There is a survey currently running on the Doomstead Diner, which asks people to make specific, numerical estimates about the timing of human extinction. It is inspired by the work of Guy McPherson, who has amassed much scientific evidence that points to very major climate disruption occurring over the next 2-3 decades, caused by multiple runaway positive feedback effects, such as Arctic methane release. Guy’s conclusion is that these changes will mean that the Earth will no longer provide a habitat for humans, leading to near-term human extinction. His reasoning, as far as I have been able to piece it together, rests on a supposition of time-invariance: the planet will be warmer than it has ever been in human experience; therefore, no humans will survive. This is far short of a proof.
I see two ways to provide a proof.
The first is based on proving the existence of an extinction mechanism. For example, humans don’t function well when atmospheric CO2 concentrations exceed 5000ppm, which cause dizziness, fainting spells and asphyxiation. Right now they are around 400ppm, going up by 2ppm every year. If that stays on track, this gives us 2300 years. However, there is not enough fossil fuels to keep burning at the same rate for another 2300 years. I am not aware of any straightforward bit of math that would conclusively demonstrate the impossibility of our continued existence.
The second is to make an inventory of all possible human habitats, and lifestyles to go with them, and then demonstrate that none of these habitats will be available in just a few decades. This is tricky, because it’s so easy to pass over some small niche that may remain survivable far into the future, and all it takes is one of these to narrowly avoid extinction. An examination of mitochondrial DNA showed that at one point the human population dwindled to just a handful, yet we are still around—numbering in the billions! Extermination is hard—ask any exterminator—and extinction is even harder.