James is a really nice guy. He was at my table in Lisbon, so be polite.
the melting and breakdown of polar ice sheets seems to be in the vicinity of a couple of degrees warming. This expectation is based on current high rates of mass loss from the ice sheets compared to relative stability through the Holocene (the past 10,000 years) and on past ice sheet response in periods such as the Pliocene (a few million years ago) when the Earth was a couple of degrees warmer than preindustrial times (and sea level up to 25m higher).
We have already had about 0.8°C warming globally, with another third of a degree locked in by the inertia of the climate system.
That leaves, somewhat optimistically, perhaps a degree or so of wiggle room. Translating that into carbon emissions, if we wish to keep the total warming below about 2°C (with 50% chance), then we have a total global carbon emission allocation of between about 800 and 1000Gt carbon.
We have already emitted about 550Gt, leaving perhaps another 250–450Gt. Current global emissions are about 10Gt per year, growing at roughly 3% per year.
That leaves a few decades at present rates before having committed to 2°C warming and crossing the expected thresholds for ice sheet disintegration. And that is for a 50% chance of not crossing the 2°C threshold. For more comfortable odds of staying within the threshold, the total carbon allocation drops and so the time to threshold is even shorter.