Physics (specifically radiation transfer mechanics) provides the compellingly supported understanding that CO2 is capable of acting like a "greenhouse" gas, the PETM is nothing more than a recent geological example of what happens when you rapidly (over the period of ~ 10k years) flood the Earth's atmosphere with climatically significant volumes of such greenhouse gases. We are currently adding CO2 to the atmosphere at ~10x the rate that occurred during the PETM and our rate of emissions is still increasing.
So, in a nutshell, certain species of forams suffered very high extinction rates. Different species on the other hand did very well. Mammals did exceptionally well and contrary to the incessant nonsense about heat killing the opposite is true. Warmth allowed plants to grow well and that allowed fauna to do well.
How do you explain that?
Paleocene
You aren't reading everything, about PETM. Acidification caused a lot of reduction, of ocean species. Survivors migrated. Dwarfism may have contributed, to diversity.
Plant diversity comes easier, than animal diversity. Plants adjust to CO2, obviously, but they develop less stomata, in times of CO2 proliferation.
But we are realeasing GHGs faster, than the volcanic eruptions of the PETM did:
PETM: Global Warming, Naturally | Weather Underground
PETM Warming vs. Current Warming
During the PETM, around 5 billion tons of CO2 was released into the atmosphere per year. The Earth warmed around 6°C (11°F) over 20,000 years, although some estimates are that the warming was more like 9°C (16°F). Using the low end of that estimated range, the globe warmed around 0.025°C every 100 years. Today, the globe is warming at least ten times as fast, anywhere from 1 to 4°C every 100 years. In 2010, our fossil fuel burning released 35 billion tons of CO2 into the atmosphere. By comparison, volcanoes release 0.2 billion tons of CO2 per year. How fast carbon enters the atmosphere translates to the how fast temperature increases, and the environmental and societal consequences of warming at such a break-neck speed could be devastating.
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How fast the methane outgasses is a factor:
Paleocene
Ocean warming due to flooding and pressure changes due to a sea-level drop may have caused clathrates to become unstable and release methane. This can take place over as short of a period as a few thousand years. The reverse process, that of fixing methane in clathrates, occurs over a larger scale of tens of thousands of years.[36]
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Here's a really good article, which tries to resolve outcomes, with disputed proxies, but the implications for our faster outgassing are clear. We re-green, or we migrate 7 billion people, to some part of the world, less affected, by faster warming, than during the PETM. Migration was the way to survival, for species affected by the PETM, so why not move to Mars, since all we have to do to get there is rocket on up there, lay down some GHGs, grow some lichens, introduce bugs, rocket all of us up there, and then we can be spacemen, who eat bugs:
RealClimate: PETM Weirdness
Temperature changes at the same time as this huge carbon spike were large too. Note that this is happening on a Paleocene background climate that we don’t fully understand either – the polar amplification in very warm paleo-climates is much larger than we’ve been able to explain using standard models. Estimates range from 5 to 9 deg C warming (with some additional uncertainty due to potential problems with the proxy data) – smaller in the tropics than at higher latitudes.
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It seems likely we will remain on Earth, where many humans will eat shit, and die, while most species become extinct because humans are collectively stupid and greedy. I wonder how long the crime-pays economy will persist? Jamie couldn't make money! What's going to happen, where idiot-traffic can't see a warmup, underway?
The first good question is, how fast will shit happen? In a geologic instant!