I like the principle of indifference. However, I think infinities are unfalsifiable in principle. In this case, does it make sense for me to attribute probabilities to them? If I did, they would be just metaphysical priors that can never be updated by any evidence.
I also believe 2 actions can have infinite expected cost-effectiveness considering all future effects, and still be comparable. Imagine actions A and B have expected cost-effectiveness of CE_A(t) and CE_B(t) considering effects in the next t years (after a given date), and that CE_A and CE_B tend to positive infinity as t tends to infinity. If the ratio R(t) = CE_A(t)/âCE_B(t) tends to:
0, B is infinitely more cost-effective than A.
N, A is N times as cost-effective as B.
Positive infinity, A is infinitely more cost-effective than B.
I am sceptical of the 1st and last cases in practice. However, in any of the above cases, there would be a clear answer about which action has the highest expected cost-effectiveness considering all future effects.
There would be a dilemma if the expected cost-effectiveness growing faster kept oscillating forever between positive and negative forever. For example, if CE_A(t) = t, and CE_B(t) = t^2*sin(t), R(t) = 1/â(t*sin(t)), which tends to 0 as t goes to infinity, but keeps oscillating between an infinitesimally positive and negative value. However, in the real world, one could reasonably assume actions grow similarly fast sufficiently far into the future when there would be exactly the same evidence about the effects of A and B? In this case, R always tends to a finite single number as t goes to infinity.
I like the principle of indifference. However, I think infinities are unfalsifiable in principle. In this case, does it make sense for me to attribute probabilities to them? If I did, they would be just metaphysical priors that can never be updated by any evidence.
St Petersburg-like lotteries, defined in terms of your Bayesian credences, donât require assigning positive probability to any possible infinities out there in the world.
Iâll leave the rest in a footnote, because itâs not that relevant to the point Iâve been making so donât plan to go further with it, but I already wrote it and it may be of interest to you.[1]
I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis. Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.
In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence weâve gathered so far is in fact consistent with it (or no less consistent than with a bounded/âfinite universe), and it remains one of the standard models used by experts. My own view is that itâs more likely than not infinite in spatial extent, because itâs the simplest model consistent with the evidence.
I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable. If we had good evidence of nonzero global curvature, that would be decent (but not overwhelming) evidence that the universe is finite/âbounded, because it would rule out the most plausible infinite models of the universe.
I agree that we can sometimes compare actions with infinite expected cost-effectiveness. Iâm most partial to expansionist views of some kind, as the most complete (given precise probabilities).
I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis.
Makes sense. I cannot imagine any evidence that would update me. However, if I did, there would be falsifiability.
Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.
The way I see it, the probability of Earth having a radius larger than X tends to 0 as X goes to infinity. So I would say the probability of Earth having an infinite radius is exactly 0.
I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable.
Do we really have any evidence that the universe is globally flat? From Wikipediaâs page on the shape of the universe:
We have evidence that the universe is close to flat. However, there are infinitely many values arbitrary close to exactly 0. So applying some sort of principle of indifference results in a probability of exactly 0 of curvature being exactly 0 (or any other sharp value)?
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.
What do you think about the St Petersburg problem now?
I think your arguments re infinities conflict with Occamâs 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 youâd guess.
A flat universe effectively has one fewer parameter and is simpler. So it shouldnât get 0 credence. Among flat universes, the bounded/âfinite 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 shouldnât 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 shouldnât this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say itâs 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 Occamâs 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 thereâs 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 Occamâs 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 shouldnât this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say itâs very weak.
Yes. Edit after Michaelâs comment just below. I would assign a probability of exactly 0 to any physical law because there are arbitrarily many physical laws arbitrarily close to any physical law. So I would also assign a probability of exactly 0 to any set of physical laws, including the set of laws involving any given number of fundamental forces.
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. Edit after Michaelâs comment just below. I would not assign a probability of exactly 1 to something falsifiable.
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 Occamâs 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. A large share could be horrified by modern secularism. It could be that every human that dies becomes a ghost indefinitely.
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.
Why do you think I should expect all distributions to have infinite or undefined expected value instead of rejecting such distributions?
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?)?
I corrected my answers in my past comment. You can see what I crossed out, and wrote after âEdit after Michaelâs comment just belowâ.
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).
For the âghosts [I mentioned in my last comment] which have exactly 0 measurable effects on the worldâ, the benefit and harm they could cause would be sufficiently small to be practically negligible.
I like the principle of indifference. However, I think infinities are unfalsifiable in principle. In this case, does it make sense for me to attribute probabilities to them? If I did, they would be just metaphysical priors that can never be updated by any evidence.
I also believe 2 actions can have infinite expected cost-effectiveness considering all future effects, and still be comparable. Imagine actions A and B have expected cost-effectiveness of CE_A(t) and CE_B(t) considering effects in the next t years (after a given date), and that CE_A and CE_B tend to positive infinity as t tends to infinity. If the ratio R(t) = CE_A(t)/âCE_B(t) tends to:
0, B is infinitely more cost-effective than A.
N, A is N times as cost-effective as B.
Positive infinity, A is infinitely more cost-effective than B.
I am sceptical of the 1st and last cases in practice. However, in any of the above cases, there would be a clear answer about which action has the highest expected cost-effectiveness considering all future effects.
There would be a dilemma if the expected cost-effectiveness growing faster kept oscillating forever between positive and negative forever. For example, if CE_A(t) = t, and CE_B(t) = t^2*sin(t), R(t) = 1/â(t*sin(t)), which tends to 0 as t goes to infinity, but keeps oscillating between an infinitesimally positive and negative value. However, in the real world, one could reasonably assume actions grow similarly fast sufficiently far into the future when there would be exactly the same evidence about the effects of A and B? In this case, R always tends to a finite single number as t goes to infinity.
St Petersburg-like lotteries, defined in terms of your Bayesian credences, donât require assigning positive probability to any possible infinities out there in the world.
Iâll leave the rest in a footnote, because itâs not that relevant to the point Iâve been making so donât plan to go further with it, but I already wrote it and it may be of interest to you.[1]
I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis. Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.
In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence weâve gathered so far is in fact consistent with it (or no less consistent than with a bounded/âfinite universe), and it remains one of the standard models used by experts. My own view is that itâs more likely than not infinite in spatial extent, because itâs the simplest model consistent with the evidence.
I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable. If we had good evidence of nonzero global curvature, that would be decent (but not overwhelming) evidence that the universe is finite/âbounded, because it would rule out the most plausible infinite models of the universe.
I agree that we can sometimes compare actions with infinite expected cost-effectiveness. Iâm most partial to expansionist views of some kind, as the most complete (given precise probabilities).
Makes sense. I cannot imagine any evidence that would update me. However, if I did, there would be falsifiability.
The way I see it, the probability of Earth having a radius larger than X tends to 0 as X goes to infinity. So I would say the probability of Earth having an infinite radius is exactly 0.
Do we really have any evidence that the universe is globally flat? From Wikipediaâs page on the shape of the universe:
We have evidence that the universe is close to flat. However, there are infinitely many values arbitrary close to exactly 0. So applying some sort of principle of indifference results in a probability of exactly 0 of curvature being exactly 0 (or any other sharp value)?
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.
What do you think about the St Petersburg problem now?
I think your arguments re infinities conflict with Occamâs 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 youâd guess.
A flat universe effectively has one fewer parameter and is simpler. So it shouldnât get 0 credence. Among flat universes, the bounded/âfinite 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 shouldnât 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 shouldnât this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say itâs 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 Occamâs 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.Edit after Michaelâs comment just below. I would assign a probability of exactly 0 to any physical law because there are arbitrarily many physical laws arbitrarily close to any physical law. So I would also assign a probability of exactly 0 to any set of physical laws, including the set of laws involving any given number of fundamental forces.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. Edit after Michaelâs comment just below. I would not assign a probability of exactly 1 to something falsifiable.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. A large share could be horrified by modern secularism. It could be that every human that dies becomes a ghost indefinitely.
Why do you think I should expect all distributions to have infinite or undefined expected value instead of rejecting such distributions?
I corrected my answers in my past comment. You can see what I crossed out, and wrote after âEdit after Michaelâs comment just belowâ.
For the âghosts [I mentioned in my last comment] which have exactly 0 measurable effects on the worldâ, the benefit and harm they could cause would be sufficiently small to be practically negligible.