maybe there is some sorta-cosistent sorta-sane notion of the “world with different pi”, but we currently don’t know how to build it and if we knew, we would have solved logical uncertainty problem.
Neither we know how to build worlds with different physiscs, don’t we? If this was the necessary condition for being able to use probability theory then we shouldn’t be able to use it neither for empiric nor for logical uncertanity.
you made all the same observations, and coin came up tail.
On the other hand, if all we need is the vague idea of how its possible to make the same observations even when situation is different, well I definetely could’ve misheard that the question is about googolth digit of pi, while in actuality the question was about some other digit or maybe some other constant.
Frankly, I’m not sure what does this speculation about alternative worlds has to do with probability theory and updating in the first place. We have a probability space that describes a particular state of our knowledge. We have Bayes theorem, that formalizes the update procedure. So what’s the problem?
We know how to construct worlds with different physics. We do it all the time. Video games, or if you don’t accept that example we can construct a world consisting of 1 bit of information and 1 time dimension. This bit flips every certain increment of time. This universe obviously has different physics than ours.
Also as the other person mentioned, a probability space is the space of all possibilities organized based on whether statement Q is true, which is isomorphic to the space all universes consistent with your previous observations. There is, as far as I am aware, no way to logic yourself into the belief that pi could somehow have a different digit in a different universe, given you use a sufficiently exclusive definition of pi(specifying the curve of the plane the circle is upon being the major example)
We know how to construct worlds with different physics. We do it all the time. Video games, or if you don’t accept that example we can construct a world consisting of 1 bit of information and 1 time dimension. This bit flips every certain increment of time.
Technically true, but irrelevant to the point I’m making.
I was talking about constructing alternative worlds similar to ours to such degree that
I can inhabit either of them
I can be reasonably uncertain which one I inhabit
Both worlds are compatible with all my observations of a particular probability experiment—a coin toss
And yet, despite all of that in one world the coin comes Heads and in the other it comes Tails.
These are the type of worlds relevant for the discussions of probability experiments. We have no idea how to construct them, when talking about empiric uncertainty, and yet we don’t mind, only demanding such level of constructivism when dealing with logical uncertainty, for some reason.
There is, as far as I am aware, no way to logic yourself into the belief that pi could somehow have a different digit in a different universe, given you use a sufficiently exclusive definition of pi(specifying the curve of the plane the circle is upon being the major example)
Accent on sufficiently exclusive definition. Likewise, we can sufficiently exculisvily define a particular coin toss in a particular world and refuse to entertain the framing of different possible worlds : “No, the question is not about an abstract coin toss that could’ve ended differently in different possible worlds, the question is about this coin toss in this world”.
It’s just pretty clear in case of empirical uncertainty, that we should not be doing it, because such level of precision doesn’t capture our knowledge state. So why are we insisting on this level of exclusivity when talking about logical uncertainty?
In other words, this seems as an isolated demand for rigor to me.
Probability space consists of three things: sample space, event space and probability function.
Sample space defines a set of possible outcomes of probability experiment, representing the knowledge state of the person participating in it. In this case its:
{Odd, Even}
For event space we can just take a superset of the sample space. And as our measure function we just need to assign probabilities to the elementary events:
P(Odd) = P(Even) = 1⁄2
Do I understand correctly that the apparent problem is in defining the probability experiment in such a way so that we could talk about Odd and Even as outcomes of it?
It’s an interesting question, but its a different, more complex problem than simply not knowing googolth digit of pi and trying to estimate whether it’s even or odd.
The reason why logical uncertainty was brought up in the first place is decision theory, to make crisp formal expression for intuitive “I cooperate with you conditional on you cooperating with me”, where “you cooperating with me” is result of analysis of probability distribution over possible algorithms which control actions of your opponent and you can’t actually run these algorithms due to computational constraints, and you want to do all this reasoning in non-arbitrary ways.
Neither we know how to build worlds with different physiscs, don’t we? If this was the necessary condition for being able to use probability theory then we shouldn’t be able to use it neither for empiric nor for logical uncertanity.
On the other hand, if all we need is the vague idea of how its possible to make the same observations even when situation is different, well I definetely could’ve misheard that the question is about googolth digit of pi, while in actuality the question was about some other digit or maybe some other constant.
Frankly, I’m not sure what does this speculation about alternative worlds has to do with probability theory and updating in the first place. We have a probability space that describes a particular state of our knowledge. We have Bayes theorem, that formalizes the update procedure. So what’s the problem?
We know how to construct worlds with different physics. We do it all the time. Video games, or if you don’t accept that example we can construct a world consisting of 1 bit of information and 1 time dimension. This bit flips every certain increment of time. This universe obviously has different physics than ours. Also as the other person mentioned, a probability space is the space of all possibilities organized based on whether statement Q is true, which is isomorphic to the space all universes consistent with your previous observations. There is, as far as I am aware, no way to logic yourself into the belief that pi could somehow have a different digit in a different universe, given you use a sufficiently exclusive definition of pi(specifying the curve of the plane the circle is upon being the major example)
Technically true, but irrelevant to the point I’m making.
I was talking about constructing alternative worlds similar to ours to such degree that
I can inhabit either of them
I can be reasonably uncertain which one I inhabit
Both worlds are compatible with all my observations of a particular probability experiment—a coin toss
And yet, despite all of that in one world the coin comes Heads and in the other it comes Tails.
These are the type of worlds relevant for the discussions of probability experiments. We have no idea how to construct them, when talking about empiric uncertainty, and yet we don’t mind, only demanding such level of constructivism when dealing with logical uncertainty, for some reason.
Accent on sufficiently exclusive definition. Likewise, we can sufficiently exculisvily define a particular coin toss in a particular world and refuse to entertain the framing of different possible worlds : “No, the question is not about an abstract coin toss that could’ve ended differently in different possible worlds, the question is about this coin toss in this world”.
It’s just pretty clear in case of empirical uncertainty, that we should not be doing it, because such level of precision doesn’t capture our knowledge state. So why are we insisting on this level of exclusivity when talking about logical uncertainty?
In other words, this seems as an isolated demand for rigor to me.
We don’t??? Probability space literally defines set of considered worlds.
Probability space consists of three things: sample space, event space and probability function.
Sample space defines a set of possible outcomes of probability experiment, representing the knowledge state of the person participating in it. In this case its:
{Odd, Even}
For event space we can just take a superset of the sample space. And as our measure function we just need to assign probabilities to the elementary events:
P(Odd) = P(Even) = 1⁄2
Do I understand correctly that the apparent problem is in defining the probability experiment in such a way so that we could talk about Odd and Even as outcomes of it?
The problem is “how to define P(P=NP|trillionth digit of pi is odd)”.
Interesting. Is there an obvious way to do that for toy examples like P(1 = 2 | 7 = 11), or something like that
It’s an interesting question, but its a different, more complex problem than simply not knowing googolth digit of pi and trying to estimate whether it’s even or odd.
The reason why logical uncertainty was brought up in the first place is decision theory, to make crisp formal expression for intuitive “I cooperate with you conditional on you cooperating with me”, where “you cooperating with me” is result of analysis of probability distribution over possible algorithms which control actions of your opponent and you can’t actually run these algorithms due to computational constraints, and you want to do all this reasoning in non-arbitrary ways.