to better understand where I’m coming from.