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4 of 4 Major rewrite. Word choice and sentences improved.

Very few people disagree with the basic fact that the greenhouse gas CO2 warms the climate, but without some kind of positive feedback mechanism, it doesn’t add very much: around 1°C-1.2°C per doubling of CO2.

The statement is correct, but very misleading. In the absence of the climate system's feedbacks, the climate sensitivity is 1 to 1.2 deg C for each doubling of CO2 (per the IPCC). However, the impact of feedbacks is greater than the direct impact of increased CO2. This can be seen without relying on climate models by looking at the powerful temperature cycles of historical ice ages (or glacial-to-interglacial transitions). These cycles are caused by variations in the Earth's orbit, which only produce a small change in incident radiation. See here for example. Without climate feedbacks, it would take a much greater change in solar output, or an enormous change in CO2 concentration, to cause an ice age. Chris Colose of Univ. of Wisconsin, Madison provides a good explanation of feedback effects:

To put this in perspective, it would take about five doublings of CO2 or a 7% increase in the total solar radiation hitting the Earth to produce the magnitude of climate change typical of glacial-to-interglacial transitions.

Thus it is clear - without relying on models - that the feedback effect must be greater than the direct effect, or else there could never have been ice ages.

The phrase "without some kind of positive feedback mechanism" gives the impression that scientists think there's an alternate mechanism that drives their projected temperatures higher. Climate sensitivity without feedbacks can be calculated fairly accurately on a single sheet of paper, (see here for example) and the result will be approximately 1 deg C. But the result is not relevant to anything, because the real world has many feedbacks; that's why the focus in climate science over the past several decades has been understanding the feedbacks and quantifying their effects. Most of them are well understood, their net effect is overwhelmingly positive (i.e. they amplify temperature fluctuations), and of course they are all based on physics, as demonstrated by Chris Colose.

According to James Hansen, the best source of information about feedbacks is not models, but paleoclimate data:s

Models are imperfect and we will never be sure that they include all important processes. Fortunately, Earth's history provides a remarkably rich record of how our planet responded to climate forcings in the past. Paleoclimate records yield, by far, our most accurate assessment of climate sensitivity and climate feedbacks.

Hansen calculates the climate sensitivity in various units, and expresses it as 2 to 4 degrees C for a doubling of CO2:

The empirical fast-feedback climate sensitivity that we infer from the LGM-Holocene comparison is thus 5°C/6.5 W/m2 ~ 3⁄4 ± 1⁄4 °C per W/m2 or 3 ± 1°C for doubled CO2. The fact that ice sheet and GHG boundary conditions are actually slow climate feedbacks is irrelevant for the purpose of evaluating the fast-feedback climate sensitivity.

This empirical climate sensitivity incorporates all fast response feedbacks in the real- world climate system, including changes of water vapor, clouds, aerosols, aerosol effects on clouds, and sea ice. In contrast to climate models, which can only approximate the physical processes and may exclude important processes, the empirical result includes all processes that exist in the real world – and the physics is exact.

Water vapor provides a great example of climate feedbacks. Because it absorbs infrared radiation very well, water vapor is a very powerful greenhouse gas. Also, warmer air holds more water vapor. So, suppose temperature increases because of higher CO2; the atmosphere will hold more water vapor, which will cause it to absorb more solar radiation, causing it to heat up still further. Calculating the impact of the CO2 while disregarding the water vapor feedback would be a significant underestimation for no good reason. Other positive climate feedbacks are also well known to climatologists, such as changes in reflectivity due to melting of snow and ice, release of CO2 from ocean water because of increasing ocean temperatures, and release of methane from melting permafrost.