There is such a strong association between the topic of ancient advanced civilizations and crazy fringe people, it's almost impossible to separate them. Even SETI has a better reputation.

But the topic seems to be worth a serious consideration:

  • Asking if there were advanced civilizations on Earth before us is a valid scientific inquiry. It is of the same validity as asking if there are advanced civilizations beyond Earth.
  • We don't know if there were advanced civilizations before us. The problem is, even if our current global civilization is destroyed, it will be hard to find any evidence of its existence only after 50k years or something. By the time, even our largest cities will be ground to dust and buried by hundreds of meters of sediments. And after some millions of years, the future archeologists will have a hard time finding anything at all, unless they're specifically looking for signatures of advanced civilizations (e.g. anomalous CO2 levels in ice cores).
  • There is a time abyss. Anatomically modern humans have been around for 300k years. That's enough time for several repetitions of our path from early agriculture to computers. And before humans, there could have been other sufficiently brainy species (e.g. Troodon dinosaurs).

So, the interesting question is, what if we're not the first AI-capable civilization on Earth?

If we ever find an unquestionable evidence of the existence of such a civilization, it could shed light on our own prospects. It's one thing to speculate about AGI destroying the world, but another thing to see an archeological evidence of such a destruction. 

And maybe the ancient genocidal AGI is still around somewhere. 

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Already downvoted enough, so I won't pile on, but I'll explain why I don't find this a valuable post.

 So, the interesting question is, what if we're not the first AI-capable civilization on Earth?

Well, no.  That's not an interesting question.  Whether or not we're not the first such civilization, we STILL experience everything exactly the same.  Reality is truth.  We can't change the past and see what's different.  There's no "what if" that makes sense for that topic.

You MIGHT ask "what are we failing to focus on that could give evidence for or against this hypothesis?"  Then go measure that and you have something to improve your modeling.  You MIGHT ask "what puzzles are present today that this possibility explains", but I suspect you'll mostly have pretty tenuous just-so stories.

Really, you should be asking (or better yet, stating, after you've spent the time studying something your intuition says is worthwhile, but most people disagree) whether there are artifacts or effects actually existent which help determine whether it's true, and what additional experiments or observations we could make.

But the topic seems to be worth a serious consideration

You don't give any reasons to give it serious consideration.  There are BILLIONS of ideas that your general points apply to - why this one?

Note: I see from your comments that you're mentioning Ancient Greece and talking about "brief sparkles of civilization".  That feels like a significant motte-and-bailey switch in a post about "AI-capable civilizations".  It's quite possible that such local and brief civilizations have happened and collapsed multiple times.  I would still ask "so what" before "what if".

Well, no.  That's not an interesting question.  Whether or not we're not the first such civilization, we STILL experience everything exactly the same.  Reality is truth.  We can't change the past and see what's different.  There's no "what if" that makes sense for that topic.

Perhaps my choice of the phrasing was sub-optimal (I'm a non-native English speaker). The intent of this question-post is to identity interesting corollaries of the hypothesis (including testable predictions).  

You don't give any reasons to give it serious consideration.

The core justification for this hypothesis is that it's basically SETI, but for civilizations in the Earth's past instead of civilizations in the outer space. And many people in the community seem to find SETI worth a serious consideration. 

 It's quite possible that such local and brief civilizations have happened and collapsed multiple times.  I would still ask "so what" before "what if".

Regarding "so what", I don't know yet (a purpose of this question-post is to explore this kind of stuff). So far I can see two interesting corollaries:

  • we can apply something like the Drake equitation or "grabby aliens" ideas, but to the hypothetical previous civilizations of the Earth. This could give us some insight about our own future
  • the possibility of the ancient genocidal AGIs still roaming around (or lying dormant in some ruins) - seems important.

Strongly upvoted because the brainy dinosaurs possibility is pretty interesting. Any remnants of their civilization such as anomalous iron deposits, would likely be buried too deep for us to stumble upon it. The minute quantities that would make it closer to the surface could be easily dismissed for a variety of factors.

Among dinosaur candidates, Troodon seems to be the most interesting:

  • a bipedal species, with agile hands and stereoscopic vision [1]
  • the structure of the teeth suggests a preference for soft food [2
  • a large brain for the animal's size [1]
  • social (it's known that their nests were shared by multiple females) [4]
  • massive concentrations of atmospheric CO2 during the time (higher than today) [3], followed by one of the largest extinction events in the Earth's history (the Cretaceous–Paleogene extinction event)

However the chart in your last link shows the peak of that concentration seems to be ~120 million years ago, with a pretty steep decline in geologic terms afterwards, ~55 million years before the K-T extinction.

Good catch! It seems that although CO2 was much higher than today, it was mostly in decline during the Troodon times. Doesn't look like an effect of a tech civilization. 

There is a time abyss. Anatomically modern humans have been around for 300k years. That's enough time for several repetitions of our path from early agriculture to computers.

That seems unlikely - our genetic diversity provides evidence of how many humans there were at any point in the past. We would notice if there were billions of humans only a couple of hundreds of thousand years ago. See https://en.m.wikipedia.org/wiki/Prehistoric_demography.

Furthermore what would they have powered their industrial revolution with? How come there's still so much surface level coal and oil? A few hundred thousands of years shouldn't be enough to regenerate that.

And before humans, there could have been other sufficiently brainy species (e.g. Troodon dinosaurs).

I'm guessing that we would still be able to notice that in the distribution of minerals - e.g. maybe all the high iron concentration ores should have been mined, but I don't really know much about this.

That seems unlikely - our genetic diversity provides evidence of how many humans there were at any point in the past. We would notice if there were billions of humans only a couple of hundreds of thousand years ago

Judging by the existence of the highly sophisticated Ancient Greek civilization (capable of devising and making the Antikythera mechanism, a mechanical computer), there is no necessity for a technological civilization to be billions of people strong. At the time, the entire population of Europe was about 34 million, comparable to the today's population of California. 

Perhaps the hypothetical previous civilization was not a global behemoth like ours, but more like the Ancient Greece.

By the way, it took Greeks only 7k years from the first stone-age agricultural societies to the Antikythera mechanism (and also to simple steam engines and all kinds of other interesting tech). 

I wounder how many such brief sparkles of civilization the Earth has seen, with the ruins buried under kilometers of sediment, or vanished without any trace. 

I'm guessing that we would still be able to notice that in the distribution of minerals - e.g. maybe all the high iron concentration ores should have been mined, but I don't really know much about this.

Troodonts became extinct about 76 million years ago. Since then, even some continental plates have been swallowed by the Earth's crust. But I agree, perhaps it's possible to infer a possible advanced civilization using geological or paleontological indicators. E.g. certain patterns of mass extinctions, an anomalous rise of CO2 etc.