Science doesn't do proof, it works by refuting false hypotheses.
The hypothesis of the climate "skeptics" is that increasing CO2 will not have a serious effect on our planet's climate and weather.
This can be refuted.
For purposes of discussion, their hypothesis needs to be given some parameters, so let us, for the sake of convenience, talk about the effect of doubling the level of atmospheric CO2 from the 1750 level of 280 ppm to 560 ppm, which will happen about 2100 if we continue with business as usual. A baby born today will be 85 years old in 2100.
A fairly simple physical calculation, based on the known physics of the interaction of the CO2 molecule and energy, concludes that a doubling of the level of CO2 will cause an increase in surface temperature in the region of 1.1C +/- 0.1C.
This increase will kick off the Three Babies, a set of positive feedbacks (together with some negative feedbacks) that will further increase the temperature at the planet's surface.
It is the magnitude of this further increase that is critical.
The "skeptics", now termed "lukewarmers" offer the hypothesis that the eventual temperature will settle down at less than two degrees C above 1750 levels.
Now hypotheses about the future cannot be tested experimentally, but a proxy for these predictions can be used: climate sensitivity. This is an indicator of how much the planet's surface temperature (call it T) increases to after any increase in heating equivalent to that caused by a doubling of CO2.
The lukewarmers claim that CS lies between the bounds of 0-2.5.
How does that figure compare with all the other work that has been done on the subject?
Take a look at this:
This represents an awful amount of work on climate sensitivity, looking at the problem in many different ways. Note the vertical blue bands: they indicate the likely eventual temperature, falling between 2 and 4.5.
Recall that lukewarmers hypothesise that equilibrium temperatures will settle at somewhere between 0-2.5. (note that there is an overlap at the 2-2.5C mark. Note that this overlap itself also invalidates the lukewarmer position of "no serious harm").
In any other area of science, this amount of work, crossing and recrossing the field, repeating the work, looking at it from different angles and different methodologies, the matter would be regarded as settled, and certainly not to be upset by a handful of studies, using a very basic computer model, which come to different conclusions.
The old joke about "Little Jimmy is the only one who is in step*" comes to mind.
But climate science is different, because its conclusions have an impact on the immensely rich and powerful fossil fuel industry.
There is very clearly a clash between the lukewarmers and the evidence produced from many other sources. The lukewarmer conclusion is based on a few studies of recent years. These studies are weakened by the limitation of their methodology; it is clearly dangerous to try to extrapolate from a few years to a much longer time. The fallacy of the Biased Sample applies.
It is the case that the evidence offered by the lukewarmers is refuted by all the work that have been previously made.
It is also the case that their highest projections are more likely to be correct. A T raised to at least 2.5C is at the top of their range of possibilities, but close to the median of the bulk of studies. 2.5C is serious. Skeptics make a big deal of uncertainty in the science, but uncertainty cuts both ways. Their range includes 2C, and that in itself is not a comfortable place to put our descendants, but they have no guarantee that the error bars may not stretch to 4C and more.
Remember also that the doubling of CO2 is not the end of the story. If we continue to burn fossil fuels, the levels of CO2, and the impacts, will continue to rise. Remember also that there is huge inertia in the system, so that even when we stabilise the levels of CO2, the impacts will continue to work, and through positive feedback loops, will continue to grow.
Here you can listen to James Hansen arguing that the 2C target is not safe, because it commits the planet to significant sea level rises. Wheat production is going to be affected adversely in a warmer world.
It is clear that the lukewarmer effort to delay any action on decarbonising the global economy is irrational and foolish. Their evidence base is limited and biased, and incompatible with a large body of science. The area of overlap between what the lukewarmers accept and what the main body of science accepts cannot in any way be said to be safe. The uncertainties implicit in their own evidence opens the gate to dangerous impacts on earth's climate.
In short, the hypothesis of the lukewarmers, that man made CO2 will not seriously harm our climate, is incompatible with the known facts. It is falsified.
*at a school parade, Jimmy is out of step. His mother sees it as him being the only one in step.