Oh no, it's by someone from Harvard!
I seriously suggest that you read the paper. It is really more of a philosophy piece attacking economics than an economic paper, but it is very convincing.
Although Weitzman was skeptical of the low discount rate and high cost of carbon from the Stern report, he sought to justify a higher cost of carbon on the premise that the uncertainty can increase the risk of a low-probability catastrophe. His first argument is that we have already exceeded the maximum carbon dioxide levels in the last 800,000 years. Conventional climate economics yields an optimal policy of stabilization at around 700 ppm while the previous high was 300 ppm. This could potentially be very dangerous.
Of course, vastly increasing carbon dioxide does not necessarily mean that global temperatures are going to be unmanageable, but the greater the concentration, the greater the risk. The IPCC had only modeled warming up to 4.5 degrees Celsius. Depending on the type of curve projecting warming, the probability of low-probability events can be significantly different. Using a pareto (power) distribution, the probability of high-warming is significantly greater than using normal distribution. This lies in the "fat tail" of a power distribution that yields higher probability for extreme events.
There are real reasons for having concern for catastrophic warming. A big concern among climate scientists is the possible release of massive amounts of methane. We have large methane deposits stuck in permafrost that may slowly leak out methane as the permafrost melts from higher temperatures. This positive feedback could be potentially devastating. This article was written before the shale gas revolution, but there is concern now about methane leakage from natural gas deposits negating the lower carbon dioxide emissions from natural gas relative to coal.
Another concern is that the damages of catastrophic warming are underrepresented in standard models. For William Nordhaus' damage function, 10 degree warming in only supposed to lead to a 19% decline in world output. Over 200 years, this would mean only an average of 2% decline in output per year, not quite world-ending. Weitzman posits that some standard assumptions for economic modeling may not hold with extreme warming. He mentions that substitutability of goods. I am not quite sure that I understand, but there is a difference between utility being multiplicatively separable and additively separable. For example, material wealth is not easily substitutable for biodiversity.
The final concern is the discount rate. Weitzman says that the choice of discount rate is largely normative (not scientific). Discounting future value means that damage in the future is not as important as damage today. If you don't know what I'm talking about, think of putting $50 in the bank today. You expect it to be worth more in 50 years, so its future value in 50 years is much higher. We will probably have a stronger, more robust economy in 50 years that is better able to deal with climate change, so there is economic sense in not doing everything today. How much we should discount though is the difference between saying that we should do nothing to having a carbon price of $300 per ton of carbon dioxide, which I think is about $3 per gallon of gas.
Weitzman's analysis also divulges into a discussion of infinity. Essentially, the possible damages from catastrophic climate change can be thought of as unbounded (eg complete destruction). Thus, people's willingness-to-pay to avoid climate change can also approach infinity (or at least a very large number). Weitzman has a "dismal theorem" on the importance of fat tails-if the probability of catastrophy is increased even slightly, then the WTP increases significantly. Some other jargon is VSL or value of a statistical life (willingness to pay).
He concludes that the risk of inaction is great if his "fat-tail" assertions are correct. Carbon dioxide emissions languish in the atmosphere for hundreds of years, so if we put ourselves on a path of destruction there is no turning back. This means that we should be more willing to start reducing emissions now if there is a higher chance of catastrophe than current estimates. Overall, a little dense, but mostly understandable and highly provocative.