This thought process suggests something very wrong. The guess "it will last again as long as it has lasted so far" doesn't give any real insight. The wall was actually as likely to end five months from when they visited it, as it was to end 500 years from then.
What this "time-wise Copernican principle" gives you is a guarantee that, if you apply this logic every time you have no other knowledge and have to guess, you will get the least mean error over all of your guesses. For some events, you'll guess that they'll end in 5 minutes, and they actually end 50 years later. For others, you'll guess they'll take another 50 years and they actually end 5 minutes later. Add these two up, and overall you get 0 - you won't have either a bias to overestimating, nor to underestimating.
But this doesn't actually give you any insight into how long the event will actually last. For a single event, with no other knowledge, the probability that it will after 1 minute is equal to the probability that it will end after the same duration that it lasted so far, and it is equal to the probability that it will end after a billion years. There is nothing at all that you can say about the probability of an event ending from pure mathematics like this - you need event-specific knowledge to draw any conclusions.
So while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation.
But you will never guess that the latest tik-tok craze will last another 50 years, and you'll never guess that Saturday Night Live (which premiered in 1075) will end 5-minutes from now. Your guesses are thus more likely to be accurate than if you ignored the information about how long something has lasted so far.
Sure, but the opposite also applies. If in 1969 you guessed that the wall would last another 20 years, then in 1989, you'll guess that the wall of Berlin will last another 40 years - when in fact it was about to fall. And in 1949, when the wall was a few months old, you'll guess that it will last for a few months at most.
So no, you're not very likely to be right at all. Now sure, if you guess "50 years" for every event, your average error rate will be even worse, across all possible events. But it is absolutely not true that it's more likely that SNL will last for another 50 years as it is that it will last for another 10 years. They are all exactly as likely, given the information we have today.
If I understand the original theory, we can work out the math with a little more detail... (For clarity, the berlin wall was erected in 1961.)
- In 1969 (8 years after the wall was erected): You'd calculate that there's a 50% chance that the wall will fall between 1972 (8x4/3=11 years) and 1993 (8x4=32 years)
- In 1989 (28 years after the wall was erected): You'd calculate that there's a 50% chance that the wall will fall between 1998 (28x4/3=37 years) and 2073 (28x4=112 years)
- In 1961 (when the wall was, say, 6 months old): You'd calculate that there's a 50% chance that the wall will fall between 1961 (0.5x4/3=0.667 years) and 1963 (0.5x4=2 years)
I found doing the math helped to point out how wide of a range the estimate provides. And 50% of the times you use this estimation method; your estimate will correctly be within this estimated range. It's also worth pointing out that, if your visit was at a random moment between 1961 and 1989, there's only a 3.6% chance that you visited in the final year of its 28 year span, and 1.8% chance that you visited in the first 6 months.
> Well, there’s nothing special about the timing of my visit. I’m just travelling—you know, Europe on five dollars a day—and I’m observing the Wall because it happens to be here.
It's relatively unlikely that you'd visit the Berlin Wall shortly after it's erected or shortly before it falls, and quite likely that you'd visit it somewhere in the middle.
No, it's exactly as likely that I'll visit it at any one time in its lifetime. Sure, if we divide its lifetime into 4 quadrants, its more likely I'm in quadrant 2-3 than in either of 1 or 4. But this is slight of hand: it's still exactly as likely that I'm in quadrant 2-3 than in quadrant (1 or 4) - or, in other words, it's as likely I'm at one of the ends of the lifetime as it is that I am in the middle.
> while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation
It's important to flag that the principle is not trite, and it is useful.
There's been a misunderstanding of the distribution after the measurement of "time taken so far", (illuminated in the other thread), which has lead to this incorrect conclusion.
To bring the core clarification from the other thread here:
The distribution is uniform before you get the measurement of time taken already. But once you get that measurement, it's no longer uniform. There's a decaying curve whose shape is defined by the time taken so far. Such that the estimate `time_left=time_so_far` is useful.
If this were actually correct, than any event ending would be a freak accident: since, according to you, the probability of something continuing increases drastically with its age. That is, according to your logic, the probability of the wall of Berlin falling within the year was at its lowest point in 1989, when it actually fell. In 1949, when it was a few months old, the probability that it would last for at least 40 years was minuscule, and that probability kept increasing rapidly until the day the wall was collapsed.
Why is the most likely time right now? What makes right now more likely than in five minutes? I guess you're saying if there's nothing that makes it more likely to fail at any time than at any other time, right now is the only time that's not precluded by it failing at other times? I.E. it can't fail twice, and if it fails right now it can't fail at any other time, but even if it would have failed in five minutes it can still fail right now first?
Yes that's pretty much it. There will be a decaying probability curve, because given you could fail at any time, you are less likely to survive for N units of time than for just 1 unit of time, etc.
> However, the average expected future lifetime increases as a thing ages, because survival is evidence of robustness.
This is a completely different argument that relies on various real-world assumptions, and has nothing to do with the Copernican principle, which is an abstract mathematical concept. And I actually think this does make sense, for many common categories of processes.
However, even this estimate is quite flawed, and many real-world processes that intuitively seem to follow it, don't. For example, looking at an individual animal, it sounds kinda right to say "if it survived this long, it means it's robust, so I should expect it will survive more". In reality, the lifetime of most animals is a binomial distribution - they either very young, because of glaring genetic defects or simply because they're small, fragile, and inexperienced ; or they die at some common age that is species dependent. For example, a humab that survived to 20 years of age has about the same chance of reaching 80 as one that survived to 60 years of age. And an alien who has no idea how long humans live and tries to apply this method may think "I met this human when they're 80 years old - so they'll probably live to be around 160".
> The wall was actually as likely to end five months from when they visited it, as it was to end 500 years from then.
I don't think this is correct; as in something that has been there for say hundreds of years had more probability to be there in a hundred years than something that has been there for a month.
What this "time-wise Copernican principle" gives you is a guarantee that, if you apply this logic every time you have no other knowledge and have to guess, you will get the least mean error over all of your guesses. For some events, you'll guess that they'll end in 5 minutes, and they actually end 50 years later. For others, you'll guess they'll take another 50 years and they actually end 5 minutes later. Add these two up, and overall you get 0 - you won't have either a bias to overestimating, nor to underestimating.
But this doesn't actually give you any insight into how long the event will actually last. For a single event, with no other knowledge, the probability that it will after 1 minute is equal to the probability that it will end after the same duration that it lasted so far, and it is equal to the probability that it will end after a billion years. There is nothing at all that you can say about the probability of an event ending from pure mathematics like this - you need event-specific knowledge to draw any conclusions.
So while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation.