Probability that a coin comes up heads is 0.5. Probability of N coins coming all up heads is 0.5^N. So what exactly was the original question in this context—are we asking whether there exist a smallest value of 0.5^N?
Well, if the universe has a finite time, if there is a smallest time unit, if the universe has finite number of elementary particles… this would provide some limit on the number of total coin flips in the universe. Even for infinite universes we could perhaps find some limit by specifying that the coin flips must happen in the same light cone...
But is this really what the original question was about? To me it seems like the question is confused. Probability is a logical construct, not something that exist, even if it is built on things that exist.
It would be like asking “what is the smallest positive rational number” with the additional constraint that a positive number must be P/Q where P and Q are numbers of pebbles in pebble heaps that exist in this universe. If there is a limited number of particles in the universe, that puts a limit on a value of Q, so there is some minimum value of 1/Q.. but what exactly does this result mean? Even if the Q really exists, the 1/Q is just a mental construct.
I’m fairly sure the original question was trying to ask about something labelled “probability” that wasn’t (exclusively) a mental construct, which is precisely why I brought up the idea of probability as a mental construct in the first place, to pre-empt confusion. Clearly I failed at that goal, though.
I’m not exactly sure what that something-labelled-”probability” was. You may well be right that the original question was simply confused. Generally when people start incorporating events in other Everett branches into their reasoning about the world I back away and leave them to it.
The OP aside, I do expect there are value of P too small for a human brain to actually represent. Given a probability like .000000001, for example, most of us either treat the probability as zero, or stop representing it in our minds as a probability at all. That is, for most of us our representation of a probability of .000000001 is just a number, indistinguishable from our representation of a temperature-difference of .000000001 degree Celsius or a mass of .000000001 grams.
So we could like exclude computations of expressions, and consider only probabilities of “basic events”, assuming that the concept shows to be coherent. We might ask about a probability of a coin flip, but not two coins. Speaking about coins, the “quantum of probability” is simply 1⁄2, end of story.
Well, I don’t even know what could be a “basic event” at the bottom level of the universe—the more I think about it, the more I realise my ignorance of quantum physics.
I don’t see where the “basic event”/”computation of expression” distinction gets us anywhere useful. As you say, even defining it clearly is problematic, and whatever definition we use it seems that any event we actually care about is not “basic.”
It also seems pretty clear to me that my mind can represent and work with probabilities smaller than 1⁄2, so restricting ourselves to domains of discourse that don’t require smaller probabilities (e.g., perfectly fair tosses of perfectly fair coins that always land on one face or the other) seems unhelpful.
Probability that a coin comes up heads is 0.5. Probability of N coins coming all up heads is 0.5^N. So what exactly was the original question in this context—are we asking whether there exist a smallest value of 0.5^N?
Well, if the universe has a finite time, if there is a smallest time unit, if the universe has finite number of elementary particles… this would provide some limit on the number of total coin flips in the universe. Even for infinite universes we could perhaps find some limit by specifying that the coin flips must happen in the same light cone...
But is this really what the original question was about? To me it seems like the question is confused. Probability is a logical construct, not something that exist, even if it is built on things that exist.
It would be like asking “what is the smallest positive rational number” with the additional constraint that a positive number must be P/Q where P and Q are numbers of pebbles in pebble heaps that exist in this universe. If there is a limited number of particles in the universe, that puts a limit on a value of Q, so there is some minimum value of 1/Q.. but what exactly does this result mean? Even if the Q really exists, the 1/Q is just a mental construct.
I’m fairly sure the original question was trying to ask about something labelled “probability” that wasn’t (exclusively) a mental construct, which is precisely why I brought up the idea of probability as a mental construct in the first place, to pre-empt confusion. Clearly I failed at that goal, though.
I’m not exactly sure what that something-labelled-”probability” was. You may well be right that the original question was simply confused. Generally when people start incorporating events in other Everett branches into their reasoning about the world I back away and leave them to it.
The OP aside, I do expect there are value of P too small for a human brain to actually represent. Given a probability like .000000001, for example, most of us either treat the probability as zero, or stop representing it in our minds as a probability at all. That is, for most of us our representation of a probability of .000000001 is just a number, indistinguishable from our representation of a temperature-difference of .000000001 degree Celsius or a mass of .000000001 grams.
So we could like exclude computations of expressions, and consider only probabilities of “basic events”, assuming that the concept shows to be coherent. We might ask about a probability of a coin flip, but not two coins. Speaking about coins, the “quantum of probability” is simply 1⁄2, end of story.
Well, I don’t even know what could be a “basic event” at the bottom level of the universe—the more I think about it, the more I realise my ignorance of quantum physics.
I don’t see where the “basic event”/”computation of expression” distinction gets us anywhere useful. As you say, even defining it clearly is problematic, and whatever definition we use it seems that any event we actually care about is not “basic.”
It also seems pretty clear to me that my mind can represent and work with probabilities smaller than 1⁄2, so restricting ourselves to domains of discourse that don’t require smaller probabilities (e.g., perfectly fair tosses of perfectly fair coins that always land on one face or the other) seems unhelpful.