If this were the case, then what is to stop me from thinking of N>3^^^^^3 impossible methods of gaining evidence (aliens from Mars, or Planet X, or the past, or from Cygnus XJ45, or another dimension...), and then claiming that since these probabilities are mutually independent, summing up the positive probabilities, and claiming evolution (or any theory) to be unlikely to be true? I mean, aside from the thing about probability theory being invalid, which I haven’t seen before.
Also, thank you Eliezer, for explaining why the argument about cryonics is logically rude. I’ve been banging my head on this exact topic with a friend for the past week and have been unable to get past that road block with her.
First, you can’t think of 3^^^^^3 ways of gaining evidence, possible or impossible, because there are not that many possible distinct states of a human brain (or of the physical universe, for that matter.)
Second, the more complex your hypothesis, the less probable it will be, so some hypotheses might only change the probability by 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3, or even less, and so it is perfectly possible to sum them all and still only move the probability by a very small amount.
You and Eliezer make good points, thank you. I just started reading about negative probabilities. I don’t believe I’ve heard of them before. Just to be clear, I never claimed that the sum of possibilities would diverge, though I don’t think I gave proper attention to the prior probability distribution summing to 1.
I did not mean to imply that I would individually think up every single impossible possibility. I figured it would be enough to hook into some countably infinite set and show that it is just a subset of all the possible impossibilities we could generate. One could simply tap into the Infinite Earths of DC Comics to construct an argument that resembles the original lack of time camera argument:
The absence of any Superman from Earths 1 through 3^^^^^^^^^^^^3 confirming evolution must elevate the probability that evolution is false.
In my obviously absurd example, I am not sure why Superman from Earth 5000 would be any more complex than Superman from Earth 500. Though I suppose the numbering system would indicate an elevated degree of difficulty crossing over, perhaps. It is true, I didn’t account for all the parallel Earths where Superman is evil, or disinterested in our Earth, or unable to get here. My mind boggles at the possibilities.
I think my original complaint remains, though. Why would the absence of evidence from something that is admittedly impossible increase the probability of something being false? I suppose I am complaining too much for such a tiny marginal increase in probability, since a random person on the street shouting “evolution is false!” is probably going to sway your opinion to a far larger degree than some 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3 event.
However, it strikes me as strange that someone should feel obligated to disbelieve (even a tiny bit) evolution on the grounds that Superman 3^^3 didn’t tell him/her that it was true.
The point was that you can easily have a sum of an infinite series that adds to some small finite amount. Assuming “Earth 5000” is defined differently from “Earth 500″ (which it must be in order to have new hypothesis), your different hypotheses will different complexities depending on the complexity of the number. Overall (but not in every single instance) the higher the number, the more complex the hypothesis, so the less the probability will be changed. There is no reason for this infinite sum not to converge to an extremely small quantity.
In any case (and this may be Peter de Blanc’s point), these probabilities are smaller than the sensitivity of the human judgement: so in fact, subjectively you don’t need to feel obliged to change your opinion at all based on them.
That makes sense. I guess as long as the sum of the infinitely many absurdly contrived possibilities remains less than rounding error and/or sensitivity of human judgment, I have no qualms with your original point.
There are counterbalancing negative possibilities, you can’t sum over just the positive ones. And since a prior probability distribution sums to 1, the contribution of even solely the positive possibilities must converge to a finite sum rather than diverging.
It happens that I’ve also been banging my head on this exact topic with a friend about a week ago and failed to get past that road block with him. Upvotes are in order.
If this were the case, then what is to stop me from thinking of N>3^^^^^3 impossible methods of gaining evidence (aliens from Mars, or Planet X, or the past, or from Cygnus XJ45, or another dimension...), and then claiming that since these probabilities are mutually independent, summing up the positive probabilities, and claiming evolution (or any theory) to be unlikely to be true? I mean, aside from the thing about probability theory being invalid, which I haven’t seen before.
Also, thank you Eliezer, for explaining why the argument about cryonics is logically rude. I’ve been banging my head on this exact topic with a friend for the past week and have been unable to get past that road block with her.
First, you can’t think of 3^^^^^3 ways of gaining evidence, possible or impossible, because there are not that many possible distinct states of a human brain (or of the physical universe, for that matter.)
Second, the more complex your hypothesis, the less probable it will be, so some hypotheses might only change the probability by 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3, or even less, and so it is perfectly possible to sum them all and still only move the probability by a very small amount.
You and Eliezer make good points, thank you. I just started reading about negative probabilities. I don’t believe I’ve heard of them before. Just to be clear, I never claimed that the sum of possibilities would diverge, though I don’t think I gave proper attention to the prior probability distribution summing to 1.
I did not mean to imply that I would individually think up every single impossible possibility. I figured it would be enough to hook into some countably infinite set and show that it is just a subset of all the possible impossibilities we could generate. One could simply tap into the Infinite Earths of DC Comics to construct an argument that resembles the original lack of time camera argument:
The absence of any Superman from Earths 1 through 3^^^^^^^^^^^^3 confirming evolution must elevate the probability that evolution is false.
In my obviously absurd example, I am not sure why Superman from Earth 5000 would be any more complex than Superman from Earth 500. Though I suppose the numbering system would indicate an elevated degree of difficulty crossing over, perhaps. It is true, I didn’t account for all the parallel Earths where Superman is evil, or disinterested in our Earth, or unable to get here. My mind boggles at the possibilities.
I think my original complaint remains, though. Why would the absence of evidence from something that is admittedly impossible increase the probability of something being false? I suppose I am complaining too much for such a tiny marginal increase in probability, since a random person on the street shouting “evolution is false!” is probably going to sway your opinion to a far larger degree than some 1/3^^^^^^^^^^^^^^^^^^^^^^^^^^^3 event.
However, it strikes me as strange that someone should feel obligated to disbelieve (even a tiny bit) evolution on the grounds that Superman 3^^3 didn’t tell him/her that it was true.
(edit: I got the Superman 500 mixed up with 5000)
The point was that you can easily have a sum of an infinite series that adds to some small finite amount. Assuming “Earth 5000” is defined differently from “Earth 500″ (which it must be in order to have new hypothesis), your different hypotheses will different complexities depending on the complexity of the number. Overall (but not in every single instance) the higher the number, the more complex the hypothesis, so the less the probability will be changed. There is no reason for this infinite sum not to converge to an extremely small quantity.
In any case (and this may be Peter de Blanc’s point), these probabilities are smaller than the sensitivity of the human judgement: so in fact, subjectively you don’t need to feel obliged to change your opinion at all based on them.
That makes sense. I guess as long as the sum of the infinitely many absurdly contrived possibilities remains less than rounding error and/or sensitivity of human judgment, I have no qualms with your original point.
There are counterbalancing negative possibilities, you can’t sum over just the positive ones. And since a prior probability distribution sums to 1, the contribution of even solely the positive possibilities must converge to a finite sum rather than diverging.
It happens that I’ve also been banging my head on this exact topic with a friend about a week ago and failed to get past that road block with him. Upvotes are in order.