What do you think about the St Petersburg problem now?
I think your arguments re infinities conflict with Occam’s razor, and the principle of indifference should be applied across models within the same complexity (or submodels), otherwise you will assign 0 or too little credence to simpler models that are special cases, e.g. parameter value=0 or effectively eliminating some type of feature. There are infinitely many ways the universe could be more complex, and arbitrarily more complex, than you’d guess.
A flat universe effectively has one fewer parameter and is simpler. So it shouldn’t get 0 credence. Among flat universes, the bounded/​finite ones also have extra parameters for the boundaries or shape of the universe (compared to something that looks like R^3), so the infinite one shouldn’t get 0 credence.
(There probably are many ways for the universe to be infinite spatially in its shape, too, but those are more complex than R^3.)
EDIT:
As an aside, there have been many cases where quantities in physics were assumed to be 0, but then turned out to be just small values, like the mass of neutrinos. It often makes sense to round a quantity to 0 for simplicity, but a sufficiently small value would explain exactly the same empirical evidence.
Note that I never suggested to assign 0 probability to anything. I think this leads to further examples to illustrate my point:
Would you assign exactly 0 probability to photons having exactly 0 mass?
Exactly 0 probability to there being no additional fundamental force, because it could just be vanishingly weak? And shouldn’t this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say it’s very weak.
Exactly 0 probability to there not being ghosts, because their effects could just be very small or rare?
I think consistently applying your arguments suggest you should assign 0 probability in these cases, and so your models blow up in complexity and you become too credulous, contrary to Occam’s razor.
What do you think about the St Petersburg problem now?
Thanks for pushing me to think about this more. I had only looked into your post a few days after you published it around 3 years ago, but just had a look again. I agree the money pump you described there does not require prospect which could have an infinite value. It only requires prospects with infinite expected value as you have been saying.
I think there is exactly 0 empirical evidence for distributions with infinite expected value for the same reasons I believe there is exactly 0 empirical evidence for infinities. As far as I can tell, exactly 100 % of the empirical evidence that could ever be gathered in principle could be exactly 100 % explained by distributions with finite expected value. Do you agree? I agree distributions should not have a maximum because one cannot be exactly 100 % confident there are not higher values. However, a lack of maximum does not imply infinite expected value.
If there’s no finite upper bound on what the value the specific probability distribution can take, how can you be 100% confident it is not a probabilistic mixture with a distribution with infinite or undefined EV (but finite for every actual value), like 0.0001% probability to it being drawn from something like a St. Petersburg lottery?
This proves too much? One could argue there is a probability above exactly 0 of any given quantity being a probabilistic mixture involving a distribution with infinite or undefined expected value. In this case, all distributions would have an infinite or undefined expected value? For me this is a bitter bullet to bite than fully rejecting distributions with infinite or undefined expected value.
I think your arguments re infinities conflict with Occam’s razor, and the principle of indifference should be applied across models within the same complexity (or submodels), otherwise you will assign 0 or too little credence to simpler models that are special cases, e.g. parameter value=0 or effectively eliminating some type of feature.
I would apply the principle of indifference to models which explain the same empirical evidence. If a curvature of 0 had a probability above 0, and the values of the curvature just above 0 followed a continuous distribution, the curvature of 0 would be infinitely more likely than a positive curvature arbitrarily close to 0. This is very counterintuitive to me because the curvatures would have an arbitrarily close explanatory power. I would rather concede all universe models are wrong with probability 1 while acknowledging simpler ones are more useful for further scientific progress all else equal.
Would you assign exactly 0 probability to photons having exactly 0 mass?
Yes. I think there will always be infinitely many values arbitrarily close to 0 which explain exactly the same empirical evidence as a value of 0.
Exactly 0 probability to there being no additional fundamental force, because it could just be vanishingly weak? And shouldn’t this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say it’s very weak.
Yes.
Exactly 0 probability to there not being ghosts [ghosts existing with probability of exactly 1], because their effects could just be very small or rare?
Yes, but the effects of the ghosts would have to be sufficiently small or rare to be unfalsifiable. I assume the existence of ghosts is falsifiable under some typical definitions. Likewise for some defitions of God.
I think consistently applying your arguments suggest you should assign 0 probability in these cases, and so your models blow up in complexity and you become too credulous, contrary to Occam’s razor.
Do you see any undesirable implications of believing in ghosts which have exactly 0 measurable effects on the world? I think this is effectively the same as not believing in such ghosts.
One could argue there is a probability above exactly 0 of any given quantity being a probabilistic mixture involving a distribution with infinite or undefined expected value. In this case, all distributions would have an infinite or undefined expected value? For me this is a bitter bullet to bite than fully rejecting distributions with infinite or undefined expected value.
I think you should just expect this, and the answer is not to deny the possibility of St Petersberg lotteries in objective quantities, but to figure out a good way to deal with them (e.g. ignore small enough probabilities, use a bounded utility function, use commitments, use bracketing of some form), or accept that they raise difficult normative problems.
Do you see any undesirable implications of believing in ghosts which have exactly 0 measurable effects on the world? I think this is effectively the same as not believing in such ghosts.
Are you saying you believe in the existence with 100% credence in anything (such as ghosts) that is not ruled out by current evidence, as long as its effects couid be arbitrarily small and are so far indistinguishable from its nonexistence? Or must it also have no important normative implications (under classical utilitarianism?)?
Ghosts could be conscious, experience pleasure and suffering and care about what you do. Some could be vengeful and want harm to fall upon those that have caused them harm in their lives (whether or not they enact it themselves). Others could want to see the happiness of loved ones. Others could want their descendants to live up to their expectations (e.g. in education, work, family, religious adherence), and not care much about their happiness. It could be that every human that dies becomes a ghost indefinitely.
What do you think about the St Petersburg problem now?
I think your arguments re infinities conflict with Occam’s razor, and the principle of indifference should be applied across models within the same complexity (or submodels), otherwise you will assign 0 or too little credence to simpler models that are special cases, e.g. parameter value=0 or effectively eliminating some type of feature. There are infinitely many ways the universe could be more complex, and arbitrarily more complex, than you’d guess.
A flat universe effectively has one fewer parameter and is simpler. So it shouldn’t get 0 credence. Among flat universes, the bounded/​finite ones also have extra parameters for the boundaries or shape of the universe (compared to something that looks like R^3), so the infinite one shouldn’t get 0 credence.
(There probably are many ways for the universe to be infinite spatially in its shape, too, but those are more complex than R^3.)
EDIT:
Note that I never suggested to assign 0 probability to anything. I think this leads to further examples to illustrate my point:
Would you assign exactly 0 probability to photons having exactly 0 mass?
Exactly 0 probability to there being no additional fundamental force, because it could just be vanishingly weak? And shouldn’t this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say it’s very weak.
Exactly 0 probability to there not being ghosts, because their effects could just be very small or rare?
I think consistently applying your arguments suggest you should assign 0 probability in these cases, and so your models blow up in complexity and you become too credulous, contrary to Occam’s razor.
Thanks for pushing me to think about this more. I had only looked into your post a few days after you published it around 3 years ago, but just had a look again. I agree the money pump you described there does not require prospect which could have an infinite value. It only requires prospects with infinite expected value as you have been saying.
I think there is exactly 0 empirical evidence for distributions with infinite expected value for the same reasons I believe there is exactly 0 empirical evidence for infinities. As far as I can tell, exactly 100 % of the empirical evidence that could ever be gathered in principle could be exactly 100 % explained by distributions with finite expected value. Do you agree? I agree distributions should not have a maximum because one cannot be exactly 100 % confident there are not higher values. However, a lack of maximum does not imply infinite expected value.
This proves too much? One could argue there is a probability above exactly 0 of any given quantity being a probabilistic mixture involving a distribution with infinite or undefined expected value. In this case, all distributions would have an infinite or undefined expected value? For me this is a bitter bullet to bite than fully rejecting distributions with infinite or undefined expected value.
I would apply the principle of indifference to models which explain the same empirical evidence. If a curvature of 0 had a probability above 0, and the values of the curvature just above 0 followed a continuous distribution, the curvature of 0 would be infinitely more likely than a positive curvature arbitrarily close to 0. This is very counterintuitive to me because the curvatures would have an arbitrarily close explanatory power. I would rather concede all universe models are wrong with probability 1 while acknowledging simpler ones are more useful for further scientific progress all else equal.
Yes. I think there will always be infinitely many values arbitrarily close to 0 which explain exactly the same empirical evidence as a value of 0.
Yes.
Yes, but the effects of the ghosts would have to be sufficiently small or rare to be unfalsifiable. I assume the existence of ghosts is falsifiable under some typical definitions. Likewise for some defitions of God.
Do you see any undesirable implications of believing in ghosts which have exactly 0 measurable effects on the world? I think this is effectively the same as not believing in such ghosts.
I think you should just expect this, and the answer is not to deny the possibility of St Petersberg lotteries in objective quantities, but to figure out a good way to deal with them (e.g. ignore small enough probabilities, use a bounded utility function, use commitments, use bracketing of some form), or accept that they raise difficult normative problems.
Are you saying you believe in the existence with 100% credence in anything (such as ghosts) that is not ruled out by current evidence, as long as its effects couid be arbitrarily small and are so far indistinguishable from its nonexistence? Or must it also have no important normative implications (under classical utilitarianism?)?
Ghosts could be conscious, experience pleasure and suffering and care about what you do. Some could be vengeful and want harm to fall upon those that have caused them harm in their lives (whether or not they enact it themselves). Others could want to see the happiness of loved ones. Others could want their descendants to live up to their expectations (e.g. in education, work, family, religious adherence), and not care much about their happiness. It could be that every human that dies becomes a ghost indefinitely.