If they interpret N in this way, then N is at least 1.
No, N is a prior. You can’t draw conclusions about what a prior is like that. N could be tiny and there could be a bunch of civilizations anyway, that’s just unlikely.
It just occurred to me that you still need some prior probability for your sentence which is smaller than 1.
Sure, prior in the sense of an estimate before you learn any of your experiences. Which clearly you’re not actually computing prior to having those experiences, but we’re talking in theory.
My personal goal would be to make SIA (or a similar principle) nothing more than a corollary of Bayesian updating, possibly together with a general theory of indexical beliefs.
SIA is just a prior over what observer one expects to end up with.
Maybe it is not just the probability that the hypothetical observer had the same observations, it’s the probability that the hypothetical observer exists and had the same observations. Not just what observations observers made is often a guess but also how many of them exist.
I’m not sure what distinction you’re drawing here. Can you give a toy problem where your description differs from mine?
So I think “has the same state of mind” is better to not exclude those freak observers to begin with, because we might be such a freak observer.
My usual definition is “subjectively indistinguishable from me”, you can substitute that above.
The sphere is of finite size and we take the probability of a cow being one-headed as the limit of the ratio as the size of the sphere goes towards infinity.
This is basically just downweighting things infinitely far away infinitely low. It’s accepting unboundedness but not infinity. Unboundedness has its own problems, but it’s more plausible than infinity.
But we would need an epistemic reason, in contrast to an instrumental reason, to a priori exclude a possibility by assigning it probability 0.
I’m not assigning it probability 0 so much as I’m denying that it’s meaningful. It doesn’t satisfy my criterion for meaning.
You seemed to specifically object to universes with finite information content on grounds that they are just (presumably periodic) “loops”.
That’s one objection among several, but the periodicity isn’t the real issue—even without that it still must repeat at some point, even if not regularly. All you really have is an irrational set of ratios between various “states of the world”, calling that “infinity” seems like a stretch.
those hypotheses are more likely to be true
What do you mean by true here?
Because lower information content means higher a priori probability.
Probability is just a means to predict the future. Probabilities attached to statements that aren’t predictive in nature are incoherent.
If you entertained the hypothesis that solipsism is true, this would not compress your evidence at all, which means the information content of that hypothesis would be very high, which means it is very improbable.
The same thing is true of the “hypothesis” that solipsism is false. It has no information content. It’s not even meaningful to say that there’s a probability that it’s true or false. Neither is a valid hypothesis.
If no external things exist, then all “y because x” statements would be false.
The problem with this line of reasoning is that we commonly use models we know are false to “explain” the world. “All models are wrong, some models are useful”.
Also re causality, Hume already pointed out we can’t know any causality claims.
Also, it’s unclear how an incoherent hypothesis can serve to “explain” anything.
I think explanations are just fine without assuming a particular metaphysics. When we say “E because H”, we just mean that our model H predicts E, which is a reason to apply H to other predictions in the future. We don’t need to assert any metaphysical statements to do that.
No, N is a prior. You can’t draw conclusions about what a prior is like that. N could be tiny and there could be a bunch of civilizations anyway, that’s just unlikely.
I just quoted the paper. It stated that N is the expected number of civilizations in the Milky Way. If that is the case, we have to account for the fact that at least one civilization exists. Which wasn’t done by the authors. Otherwise N is just the expected number of civilizations in the Milky Way under the assumption we didn’t knew that we existed.
Sure, prior in the sense of an estimate before you learn any of your experiences. Which clearly you’re not actually computing prior to having those experiences, but we’re talking in theory.
“before you learn any experience”? I.e. before you know you exist? Before you exist? Before the “my” refers to anything? You seem to require exactly what I suspected: a non-indexical version of your statement.
SIA is just a prior over what observer one expects to end up with.
There are infinitely many possible priors. One would need a justification that the SIA prior is more rational than the alternatives. FNC made much progress in this direction by only using Bayesian updating and no special prior like SIA. Unfortunately there are problems with this approach. But I think those can be fixed without needing to “assume” some prior.
This is basically just downweighting things infinitely far away infinitely low.
All things in the universe get weighted and all get weighted equally. Things just get weighted in a particular order, nearer things get weighted “earlier” so to speak (not in a temporal sense), but not with more weight.
It’s accepting unboundedness but not infinity. Unboundedness has its own problems, but it’s more plausible than infinity.
“Unboundednes” is means usually something else. A universe with a sphere or torus topology is unbounded but finite in size. I’m talking about a plane topology universe here which is both unbounded and infinitely large.
But you seem to have something like hyperreal numbers in mind when you talk about infinity. Hyperreal numbers include “infinite numbers” (the first is called omega) which are larger than any real number. But if cosmologists talk about a universe which is spatially infinite, they only say that for any positive real number n, there is a place in the universe which is at least n+1 light-years away. They do not say “there is something which is omega light-years away”. They do not treat infinite as a (kind of) number. That’s more of a game played by some mathematicians who sometimes like to invent new numbers.
I’m not sure what distinction you’re drawing here. Can you give a toy problem where your description differs from mine?
You might be certain that 100 observers exist in the universe. You are not sure who might be you, but one of the observers you regard as twice as likely to be you as each of the other ones, so you weigh it twice a strong.
But you may also be uncertain of how many observers exist. Say you are equally uncertain about the existence of each of 99 and twice as certain about the existence of a hundredth one. Then you weigh it twice as strong. (I’m not quite sure whether this is right.)
That’s one objection among several, but the periodicity isn’t the real issue—even without that it still must repeat at some point, even if not regularly.
Even in a finite universe there might be repetition. Possibly our universe is finite and contains not only Earth but also a planet we might call Twin-Earth very far away from Earth. Twin-Earth is a perfect duplicate of Earth. It’s even called “Earth” by twin-earthlings. If a person X on Earth moves only his left arm, Twin-X on Twin-Earth also moves only his left arm. But this is merely (perfect) correlation, there is no stronger form of dependence, like counterfactual dependence. If X had moved his right arm instead, Twin-X still had moved only his left arm. This could not be the case if X and Twin-X were identical. Also, if X hurts his foot, Twin-X will also hurt his foot, but X will only feel the pain caused by X’s foot and not the pain caused by the foot of Twin-X. They don’t share a single mind.
All you really have is an irrational set of ratios between various “states of the world”, calling that “infinity” seems like a stretch.
I would rather say that it’s a stretch to regard infinity as a ordinary number, as you are apparently doing. The limit view of infinity doesn’t do this. “Infinity” then just means that for any real number there is another real number which is larger (or smaller).
those hypotheses are more likely to be true
What do you mean by true here?
What we usually mean. But you can remove “to be true” here and the meaning of the sentence stays the same.
Probability is just a means to predict the future.
We can perfectly well (and do all the time) make probabilistic statements about the present or the past. I suggest to regard probability not so much as a “means” but as a measure of uncertainty, where P(A)=1/2 means I am (or perhaps: I should be) perfectly uncertain whether A or not A. This has nothing to do with predictions. (But as I said, the hypothesis of an infinite universe makes predictions anyway.)
Probabilities attached to statements that aren’t predictive in nature are incoherent.
Where is the supposed “incoherence” here?
The best characterization of incoherence I know treats it as a generalization of logical contradiction: A and B are (to some degree) incoherent if P(A and B) < P(A)*P(B). Negative statistical dependence. I.e. each one is evidence against the other. But you seem to mean something else.
The same thing is true of the “hypothesis” that solipsism is false. It has no information content.
It is verified by just a single non-mental object. It has information content, just a very low one. Not as low as “something exists” (because this is also verified by mental objects) but still quite low. Only tautologies have no (i.e. zero) information content.
The problem with this line of reasoning is that we commonly use models we know are false to “explain” the world. “All models are wrong, some models are useful”.
The common answer to that is that Newton’s theory of gravity isn’t so much wrong as it is somewhat inaccurate. A special case of Einstein’s more accurate theory. A measure of (in)accuracy is generalization error in statistics. Low generalization error seems to be for many theories what truth is for ordinary statements. And if we would say of an ordinary statement A that it is “more likely” than an other statement B we would say that a theory X has a “lower expected” generalization error than a theory Y.
Also re causality, Hume already pointed out we can’t know any causality claims.
Well, not only that! Hume also said that no sort of inductive inference is justified, probabilistic or not, so all predictions would be out of the window, not just ones about causal relationships. Because the evidence is almost always consistent with lots of possible but incompatible predictions. I would say that an objective a priori probability distribution over hypotheses (i.e. all possible statements) based on information content solves the problem. For indexical hypotheses I’m not quite certain yet, maybe there is something similar objective for an improved version of SIA. If there is no objective first prior then Hume is right and verificationism is wrong. What you predict would rely on an arbitrary choice of prior probabilities.
I think explanations are just fine without assuming a particular metaphysics. When we say “E because H”, we just mean that our model H predicts E, which is a reason to apply H to other predictions in the future. We don’t need to assert any metaphysical statements to do that.
That doesn’t work for many reasons. Some barometer reading predicts a storm, but it doesn’t explain it. Rather there is a common explanation for both the barometer reading and the storm: air pressure.
Also, explanation (because statements) are asymmetric. If B because A then not A because B. But prediction is symmetric: If A is evidence for B, then B is evidence for A. Because one is evidence for the other if both are positively probabilistically dependent (“correlated”). P(A|B) > P(A) implies P(B|A) > P(B). The rain predicts the wet street, so the wet street predicts the rain. The rain explains the wet street, so the wet street doesn’t explain the rain.
There are even some cases where H explains E but H and E don’t predict each other,
i.e. they are not positively statistically dependent. These cases are known as Simpson’s paradox.
I just quoted the paper. It stated that N is the expected number of civilizations in the Milky Way. If that is the case, we have to account for the fact that at least one civilization exists. Which wasn’t done by the authors. Otherwise N is just the expected number of civilizations in the Milky Way under the assumption we didn’t knew that we existed.
The update we need to do is not equivalent to assuming N is at least one, because as I said, N being less than one is consistent with our experiences.
“before you learn any experience”? I.e. before you know you exist? Before you exist? Before the “my” refers to anything?
Yes, it gets awkward if you try to interpret the prior literally. Don’t do that, just apply the updating rules.
There are infinitely many possible priors. One would need a justification that the SIA prior is more rational than the alternatives.
SIA as a prior just says it’s equally likely for you to be one of two observers that are themselves equally likely to exist. Any alternative will necessarily say that in at least one such case, you’re more likely to be one observer than the other, which violates the indifference principle.
You might be certain that 100 observers exist in the universe. You are not sure who might be you, but one of the observers you regard as twice as likely to be you as each of the other ones, so you weigh it twice a strong.
But you may also be uncertain of how many observers exist. Say you are equally uncertain about the existence of each of 99 and twice as certain about the existence of a hundredth one. Then you weigh it twice as strong.
I’m not sure where my formulation is supposed to diverge here.
“Infinity” then just means that for any real number there is another real number which is larger (or smaller).
Well, this is possible without even letting the reals be unbounded. For any real number under 2, there’s another real number under 2 that’s greater than it.
We can perfectly well (and do all the time) make probabilistic statements about the present or the past.
And those statements are meaningless except insofar as they imply predictions about the future.
Where is the supposed “incoherence” here?
The statement lacks informational content.
It is verified by just a single non-mental object.
I don’t know what this is supposed to mean. What experience does the statement imply?
Low generalization error seems to be for many theories what truth is for ordinary statements.
Sure, I have no problem with calling your theory true once it’s shown strong predictive ability. But don’t confuse that with there being some territory out there that the theory somehow corresponds to.
objective a priori probability distribution over hypotheses (i.e. all possible statements) based on information content
Yes, this is SIA + Solomonoff universal prior, as far as I’m concerned. And this prior doesn’t require calling any of the hypotheses “true”, the prior is only used for prediction. Solomonoff aggregates a large number of hypotheses, none of which are “true”.
Some barometer reading predicts a storm, but it doesn’t explain it.
The reading isn’t a model. You can turn it into a model, and then it would indeed explain the storm, while air pressure would explain it better, by virtue of explaining other things as well and being part of a larger model that explains many things simply (such as how barometers are constructed.)
prediction is symmetric:
A model isn’t an experience, and can’t get conditioned on. There is no symmetry between models and experiences in my ontology.
The experience of rain doesn’t explain the experience of the wet street—rather, a model of rain explains / predicts both experiences.
No, N is a prior. You can’t draw conclusions about what a prior is like that. N could be tiny and there could be a bunch of civilizations anyway, that’s just unlikely.
Sure, prior in the sense of an estimate before you learn any of your experiences. Which clearly you’re not actually computing prior to having those experiences, but we’re talking in theory.
SIA is just a prior over what observer one expects to end up with.
I’m not sure what distinction you’re drawing here. Can you give a toy problem where your description differs from mine?
My usual definition is “subjectively indistinguishable from me”, you can substitute that above.
This is basically just downweighting things infinitely far away infinitely low. It’s accepting unboundedness but not infinity. Unboundedness has its own problems, but it’s more plausible than infinity.
I’m not assigning it probability 0 so much as I’m denying that it’s meaningful. It doesn’t satisfy my criterion for meaning.
That’s one objection among several, but the periodicity isn’t the real issue—even without that it still must repeat at some point, even if not regularly. All you really have is an irrational set of ratios between various “states of the world”, calling that “infinity” seems like a stretch.
What do you mean by true here?
Probability is just a means to predict the future. Probabilities attached to statements that aren’t predictive in nature are incoherent.
The same thing is true of the “hypothesis” that solipsism is false. It has no information content. It’s not even meaningful to say that there’s a probability that it’s true or false. Neither is a valid hypothesis.
The problem with this line of reasoning is that we commonly use models we know are false to “explain” the world. “All models are wrong, some models are useful”.
Also re causality, Hume already pointed out we can’t know any causality claims.
Also, it’s unclear how an incoherent hypothesis can serve to “explain” anything.
I think explanations are just fine without assuming a particular metaphysics. When we say “E because H”, we just mean that our model H predicts E, which is a reason to apply H to other predictions in the future. We don’t need to assert any metaphysical statements to do that.
I just quoted the paper. It stated that N is the expected number of civilizations in the Milky Way. If that is the case, we have to account for the fact that at least one civilization exists. Which wasn’t done by the authors. Otherwise N is just the expected number of civilizations in the Milky Way under the assumption we didn’t knew that we existed.
“before you learn any experience”? I.e. before you know you exist? Before you exist? Before the “my” refers to anything? You seem to require exactly what I suspected: a non-indexical version of your statement.
There are infinitely many possible priors. One would need a justification that the SIA prior is more rational than the alternatives. FNC made much progress in this direction by only using Bayesian updating and no special prior like SIA. Unfortunately there are problems with this approach. But I think those can be fixed without needing to “assume” some prior.
All things in the universe get weighted and all get weighted equally. Things just get weighted in a particular order, nearer things get weighted “earlier” so to speak (not in a temporal sense), but not with more weight.
“Unboundednes” is means usually something else. A universe with a sphere or torus topology is unbounded but finite in size. I’m talking about a plane topology universe here which is both unbounded and infinitely large.
But you seem to have something like hyperreal numbers in mind when you talk about infinity. Hyperreal numbers include “infinite numbers” (the first is called omega) which are larger than any real number. But if cosmologists talk about a universe which is spatially infinite, they only say that for any positive real number n, there is a place in the universe which is at least n+1 light-years away. They do not say “there is something which is omega light-years away”. They do not treat infinite as a (kind of) number. That’s more of a game played by some mathematicians who sometimes like to invent new numbers.
You might be certain that 100 observers exist in the universe. You are not sure who might be you, but one of the observers you regard as twice as likely to be you as each of the other ones, so you weigh it twice a strong.
But you may also be uncertain of how many observers exist. Say you are equally uncertain about the existence of each of 99 and twice as certain about the existence of a hundredth one. Then you weigh it twice as strong. (I’m not quite sure whether this is right.)
Even in a finite universe there might be repetition. Possibly our universe is finite and contains not only Earth but also a planet we might call Twin-Earth very far away from Earth. Twin-Earth is a perfect duplicate of Earth. It’s even called “Earth” by twin-earthlings. If a person X on Earth moves only his left arm, Twin-X on Twin-Earth also moves only his left arm. But this is merely (perfect) correlation, there is no stronger form of dependence, like counterfactual dependence. If X had moved his right arm instead, Twin-X still had moved only his left arm. This could not be the case if X and Twin-X were identical. Also, if X hurts his foot, Twin-X will also hurt his foot, but X will only feel the pain caused by X’s foot and not the pain caused by the foot of Twin-X. They don’t share a single mind.
I would rather say that it’s a stretch to regard infinity as a ordinary number, as you are apparently doing. The limit view of infinity doesn’t do this. “Infinity” then just means that for any real number there is another real number which is larger (or smaller).
What we usually mean. But you can remove “to be true” here and the meaning of the sentence stays the same.
We can perfectly well (and do all the time) make probabilistic statements about the present or the past. I suggest to regard probability not so much as a “means” but as a measure of uncertainty, where P(A)=1/2 means I am (or perhaps: I should be) perfectly uncertain whether A or not A. This has nothing to do with predictions. (But as I said, the hypothesis of an infinite universe makes predictions anyway.)
Where is the supposed “incoherence” here?
The best characterization of incoherence I know treats it as a generalization of logical contradiction: A and B are (to some degree) incoherent if P(A and B) < P(A)*P(B). Negative statistical dependence. I.e. each one is evidence against the other. But you seem to mean something else.
It is verified by just a single non-mental object. It has information content, just a very low one. Not as low as “something exists” (because this is also verified by mental objects) but still quite low. Only tautologies have no (i.e. zero) information content.
The common answer to that is that Newton’s theory of gravity isn’t so much wrong as it is somewhat inaccurate. A special case of Einstein’s more accurate theory. A measure of (in)accuracy is generalization error in statistics. Low generalization error seems to be for many theories what truth is for ordinary statements. And if we would say of an ordinary statement A that it is “more likely” than an other statement B we would say that a theory X has a “lower expected” generalization error than a theory Y.
Well, not only that! Hume also said that no sort of inductive inference is justified, probabilistic or not, so all predictions would be out of the window, not just ones about causal relationships. Because the evidence is almost always consistent with lots of possible but incompatible predictions. I would say that an objective a priori probability distribution over hypotheses (i.e. all possible statements) based on information content solves the problem. For indexical hypotheses I’m not quite certain yet, maybe there is something similar objective for an improved version of SIA. If there is no objective first prior then Hume is right and verificationism is wrong. What you predict would rely on an arbitrary choice of prior probabilities.
That doesn’t work for many reasons. Some barometer reading predicts a storm, but it doesn’t explain it. Rather there is a common explanation for both the barometer reading and the storm: air pressure.
Also, explanation (because statements) are asymmetric. If B because A then not A because B. But prediction is symmetric: If A is evidence for B, then B is evidence for A. Because one is evidence for the other if both are positively probabilistically dependent (“correlated”). P(A|B) > P(A) implies P(B|A) > P(B). The rain predicts the wet street, so the wet street predicts the rain. The rain explains the wet street, so the wet street doesn’t explain the rain.
There are even some cases where H explains E but H and E don’t predict each other, i.e. they are not positively statistically dependent. These cases are known as Simpson’s paradox.
The update we need to do is not equivalent to assuming N is at least one, because as I said, N being less than one is consistent with our experiences.
Yes, it gets awkward if you try to interpret the prior literally. Don’t do that, just apply the updating rules.
SIA as a prior just says it’s equally likely for you to be one of two observers that are themselves equally likely to exist. Any alternative will necessarily say that in at least one such case, you’re more likely to be one observer than the other, which violates the indifference principle.
I’m not sure where my formulation is supposed to diverge here.
Well, this is possible without even letting the reals be unbounded. For any real number under 2, there’s another real number under 2 that’s greater than it.
And those statements are meaningless except insofar as they imply predictions about the future.
The statement lacks informational content.
I don’t know what this is supposed to mean. What experience does the statement imply?
Sure, I have no problem with calling your theory true once it’s shown strong predictive ability. But don’t confuse that with there being some territory out there that the theory somehow corresponds to.
Yes, this is SIA + Solomonoff universal prior, as far as I’m concerned. And this prior doesn’t require calling any of the hypotheses “true”, the prior is only used for prediction. Solomonoff aggregates a large number of hypotheses, none of which are “true”.
The reading isn’t a model. You can turn it into a model, and then it would indeed explain the storm, while air pressure would explain it better, by virtue of explaining other things as well and being part of a larger model that explains many things simply (such as how barometers are constructed.)
A model isn’t an experience, and can’t get conditioned on. There is no symmetry between models and experiences in my ontology.
The experience of rain doesn’t explain the experience of the wet street—rather, a model of rain explains / predicts both experiences.