I don’t think so. The “immeasurability” of the future that Vaden has highlighted has nothing to do with the literal finiteness of the timeline of the universe. It has to do, rather, with the set of all possible futures (which is provably infinite). This set is immeasurable in the mathematical sense of lacking sufficient structure to be operated upon with a well-defined probability measure. Let me turn the question around on you: Suppose we knew that the time-horizon of the universe was finite, can you write out the sample space, $\sigma$-algebra, and measure which allows us to compute over possible futures?
It certainly not obvious that the universe is infinite in the sense that you suggest. Certainly nothing is “provably infinite” with our current knowledge. Furthermore, although we may not be certain about the properties of our own universe, we can easily imagine worlds rich enough to contain moral agents yet which remain completely finite. For instance, you could image a cellular automata with a finite grid size and which only lasted for a finite duration.
However, perhaps the more important consideration is the in principle set of possible futures that we must consider when doing EV calculations, rather than the universe we actually inhabit, since even if our universe is finite we would never be able to convince our selves of this with certainty. Is it this set of possible futures that you think suffers from “immeasurability”?
if we helped ourselves to some cast-iron guarantees about the size and future lifespan of the universe (and made some assumptions about quantization) then we’d know that the set of possible futures was smaller than a particular finite number (since there would only be a finite number of time steps and a finite number of ways of arranging all particles at each time step). Then even if I can’t write it down, in principle someone could write it down, and the mathematical worries about undefined expectations go away.
It certainly not obvious that the universe is infinite in the sense that you suggest. Certainly nothing is “provably infinite” with our current knowledge. Furthermore, although we may not be certain about the properties of our own universe, we can easily imagine worlds rich enough to contain moral agents yet which remain completely finite. For instance, you could image a cellular automata with a finite grid size and which only lasted for a finite duration.
Aarrrgggggg was trying to resist weighing in again … but I think there’s some misunderstanding of my argument here. I wrote:
The set of all possible futures is infinite, regardless of whether we consider the life of the universe to be infinite. Why is this? Add to any finite set of possible futures a future where someone spontaneously shouts “1”!, and a future where someone spontaneously shouts “2”!, and a future where someone spontaneously shouts “3!” (italics added)
A few comments:
We’re talking about possible universes, not actual ones, so cast-iron guarantees about the size and future lifespan of the universe are irrelevant (and impossible anyway).
I intentionally framed it as someone shouting a natural number in order to circumvent any counterargument based on physical limits of the universe. If someone can think it, they can shout it.
The set of possible futures is provably infinite because the “shouting a natural number” argument established a one-to-one correspondence between the set of possible (triple emphasis on the word * possible * ) futures, and the set of natural numbers, which are provably infinite (see proof here ).
I’m not using fancy or exotic mathematics here, as Owen can verify. Putting sets in one-to-one correspondence with the natural numbers is the standard way one proves a set is countably infinite. (See https://en.wikipedia.org/wiki/Countable_set).
Physical limitations regarding the largest number that can be physically instantiated are irrelevant to answering the question “is this set finite or infinite”? Mathematicians do not say the set of natural numbers are finite because there are a finite number of particles in the universe. We’re approaching numerology territory here...
Okay this will hopefully be my last comment, because I’m really not trying to be a troll in the forum or anything. But please represent my argument accurately!
You really don’t seem like a troll! I think the discussion in the comments on this post is a very valuable conversation and I’ve been following it closely. I think it would be helpful for quite a few people for you to keep responding to comments
Of course, it’s probably a lot of effort to keep replying carefully to things, so understandable if you don’t have time :)
I second what Alex has said about this discussion being very valuable pushback against ideas that have got some traction—at the moment I think that strong longtermism seems right, but it’s important to know if I’m mistaken! So thank you for writing the post & taking some time to engage in the comments.
On this specific question, I have either misunderstood your argument or think it might be mistaken. I think your argument is “even if we assume that the life of the universe is finite, there are still infinitely many possible futures—for example, the infinite different possible universes where someone shouts a different natural number”.
But I think this is mistaken, because the universe will end before you finish shouting most natural numbers. In fact, there would only be finitely many natural numbers you could finish shouting before the universe ends, so this doesn’t show there are infinitely many possible universes. (Of course, there might be other arguments for infinite possible futures.)
More generally, I think I agree with Owen’s point that if we make the (strong) assumption the universe is finite in duration and finite in possible states, and can quantise time, then it follows that there are only finite possible universes, so we can in principle compute expected value.
So I’d be especially interested if you have any thoughts on whether expected value is in practice an inappropriate tool to use (e.g. with subjective probabilities) even assuming in principle it is computable. For example, I’d love to hear when (if at all) you think we should use expected value reasoning, and how we should make decisions when we shouldn’t.
On this specific question, I have either misunderstood your argument or think it might be mistaken. I think your argument is “even if we assume that the life of the universe is finite, there are still infinitely many possible futures—for example, the infinite different possible universes where someone shouts a different natural number”.
But I think this is mistaken, because the universe will end before you finish shouting most natural numbers. In fact, there would only be finitely many natural numbers you could finish shouting before the universe ends, so this doesn’t show there are infinitely many possible universes.
Yup you’ve misunderstood the argument. When we talk about the set of all future possibilities, we don’t line up all the possible futures and iterate through them sequentially. For example, if we say it’s possible tomorrow might either rain, snow, or hail, we * aren’t * saying that it will first rain, then snow, then hail. Only one of them will actually happen.
Rather we are discussing the set of possibilities {rain, snow, hail}, which has no intrinsic order, and in this case has a cardinality of 3.
Similarly with the set of all possible futures. If we let fi represent a possible future where someone shouts the number i, then the set of all possible futures is {f1, f2, f3, … }, which has cardinality ∞ and again no intrinsic ordering. We aren’t saying here that a single person will shout all numbers between 1 and ∞, because as with the weather example, we’re talking about what might possibly happen, not what actually happens.
More generally, I think I agree with Owen’s point that if we make the (strong) assumption the universe is finite in duration and finite in possible states, and can quantise time, then it follows that there are only finite possible universes, so we can in principle compute expected value.
No this is wrong. We don’t consider physical constraints when constructing the set of future possibilities—physical constraints come into the picture later. So in the weather example, we could include into our set of future possibilities something absurd, and which violates known laws of physics. For example we are free to construct a set like {rain, snow, hail, rains_frogs}.
Then we factor in physical constraints by assigning probability 0 to the absurd scenario. For example our probabilities might be {0.2,0.4,0.4,0}.
But no laws of physics are being violated with the scenario “someone shouts the natural number i”. This is why this establishes a one-to-one correspondence between the set of future possibilities and the natural numbers, and why we can say the set of future possibilities is (at least) countably infinite. (You could establish that the set of future possibilities is uncountably infinite as well by having someone shout a single digit in Cantor’s diagonal argument, but that’s beyond what is necessary to show that EVs are undefined.
For example, I’d love to hear when (if at all) you think we should use expected value reasoning, and how we should make decisions when we shouldn’t.
Yes I think that the EV style-reasoning popular on this forum should be dropped entirely because it leads to absurd conclusions, and basically forces people to think along a single dimension.
So for example I’ll produce some ridiculous future scenario (Vaden’s x-risk: In the year 254 012 412 there will be a war over blueberries in the Qualon region of delta quadrant , which causes an unfathomable amount of infinite suffering ) and then say: great, you’re free to set your credence about this scenario as high or as low as you like.
But now I’ve trapped you! Because I’ve forced you to think about the scenario only in terms of a single 1 dimensional credence-slider. Your only move is to set your credence-slider really really small, and I’ll set my suffering-slider really really high, and then using EVs, get you to dedicate your income and the rest of your life to Blueberry-Safety research.
Note also that EV style reasoning is only really popular in this community. No other community of researchers reasons in this way, and they’re able to make decisions just fine. How would any other community reason about my scenario? They would reject it as absurd and be done with it. Not think along a single axis (low credence/high credence).
That’s the informal answer, anyway. Realizing that other communities don’t reason in this way and are able to make decisions just fine should at least be a clue that dropping EV style arguments isn’t going to result in decision-paralysis.
The more formal answer is to consider using an entirely different epistemology, which doesn’t deal with EVs at all. This is what my vague comments about the ‘framework’ were eluding to in the piece. Specifically, I have in mind Karl Popper’s critical rationalism, which is at the foundation of modern science. CR is about much more than that, however. I discuss what a CR approach to decision making would look like in this piece if you want some longer thoughts on it.
But anyway, I digress… I don’t expect people to jettison their entire worldview just because some random dude on the internet tells them to. But for anyone reading who might be curious to know where I’m getting a lot of these ideas from (few are original to me), I’d recommend Conjectures and Refutations. If you want to know what an alternative to EV style reasoning looks like, the answers are in that book.
(Note: This is a book many people haven’t read because think they already know the gist. “Oh, C&R! That’s the book about falsification, right?” It’s about much much more than that :) )
Hi Vaden, thanks again for posting this! Great to see this discussion. I wanted to get further along C&R before replying, but:
no laws of physics are being violated with the scenario “someone shouts the natural number i”. This is why this establishes a one-to-one correspondence between the set of future possibilities and the natural numbers
If we’re assuming that time is finite and quantized, then wouldn’t these assumptions (or, alternatively, finite time + the speed of light) imply a finite upper bound on how many syllables someone can shout before the end of the universe (and therefore a finite upper bound on the size of the set of shoutable numbers)? I thought Isaac was making this point; not that it’s physically impossible to shout all natural numbers sequentially, but that it’s physically impossible to shout any of the natural numbers (except for a finite subset).
(Although this may not be crucial, since I think you can still validly make the point that Bayesians don’t have the option of, say, totally ruling out faster-than-light number-pronunciation as absurd.)
Note also that EV style reasoning is only really popular in this community. No other community of researchers reasons in this way, and they’re able to make decisions just fine.
Are they? I had the impression that most communities of researchers are more interested in finding interesting truths than in making decisions, while most communities of decision makers severely neglect large-scale problems. (Maybe there’s better ways to account for scope than EV, but I’d hesitate to look for them in conventional decision making.)
It certainly not obvious that the universe is infinite in the sense that you suggest. Certainly nothing is “provably infinite” with our current knowledge. Furthermore, although we may not be certain about the properties of our own universe, we can easily imagine worlds rich enough to contain moral agents yet which remain completely finite. For instance, you could image a cellular automata with a finite grid size and which only lasted for a finite duration.
However, perhaps the more important consideration is the in principle set of possible futures that we must consider when doing EV calculations, rather than the universe we actually inhabit, since even if our universe is finite we would never be able to convince our selves of this with certainty. Is it this set of possible futures that you think suffers from “immeasurability”?
Aarrrgggggg was trying to resist weighing in again … but I think there’s some misunderstanding of my argument here. I wrote:
A few comments:
We’re talking about possible universes, not actual ones, so cast-iron guarantees about the size and future lifespan of the universe are irrelevant (and impossible anyway).
I intentionally framed it as someone shouting a natural number in order to circumvent any counterargument based on physical limits of the universe. If someone can think it, they can shout it.
The set of possible futures is provably infinite because the “shouting a natural number” argument established a one-to-one correspondence between the set of possible (triple emphasis on the word * possible * ) futures, and the set of natural numbers, which are provably infinite (see proof here ).
I’m not using fancy or exotic mathematics here, as Owen can verify. Putting sets in one-to-one correspondence with the natural numbers is the standard way one proves a set is countably infinite. (See https://en.wikipedia.org/wiki/Countable_set).
Physical limitations regarding the largest number that can be physically instantiated are irrelevant to answering the question “is this set finite or infinite”? Mathematicians do not say the set of natural numbers are finite because there are a finite number of particles in the universe. We’re approaching numerology territory here...
Okay this will hopefully be my last comment, because I’m really not trying to be a troll in the forum or anything. But please represent my argument accurately!
You really don’t seem like a troll! I think the discussion in the comments on this post is a very valuable conversation and I’ve been following it closely. I think it would be helpful for quite a few people for you to keep responding to comments
Of course, it’s probably a lot of effort to keep replying carefully to things, so understandable if you don’t have time :)
I second what Alex has said about this discussion being very valuable pushback against ideas that have got some traction—at the moment I think that strong longtermism seems right, but it’s important to know if I’m mistaken! So thank you for writing the post & taking some time to engage in the comments.
On this specific question, I have either misunderstood your argument or think it might be mistaken. I think your argument is “even if we assume that the life of the universe is finite, there are still infinitely many possible futures—for example, the infinite different possible universes where someone shouts a different natural number”.
But I think this is mistaken, because the universe will end before you finish shouting most natural numbers. In fact, there would only be finitely many natural numbers you could finish shouting before the universe ends, so this doesn’t show there are infinitely many possible universes. (Of course, there might be other arguments for infinite possible futures.)
More generally, I think I agree with Owen’s point that if we make the (strong) assumption the universe is finite in duration and finite in possible states, and can quantise time, then it follows that there are only finite possible universes, so we can in principle compute expected value.
So I’d be especially interested if you have any thoughts on whether expected value is in practice an inappropriate tool to use (e.g. with subjective probabilities) even assuming in principle it is computable. For example, I’d love to hear when (if at all) you think we should use expected value reasoning, and how we should make decisions when we shouldn’t.
Hey Issac,
Yup you’ve misunderstood the argument. When we talk about the set of all future possibilities, we don’t line up all the possible futures and iterate through them sequentially. For example, if we say it’s possible tomorrow might either rain, snow, or hail, we * aren’t * saying that it will first rain, then snow, then hail. Only one of them will actually happen.
Rather we are discussing the set of possibilities {rain, snow, hail}, which has no intrinsic order, and in this case has a cardinality of 3.
Similarly with the set of all possible futures. If we let fi represent a possible future where someone shouts the number i, then the set of all possible futures is {f1, f2, f3, … }, which has cardinality ∞ and again no intrinsic ordering. We aren’t saying here that a single person will shout all numbers between 1 and ∞, because as with the weather example, we’re talking about what might possibly happen, not what actually happens.
No this is wrong. We don’t consider physical constraints when constructing the set of future possibilities—physical constraints come into the picture later. So in the weather example, we could include into our set of future possibilities something absurd, and which violates known laws of physics. For example we are free to construct a set like {rain, snow, hail, rains_frogs}.
Then we factor in physical constraints by assigning probability 0 to the absurd scenario. For example our probabilities might be {0.2,0.4,0.4,0}.
But no laws of physics are being violated with the scenario “someone shouts the natural number i”. This is why this establishes a one-to-one correspondence between the set of future possibilities and the natural numbers, and why we can say the set of future possibilities is (at least) countably infinite. (You could establish that the set of future possibilities is uncountably infinite as well by having someone shout a single digit in Cantor’s diagonal argument, but that’s beyond what is necessary to show that EVs are undefined.
Yes I think that the EV style-reasoning popular on this forum should be dropped entirely because it leads to absurd conclusions, and basically forces people to think along a single dimension.
So for example I’ll produce some ridiculous future scenario (Vaden’s x-risk: In the year 254 012 412 there will be a war over blueberries in the Qualon region of delta quadrant , which causes an unfathomable amount of infinite suffering ) and then say: great, you’re free to set your credence about this scenario as high or as low as you like.
But now I’ve trapped you! Because I’ve forced you to think about the scenario only in terms of a single 1 dimensional credence-slider. Your only move is to set your credence-slider really really small, and I’ll set my suffering-slider really really high, and then using EVs, get you to dedicate your income and the rest of your life to Blueberry-Safety research.
Note also that EV style reasoning is only really popular in this community. No other community of researchers reasons in this way, and they’re able to make decisions just fine. How would any other community reason about my scenario? They would reject it as absurd and be done with it. Not think along a single axis (low credence/high credence).
That’s the informal answer, anyway. Realizing that other communities don’t reason in this way and are able to make decisions just fine should at least be a clue that dropping EV style arguments isn’t going to result in decision-paralysis.
The more formal answer is to consider using an entirely different epistemology, which doesn’t deal with EVs at all. This is what my vague comments about the ‘framework’ were eluding to in the piece. Specifically, I have in mind Karl Popper’s critical rationalism, which is at the foundation of modern science. CR is about much more than that, however. I discuss what a CR approach to decision making would look like in this piece if you want some longer thoughts on it.
But anyway, I digress… I don’t expect people to jettison their entire worldview just because some random dude on the internet tells them to. But for anyone reading who might be curious to know where I’m getting a lot of these ideas from (few are original to me), I’d recommend Conjectures and Refutations. If you want to know what an alternative to EV style reasoning looks like, the answers are in that book.
(Note: This is a book many people haven’t read because think they already know the gist. “Oh, C&R! That’s the book about falsification, right?” It’s about much much more than that :) )
Hi Vaden, thanks again for posting this! Great to see this discussion. I wanted to get further along C&R before replying, but:
If we’re assuming that time is finite and quantized, then wouldn’t these assumptions (or, alternatively, finite time + the speed of light) imply a finite upper bound on how many syllables someone can shout before the end of the universe (and therefore a finite upper bound on the size of the set of shoutable numbers)? I thought Isaac was making this point; not that it’s physically impossible to shout all natural numbers sequentially, but that it’s physically impossible to shout any of the natural numbers (except for a finite subset).
(Although this may not be crucial, since I think you can still validly make the point that Bayesians don’t have the option of, say, totally ruling out faster-than-light number-pronunciation as absurd.)
Are they? I had the impression that most communities of researchers are more interested in finding interesting truths than in making decisions, while most communities of decision makers severely neglect large-scale problems. (Maybe there’s better ways to account for scope than EV, but I’d hesitate to look for them in conventional decision making.)