I believe (without proof) that there is no God. I strongly dislike it when people like Dragonlord are dismissed when they argue that people like me also are, in a sense, people of faith. When I read that kind of dismissal, it certainly feels like I am being defined out of atheism.
As I understand it, Dragonlord was not trying to refute atheism by making a definition. What he said was in no way a refutation of my brand of atheism, nor yours. It was merely a charge that an atheist who considers himself by definition epistemically superior to a theist is something of a hypocrite. As it happens, I agree.
I find your “you can’t refute atheism by definition” to be both ambiguous and ironic. Clearly, what you mean to argue is that it is impossible to refute a position (such as atheism) by making a definition. But, it can also be read as the exact opposite: “By definition, you can’t refute atheism.” And it certainly appears to me, as well as to most theists, that this is exactly what far too many atheists are doing these days.
I have been an atheist longer than most posters here have been alive. At the time I began calling myself an atheist, the definition was that an atheist is someone who believes that there is no God. A theist believes there is a God. An agnostic is unsure or unwilling to commit. But, as a result of jibes like Dragonlord’s, our brave modern-day atheists have redefined the term to include almost everyone formerly called agnostics. They are so frightened of being soiled by that word “faith” that they deny having any beliefs at all regarding the existence of a deity.
I believe (without proof) that there is no God. I strongly dislike it when people like Dragonlord are dismissed when they argue that people like me also are, in a sense, people of faith. When I read that kind of dismissal, it certainly feels like I am being defined out of atheism.
You’re not being defined out—other people are being defined in. But you’re correct: I don’t actually address Dragonlord’s point (which I assume regards the matter of the burden of proof) in any fashion by nitpicking the definition of “agnostic” and “atheism”. I am writing a new reply.
By “without proof” do you mean that you don’t have 100% evidence or by without proof you mean that you are believing this even though you don’t have evidence for the claim? If the first, there’s no problem as long as your probability estimate for some form of deity isn’t 0. If the second then whether or not you are an atheist there’s a serious gap in your rationality.
Incidentally, it might help in this discussion to taboo the word the atheist.
Reasons for belief. JoshuaZ used the phrase “100% evidence”, which strikes me as meaningless. He also asked whether I believed something with no evidence. That strikes me as an absurd question. Evidence is always present—the evidence might easily be inadequate, and conceivably could be balanced. But “don’t have evidence”?
I think that if he tries to ask his questions using different words, he will find that he already knows the answers and that my previous responses provide the answers.
There are multiple interpretations I have of what Dragonlord meant. The statement wasn’t very clear. Here are three translations:
In the first, Dragonlord was defining “atheist” as someone who is 100% certain that there is no god and then meant something something like “A 100% claim that there is no God is just as irrational as someone who claims to believe in God.” And then he implicitly defined the term “agnostic” as anyone assigning a probability to God’s existence that isn’t 0 or 1.
In the second, Dragonlord was defining atheist as someone who assigns a very low probability to God’s existence. He then meant something like: “Anyone who makes a strong claim about the probability of God’s existence has insufficient evidence either way and so is using “faith” to push their probability estimate in a direction unjustified by evidence.” And then he implicitly identified agnostic as people with a middling probability.
In the third, Dragonlord identified atheism the same way as in the second but then meant something like “I am uncomfortable with people making strong claims about this question, so I am going to declare that everyone making strong claims about this question are being irrational in the same way.” And then he identified agnostics as people who aren’t making him uncomfortable with strong claims about the existence of a deity.
Ah! Very good. Thank you. That exercise was more productive than I expected.
My original interpretation (and the interpretation I still hold) was that he was saying either the first or the second. And, based on that interpretation, I felt that Dragonlord’s statement was reasonable and perhaps even defensible. And I felt that the downvoting was unfair. I realize that RobinZ also thought the downvoting was unfair, but I thought that RobinZ’s defense of Dragonlord (“The poor guy just doesn’t understand the definitions”, in effect) was worse than useless, because it didn’t respond to what Dragonlord was actually trying to say.
And I thank you for making clear what he apparently really was saying. And even if Dragonlord actually meant your third version, he should still have received a substantive response rather than a lecture on the modern usage of the word “atheist” among atheists.
Well, part of the response about how people use the word atheist might be something like simply clarifying that people aren’t making an 100% claim when they identify as atheists. That’s quite relevant if he meant the first interpretation. I’m not so sure that anything discussed is that relevant to interpretation 2.
Actually, I am making a 100% claim. Certainly a claim as strong as the claim of most theists, and they would also admit to being 100% certain. Call me irrational, if you wish. Remind me of Cromwell’s rule. But don’t ask me to admit that I made a mistake until you can prove to me that God exists.
Call me irrational, if you wish. Remind me of Cromwell’s rule.
Okay, I’ll do so, because it genuinely seems like you need it: You are, in this particular instance, being irrational, and Cromwell’s rule (or more specifically, the way 0% or 100% priors conflict with Bayesian updating) does indeed explain why.
You clearly know what mistake it is you’re making; why are you stopping right before the part where you correct that mistake?
You clearly know what mistake it is you’re making; why are you stopping right before the part where you correct that mistake?
Because:
Just because I understand why you are calling it a mistake doesn’t mean that I need to either agree with you that it is a mistake or argue with you.
I have already suggested elsewhere that my preference would be to assign an infinitesimal probability, rather than a zero probability, to God’s existence, but that spoilsport Jaynes doesn’t want to let me.
You Bayes purists keep telling me that the mistake, once made, is very difficult to correct, and I have enough difficulties in my life right now. Seems to me that I may as well just continue with my current probability assignments until I die. At which time I may either be proven right after all, or else I will be presented with ample time to correct my mistake.
or self I will be presented with ample time to correct my mistake.
I’m confused by this. You seem to be implying that you use a different epistemology if there’s life after death than you do during life. How do you justify this? If you aren’t doing so, how can you then update in the way described?
I have already suggested elsewhere that my preference would be to assign an infinitesimal probability, rather than a zero probability, to God’s existence, but that spoilsport Jaynes doesn’t want to let me.
Does setting an infinitesimal probability preclude updates to non-infinitesimal probabilities?
Ok. This is a real problem then. Because then there’s no way to update based on new evidence. Indeed, I suspect that there’s probably some set of evidence which would make you change your mind. If that’s the case, then you can’t have a prior of 1 even if you round up to that. Note incidentally the most of the theists who make that claim are demonstrably incorrect based on their actual behavior which demonstrates a lot more uncertainty about their beliefs than they profess.
I understand that I cannot be a true Bayesian and assign a probability of 0 to something unless it is logically impossible. But I’m not sure why you say I can’t update. I can update everything except my belief (disbelief) in a deity. And I don’t expect I will ever have to do that.
And the way I figure it, if I ever do encounter overwhelming proof of God’s existence, I am going to have bigger problems than a need to back out all of the Bayesian updating I have done since I became an atheist and start from scratch.
But I have another reason for being less worried than you think I ought to be about probabilities of 0 and 1. I am looking into Abraham Robinson’s non-standard analysis and so actually my true level of belief in God is not exactly 0, it is (literally) infinitesimal. Now all I need are some Bayesian updating rules for how to handle ‘black swan’ events.
I think I can explain both sides of this disagreement.
A Bayesian is someone who is, at some level of abstraction, keeping track of their entire hypothesis space. “Updating,” for a Bayesian, means paying more attention to the part of yourself that predicted what you just saw happen. If a Bayesian says that they updated towards believing in God, they mean that they are paying more attention to the small part of themselves that already believed in God. “I assign a probability of 0 to God” means that no part of you believes in God, so there is no possibility of updating. You can’t shift attention towards a part of yourself that isn’t there.
Humans who say that they assign a probability of 0 to God, but who also claim that they could update, are not keeping track of the entire hypothesis space within their own brains. They are also using other brains to do that. There is no part of these humans that believes in God, but they would be able to copy this part from other humans if they ever needed to. These humans do not see themselves as trying to create a complete Bayesian within their own single brain; rather, they are trying to be part of a multiple-brain process that is doing something Bayesian. It does not matter as much to them which particular brains are paying attention to God.
I like your model, or something similar. The way I would do it, though, is to claim that my own decision-making is done by a committee of rational Bayesian minds—all resident within my head. Right now, the chairman of the committee assigns 0 to the probability that God exists. However, should we have a black swan event like a Rapture, then the chairman is helpless—he cannot update. No problem, though, because the other members of the committee were not so foolish as to become strong atheists. They will simply assassinate the now-useless former chairman of the Committee for Allowing Perplexed to Exhibit Rationality (CAPER) and replace him with a theist. The mind is under new management, but the body marches on.
Indirect response:
Perhaps you should discuss my level of confidence with Tim Tyler. When you two reach consensus regarding my level of confidence, then come back and challenge me about it.
Direct response: Do you have some point in making your observation?
But I have another reason for being less worried than you think I ought to be about probabilities of 0 and 1. I am looking into Abraham Robinson’s non-standard analysis and so actually my true level of belief in God is not exactly 0, it is (literally) infinitesimal.
Now this is a clear rationalization. You are giving a justification by analogy, not quite a mysterious answer to a mysterious question (because non-standard analysis is not mysterious, on the other hand Bayesian epistemology based on non-standard analysis is), but epistemically the same error (with other examples being, seeking answers in complexity and randomness).
Just to be clear, I am not using my research program into non-standard analysis to justify my “carelessness” in becoming a strong atheist when the evidence only forces me into weak atheism. That carelessness happened fifty years ago, before I had even heard of Robinson.
What I may be rationalizing is my lack of concern regarding my violation of Cromwell’s rule. And I don’t see this as a mysterious answer. It is pretty straightforward. The set of ‘worlds’ in which God exists is not empty, but it is a set of measure zero. Using standard analysis, I am forced to assign probability zero to this event, and hence I have no way to update. Using non-standard analysis, I may be able to assign an infinitesimal probability to the God Hypothesis, and then (details not yet worked out) have the arithmetic work should proof of God’s existence somehow appear and I be forced to reassign measures to the remaining possible worlds (no longer a set of measure zero).
To be clear, would you actually bet an unbounded amount of money or resources or other things that are valuable to you (for instance, your life or your children’s lives or the entire human race) against the existence of any god, for a payoff of $1 if you are right? That’s the sort of thing that you should be able to calmly and confidently do if you really have infinitesimal credence in the thing you’re betting against.
To be clear, would you actually bet an unbounded amount of money or resources or other things that are valuable to you (for instance, your life or your children’s lives or the entire human race) against the existence of any god, for a payoff of $1 if you are right?
Yes. PM me and I will provide the address where you should send the money.
I understand that I cannot be a true Bayesian and assign a probability of 0 to something unless it is logically impossible.
Normally, you won’t be able to learn that it’s logically impossible, because you can only ever use potentially faulty calculators to come to that conclusion, so there goes this clause. It could be directly in your prior though.
And the way I figure it, if I ever do encounter overwhelming proof of God’s existence, I am going to have bigger problems than a need to back out all of the Bayesian updating I have done since I became an atheist and start from scratch.
What will you do if you encounter evidence of God’s existence that is significant but not overwhelming?
That’s the major concern I have with Bayes-breaking priors. You don’t really need Bayes for cases where there is overwhelming evidence, but those are pretty dang uncommon. Bayes is very helpful when dealing with evidence at everyday levels of moderate or slight significance.
What will you do if you encounter evidence of God’s existence that is significant but not overwhelming?
Well, suppose I currently assess the odds of God’s existence at epsilon. If I encounter evidence with an odds ratio of a million to one, then I update to 1,000,000 * epsilon.
If epsilon is required to be a standard real number, then I am forced to either make epsilon non-zero (but less than 1 ppm), or make it zero and stop calling myself a Bayesian.
But if epsilon is allowed to be a non-standard real—specifically, an infinitesimal—then I think I can have my atheist cake and be a Bayesian too.
Perhaps this example might help. Suppose I tell you that I am thinking of a random point in the closed unit square. You choose a uniform prior. That means you believe that the probability that my point is on the boundary of the square is zero. So, what do you, as a Bayesian, do when I inform you that the point is indeed on the boundary and ask you for the probability that it is on the bottom edge?
Either you had to initially assign a finite probability to the point being on the boundary
(and also a finite probability to it having an x coordinate of exactly 0.5, etc.) or else you find some way of claiming that the probability of the point is infinitesimal—that is, if you are forced to pick a real number, you will pick 0, but you refuse to be forced to pick a real number.
Perhaps this example might help. Suppose I tell you that I am thinking of a random point in the closed unit square. You choose a uniform prior. That means you believe that the probability that my point is on the boundary of the square is zero. So, what do you, as a Bayesian, do when I inform you that the point is indeed on the boundary and ask you for the probability that it is on the bottom edge?
For any probablity p strictly between 0 and 1, and any distance r greater than 0, there exists a finite amount of evidence E that would convince a Bayesian that your point is within the distance r of the boundary with probablity greater than p.
Do you think that propositions about God are part of an uncountably large space? Is there a reasonable notion of “similar” such that you could be convinced with finite evidence that there is a true proposition arbitrarily “similar” to a proposition that a given God exists?
I think we need to taboo the word “finite”. And stick to my example of the square for the time being.
If you had a uniform prior over the square, and then I inform you that my “random point” is on the edge, have I provided you with a ‘finite’ or an ‘infinite’ amount of evidence? A case could be made, I think, for either answer.
The same applies for the amount of evidence required to demonstrate something similar to the proposition that God exists, for many reasonable values of ‘similar’.
Notice that “amount of evidence” is not just a property of the evidence. It also depends on what your prior was for receiving that evidence. It is a subjective number.
And stick to my example of the square for the time being.
No. That example was a metaphor, it is reasonable to explore how its features correlates back to features of the question of interest, which is if it makes sense for you to assign infinitesimal probability to propositions about God.
If you had a uniform prior over the square, and then I inform you that my “random point” is on the edge, have I provided you with a ‘finite’ or an ‘infinite’ amount of evidence? A case could be made, I think, for either answer.
Then I would update my differential probability distribution using prior conditional probablities that you would make such a claim given that your “random point” is any particular point in the square. This could cause me to conclude that your point is very close to the border with high probability, but not to concentrate all my probablity onto the border itself, which would require that I had infinite information about under which conditions you would make such a statement.
You have not answered my question about if the proposition about God is part of an uncountable space. The rest of this only matters if your answer is yes.
You have not answered my question about if the proposition about God is part of an uncountable space. The rest of this only matters if your answer is yes.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
Ah! But perhaps you are suggesting that I can only formulate a countable number of sentences in my logic and hence that I should be using some kind of Solomonoff prior which necessarily forces a finite prior for the God Hypothesis—assuming that I can express it. Is that what you are getting at? If so, I’m not sure exactly how the hypothesis that some kind of god exists can be expressed properly in any axiomatizable logic.
Yes, but in a countable measure space the measure is determined entirely by the measures on the points, hence there is no problem with making the interpretation “probability 0 = impossible”, and this sort of weirdness does not occur.
Countability is not precisely the condition needed to avoid this, but it’s certainly a sufficient condition.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
Measure theory tends to be a lot simpler with countable sets.
But perhaps you are suggesting that I can only formulate a countable number of sentences in my logic and hence that I should be using some kind of Solomonoff prior which necessarily forces a finite prior for the God Hypothesis—assuming that I can express it. Is that what you are getting at?
No, although, if you answer that the space of propositions is countable, then I would argue that all propositions in that space should have a real probability between 0 and 1.
I would like you to answer the question, rather than speculating on hidden meaning in the question, so that I can know what kind of probability distributions we should be talking about.
Ok, I don’t know the cardinality of the space we are talking about, but since I have trouble imagining a language permitting an uncountable number of sentences, lets assume that the space is countable. What are the consequences of that?
If the space is countable, then as long as you can order the propositions in some way, say by complexity, you can assign non-zero probability to every proposition so the total adds up to 1, so you don’t have the same excuse you have in the case of predicting which point in a continuous space is special for using infinitesimal probabilities.
Ok, that makes sense. As I told JoshuaZ, I need to retire and lick my wounds at the very least. I seem to recall that in Nelson’s version of non-standard analysis, there could be infinitesimals even in systems of countable cardinality, but I need to check that and decide whether it matters in this case.
I seem to recall that in Nelson’s version of non-standard analysis, there could be infinitesimals even in systems of countable cardinality, but I need to check that and decide whether it matters in this case.
Sorry, what do you mean by this? We’re talking about the cardinality of the set the measure is on; this sounds like you’re talking about the cardinality of its target space? (Where values of measures are somehow generalized appropriately… let’s not worry about how.) It’s easy to put an order on, say, Q[t] so as to make t infinitesimal but I don’t see what that has to do with this. Or is that not what you meant?
We’re talking about the cardinality of the set the measure is on.
So am I. But I may be confused about what cardinality even means in Nelson’s internal set theory.
Let me give a simple example of the kind of thing I am thinking about. Consider the space of ordered pairs (a,n) where a is either 0 or 1 and n is a non-negative integer, i.e. an element of {1,2,...}. To each such pair with a=0, associate the measure M(0,n)= 1/2^n. To each such pair with a=1 associate the “infinitesimal measure” M(1,n)=M(0,n)/omega where omega is taken to be indefinitely large.
So, the total measure of this space is 1 unit, and all but an infinitesimal portion of that total measure is associated with the portion of the space with a=0.
I claim that in some sense P(a=1) = 0 but P(n=2 | a=1) = 1⁄4.
The analogy here is that the assertion a=1 corresponds to the assertion that God exists. The probability is infinitesimal, yet Bayesian updating is possible (in some sense). And yet the space of all events is countable.
Yes. Definitely. Sorry that was unclear. And infinitesimal measures result in probabilities which are zero in some sense, but not exactly zero in a different sense.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
The two are related. Most relevantly, if my set is countable then I must have some singletons with non-zero measure. Moreover, the subset of points who have zero measure itself has zero measure, so they don’t matter at all. It is only in higher cardinality sets that you can have a collection of points each with zero measure that still have positive measure.
Ok, I can see that this tends to rule out my use of the unit-square analogy to justify my suggestion that the probability of the God Hypothesis is infinitesimal. I’m going to have to look more closely at the math, and in particular at my references for non-standard analysis to see whether any of my intuitions can be saved.
I’m curious then. Suppose that every child under five years or so old dissappeared as did most evangelical Christians. Would you then assign a chance of zero that the Rapture had just taken place?
Actually, the etymology of agnosticism is rejection of gnosis, which is rather broader than the gnostics. We know that because Huxley said so when he coined the word in 1869. That’s also exactly the meaning RobinZ gave.
I believe (without proof) that there is no God. I strongly dislike it when people like Dragonlord are dismissed when they argue that people like me also are, in a sense, people of faith. When I read that kind of dismissal, it certainly feels like I am being defined out of atheism.
As I understand it, Dragonlord was not trying to refute atheism by making a definition. What he said was in no way a refutation of my brand of atheism, nor yours. It was merely a charge that an atheist who considers himself by definition epistemically superior to a theist is something of a hypocrite. As it happens, I agree.
I find your “you can’t refute atheism by definition” to be both ambiguous and ironic. Clearly, what you mean to argue is that it is impossible to refute a position (such as atheism) by making a definition. But, it can also be read as the exact opposite: “By definition, you can’t refute atheism.” And it certainly appears to me, as well as to most theists, that this is exactly what far too many atheists are doing these days.
I have been an atheist longer than most posters here have been alive. At the time I began calling myself an atheist, the definition was that an atheist is someone who believes that there is no God. A theist believes there is a God. An agnostic is unsure or unwilling to commit. But, as a result of jibes like Dragonlord’s, our brave modern-day atheists have redefined the term to include almost everyone formerly called agnostics. They are so frightened of being soiled by that word “faith” that they deny having any beliefs at all regarding the existence of a deity.
That is what I am arguing about.
You’re not being defined out—other people are being defined in. But you’re correct: I don’t actually address Dragonlord’s point (which I assume regards the matter of the burden of proof) in any fashion by nitpicking the definition of “agnostic” and “atheism”. I am writing a new reply.
Ok. We cool now.
By “without proof” do you mean that you don’t have 100% evidence or by without proof you mean that you are believing this even though you don’t have evidence for the claim? If the first, there’s no problem as long as your probability estimate for some form of deity isn’t 0. If the second then whether or not you are an atheist there’s a serious gap in your rationality.
Incidentally, it might help in this discussion to taboo the word the atheist.
I would prefer to taboo the word ‘evidence’.
If you want to taboo ‘atheist’, please provide your translation of what Dragonlord said to begin this discussion.
What does that leave us with, precisely, on a rationalist website?
Reasons for belief. JoshuaZ used the phrase “100% evidence”, which strikes me as meaningless. He also asked whether I believed something with no evidence. That strikes me as an absurd question. Evidence is always present—the evidence might easily be inadequate, and conceivably could be balanced. But “don’t have evidence”?
I think that if he tries to ask his questions using different words, he will find that he already knows the answers and that my previous responses provide the answers.
There are multiple interpretations I have of what Dragonlord meant. The statement wasn’t very clear. Here are three translations:
In the first, Dragonlord was defining “atheist” as someone who is 100% certain that there is no god and then meant something something like “A 100% claim that there is no God is just as irrational as someone who claims to believe in God.” And then he implicitly defined the term “agnostic” as anyone assigning a probability to God’s existence that isn’t 0 or 1.
In the second, Dragonlord was defining atheist as someone who assigns a very low probability to God’s existence. He then meant something like: “Anyone who makes a strong claim about the probability of God’s existence has insufficient evidence either way and so is using “faith” to push their probability estimate in a direction unjustified by evidence.” And then he implicitly identified agnostic as people with a middling probability.
In the third, Dragonlord identified atheism the same way as in the second but then meant something like “I am uncomfortable with people making strong claims about this question, so I am going to declare that everyone making strong claims about this question are being irrational in the same way.” And then he identified agnostics as people who aren’t making him uncomfortable with strong claims about the existence of a deity.
Ah! Very good. Thank you. That exercise was more productive than I expected.
My original interpretation (and the interpretation I still hold) was that he was saying either the first or the second. And, based on that interpretation, I felt that Dragonlord’s statement was reasonable and perhaps even defensible. And I felt that the downvoting was unfair. I realize that RobinZ also thought the downvoting was unfair, but I thought that RobinZ’s defense of Dragonlord (“The poor guy just doesn’t understand the definitions”, in effect) was worse than useless, because it didn’t respond to what Dragonlord was actually trying to say.
And I thank you for making clear what he apparently really was saying. And even if Dragonlord actually meant your third version, he should still have received a substantive response rather than a lecture on the modern usage of the word “atheist” among atheists.
Well, part of the response about how people use the word atheist might be something like simply clarifying that people aren’t making an 100% claim when they identify as atheists. That’s quite relevant if he meant the first interpretation. I’m not so sure that anything discussed is that relevant to interpretation 2.
Actually, I am making a 100% claim. Certainly a claim as strong as the claim of most theists, and they would also admit to being 100% certain. Call me irrational, if you wish. Remind me of Cromwell’s rule. But don’t ask me to admit that I made a mistake until you can prove to me that God exists.
Okay, I’ll do so, because it genuinely seems like you need it: You are, in this particular instance, being irrational, and Cromwell’s rule (or more specifically, the way 0% or 100% priors conflict with Bayesian updating) does indeed explain why.
You clearly know what mistake it is you’re making; why are you stopping right before the part where you correct that mistake?
Because:
Just because I understand why you are calling it a mistake doesn’t mean that I need to either agree with you that it is a mistake or argue with you.
I have already suggested elsewhere that my preference would be to assign an infinitesimal probability, rather than a zero probability, to God’s existence, but that spoilsport Jaynes doesn’t want to let me.
You Bayes purists keep telling me that the mistake, once made, is very difficult to correct, and I have enough difficulties in my life right now. Seems to me that I may as well just continue with my current probability assignments until I die. At which time I may either be proven right after all, or else I will be presented with ample time to correct my mistake.
I’m confused by this. You seem to be implying that you use a different epistemology if there’s life after death than you do during life. How do you justify this? If you aren’t doing so, how can you then update in the way described?
More time to think things through carefully. But, at least to some extent, I was being facetious in my point #3.
Does setting an infinitesimal probability preclude updates to non-infinitesimal probabilities?
Ok. This is a real problem then. Because then there’s no way to update based on new evidence. Indeed, I suspect that there’s probably some set of evidence which would make you change your mind. If that’s the case, then you can’t have a prior of 1 even if you round up to that. Note incidentally the most of the theists who make that claim are demonstrably incorrect based on their actual behavior which demonstrates a lot more uncertainty about their beliefs than they profess.
I understand that I cannot be a true Bayesian and assign a probability of 0 to something unless it is logically impossible. But I’m not sure why you say I can’t update. I can update everything except my belief (disbelief) in a deity. And I don’t expect I will ever have to do that.
And the way I figure it, if I ever do encounter overwhelming proof of God’s existence, I am going to have bigger problems than a need to back out all of the Bayesian updating I have done since I became an atheist and start from scratch.
But I have another reason for being less worried than you think I ought to be about probabilities of 0 and 1. I am looking into Abraham Robinson’s non-standard analysis and so actually my true level of belief in God is not exactly 0, it is (literally) infinitesimal. Now all I need are some Bayesian updating rules for how to handle ‘black swan’ events.
I think I can explain both sides of this disagreement.
A Bayesian is someone who is, at some level of abstraction, keeping track of their entire hypothesis space. “Updating,” for a Bayesian, means paying more attention to the part of yourself that predicted what you just saw happen. If a Bayesian says that they updated towards believing in God, they mean that they are paying more attention to the small part of themselves that already believed in God. “I assign a probability of 0 to God” means that no part of you believes in God, so there is no possibility of updating. You can’t shift attention towards a part of yourself that isn’t there.
Humans who say that they assign a probability of 0 to God, but who also claim that they could update, are not keeping track of the entire hypothesis space within their own brains. They are also using other brains to do that. There is no part of these humans that believes in God, but they would be able to copy this part from other humans if they ever needed to. These humans do not see themselves as trying to create a complete Bayesian within their own single brain; rather, they are trying to be part of a multiple-brain process that is doing something Bayesian. It does not matter as much to them which particular brains are paying attention to God.
I like your model, or something similar. The way I would do it, though, is to claim that my own decision-making is done by a committee of rational Bayesian minds—all resident within my head. Right now, the chairman of the committee assigns 0 to the probability that God exists. However, should we have a black swan event like a Rapture, then the chairman is helpless—he cannot update. No problem, though, because the other members of the committee were not so foolish as to become strong atheists. They will simply assassinate the now-useless former chairman of the Committee for Allowing Perplexed to Exhibit Rationality (CAPER) and replace him with a theist. The mind is under new management, but the body marches on.
“And I don’t expect I will ever have to do that.”
You do not sound 100% certain.
Indirect response: Perhaps you should discuss my level of confidence with Tim Tyler. When you two reach consensus regarding my level of confidence, then come back and challenge me about it.
Direct response: Do you have some point in making your observation?
Now this is a clear rationalization. You are giving a justification by analogy, not quite a mysterious answer to a mysterious question (because non-standard analysis is not mysterious, on the other hand Bayesian epistemology based on non-standard analysis is), but epistemically the same error (with other examples being, seeking answers in complexity and randomness).
Just to be clear, I am not using my research program into non-standard analysis to justify my “carelessness” in becoming a strong atheist when the evidence only forces me into weak atheism. That carelessness happened fifty years ago, before I had even heard of Robinson.
What I may be rationalizing is my lack of concern regarding my violation of Cromwell’s rule. And I don’t see this as a mysterious answer. It is pretty straightforward. The set of ‘worlds’ in which God exists is not empty, but it is a set of measure zero. Using standard analysis, I am forced to assign probability zero to this event, and hence I have no way to update. Using non-standard analysis, I may be able to assign an infinitesimal probability to the God Hypothesis, and then (details not yet worked out) have the arithmetic work should proof of God’s existence somehow appear and I be forced to reassign measures to the remaining possible worlds (no longer a set of measure zero).
To be clear, would you actually bet an unbounded amount of money or resources or other things that are valuable to you (for instance, your life or your children’s lives or the entire human race) against the existence of any god, for a payoff of $1 if you are right? That’s the sort of thing that you should be able to calmly and confidently do if you really have infinitesimal credence in the thing you’re betting against.
Yes. PM me and I will provide the address where you should send the money.
Normally, you won’t be able to learn that it’s logically impossible, because you can only ever use potentially faulty calculators to come to that conclusion, so there goes this clause. It could be directly in your prior though.
What will you do if you encounter evidence of God’s existence that is significant but not overwhelming?
That’s the major concern I have with Bayes-breaking priors. You don’t really need Bayes for cases where there is overwhelming evidence, but those are pretty dang uncommon. Bayes is very helpful when dealing with evidence at everyday levels of moderate or slight significance.
Well, suppose I currently assess the odds of God’s existence at epsilon. If I encounter evidence with an odds ratio of a million to one, then I update to 1,000,000 * epsilon.
If epsilon is required to be a standard real number, then I am forced to either make epsilon non-zero (but less than 1 ppm), or make it zero and stop calling myself a Bayesian.
But if epsilon is allowed to be a non-standard real—specifically, an infinitesimal—then I think I can have my atheist cake and be a Bayesian too.
Perhaps this example might help. Suppose I tell you that I am thinking of a random point in the closed unit square. You choose a uniform prior. That means you believe that the probability that my point is on the boundary of the square is zero. So, what do you, as a Bayesian, do when I inform you that the point is indeed on the boundary and ask you for the probability that it is on the bottom edge?
Either you had to initially assign a finite probability to the point being on the boundary (and also a finite probability to it having an x coordinate of exactly 0.5, etc.) or else you find some way of claiming that the probability of the point is infinitesimal—that is, if you are forced to pick a real number, you will pick 0, but you refuse to be forced to pick a real number.
For any probablity p strictly between 0 and 1, and any distance r greater than 0, there exists a finite amount of evidence E that would convince a Bayesian that your point is within the distance r of the boundary with probablity greater than p.
Do you think that propositions about God are part of an uncountably large space? Is there a reasonable notion of “similar” such that you could be convinced with finite evidence that there is a true proposition arbitrarily “similar” to a proposition that a given God exists?
I think we need to taboo the word “finite”. And stick to my example of the square for the time being.
If you had a uniform prior over the square, and then I inform you that my “random point” is on the edge, have I provided you with a ‘finite’ or an ‘infinite’ amount of evidence? A case could be made, I think, for either answer.
The same applies for the amount of evidence required to demonstrate something similar to the proposition that God exists, for many reasonable values of ‘similar’.
Notice that “amount of evidence” is not just a property of the evidence. It also depends on what your prior was for receiving that evidence. It is a subjective number.
No. That example was a metaphor, it is reasonable to explore how its features correlates back to features of the question of interest, which is if it makes sense for you to assign infinitesimal probability to propositions about God.
Then I would update my differential probability distribution using prior conditional probablities that you would make such a claim given that your “random point” is any particular point in the square. This could cause me to conclude that your point is very close to the border with high probability, but not to concentrate all my probablity onto the border itself, which would require that I had infinite information about under which conditions you would make such a statement.
You have not answered my question about if the proposition about God is part of an uncountable space. The rest of this only matters if your answer is yes.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
Ah! But perhaps you are suggesting that I can only formulate a countable number of sentences in my logic and hence that I should be using some kind of Solomonoff prior which necessarily forces a finite prior for the God Hypothesis—assuming that I can express it. Is that what you are getting at? If so, I’m not sure exactly how the hypothesis that some kind of god exists can be expressed properly in any axiomatizable logic.
Yes, but in a countable measure space the measure is determined entirely by the measures on the points, hence there is no problem with making the interpretation “probability 0 = impossible”, and this sort of weirdness does not occur.
Countability is not precisely the condition needed to avoid this, but it’s certainly a sufficient condition.
Uh, what sort of weirdness does not occur?
Measure theory tends to be a lot simpler with countable sets.
No, although, if you answer that the space of propositions is countable, then I would argue that all propositions in that space should have a real probability between 0 and 1.
I would like you to answer the question, rather than speculating on hidden meaning in the question, so that I can know what kind of probability distributions we should be talking about.
Ok, I don’t know the cardinality of the space we are talking about, but since I have trouble imagining a language permitting an uncountable number of sentences, lets assume that the space is countable. What are the consequences of that?
If the space is countable, then as long as you can order the propositions in some way, say by complexity, you can assign non-zero probability to every proposition so the total adds up to 1, so you don’t have the same excuse you have in the case of predicting which point in a continuous space is special for using infinitesimal probabilities.
Ok, that makes sense. As I told JoshuaZ, I need to retire and lick my wounds at the very least. I seem to recall that in Nelson’s version of non-standard analysis, there could be infinitesimals even in systems of countable cardinality, but I need to check that and decide whether it matters in this case.
Sorry, what do you mean by this? We’re talking about the cardinality of the set the measure is on; this sounds like you’re talking about the cardinality of its target space? (Where values of measures are somehow generalized appropriately… let’s not worry about how.) It’s easy to put an order on, say, Q[t] so as to make t infinitesimal but I don’t see what that has to do with this. Or is that not what you meant?
So am I. But I may be confused about what cardinality even means in Nelson’s internal set theory.
Let me give a simple example of the kind of thing I am thinking about. Consider the space of ordered pairs (a,n) where a is either 0 or 1 and n is a non-negative integer, i.e. an element of {1,2,...}. To each such pair with a=0, associate the measure M(0,n)= 1/2^n. To each such pair with a=1 associate the “infinitesimal measure” M(1,n)=M(0,n)/omega where omega is taken to be indefinitely large.
So, the total measure of this space is 1 unit, and all but an infinitesimal portion of that total measure is associated with the portion of the space with a=0.
I claim that in some sense P(a=1) = 0 but P(n=2 | a=1) = 1⁄4.
The analogy here is that the assertion a=1 corresponds to the assertion that God exists. The probability is infinitesimal, yet Bayesian updating is possible (in some sense). And yet the space of all events is countable.
Ah, so by “there could be infinitesimals” you meant “there could be things of infinitesimal measure”.
Yes. Definitely. Sorry that was unclear. And infinitesimal measures result in probabilities which are zero in some sense, but not exactly zero in a different sense.
The two are related. Most relevantly, if my set is countable then I must have some singletons with non-zero measure. Moreover, the subset of points who have zero measure itself has zero measure, so they don’t matter at all. It is only in higher cardinality sets that you can have a collection of points each with zero measure that still have positive measure.
Ok, I can see that this tends to rule out my use of the unit-square analogy to justify my suggestion that the probability of the God Hypothesis is infinitesimal. I’m going to have to look more closely at the math, and in particular at my references for non-standard analysis to see whether any of my intuitions can be saved.
I’m curious then. Suppose that every child under five years or so old dissappeared as did most evangelical Christians. Would you then assign a chance of zero that the Rapture had just taken place?
It seems as though you are being overconfident.
What’s the deal with that? Are you trying to signal something?
Exactly—which is part of why I don’t get it. A 50 year-old habit, maybe?
However, there doesn’t seem much reason to speculate when we can just ask you.
The etymology fairly strongly suggests that a-theism is a lack of belief in theism, and a-gnosticism is a lack of belief in gnosticism.
Actually, the etymology of agnosticism is rejection of gnosis, which is rather broader than the gnostics. We know that because Huxley said so when he coined the word in 1869. That’s also exactly the meaning RobinZ gave.