The furious majesty of a thunderstorm defies computer simulation. In a world divided up into 10,000 square kilometer grids to make the 510 million square kilometer Earth digestible to a computer, a thundercloud that rains over two square kilometers remains too small to properly calculate in a climate simulation—as does even a hurricane like Sandy that sprawled over 280 kilometers of ocean and land in 2012.

Clouds control climate. Even if they could be correctly accounted for in computer simulations, there are all the complexities in the types of clouds, their height in the atmosphere, even the composition and shape of droplets in the cloud. Climate models struggle to simulate hurricanes, some of the biggest cloud systems, let alone lonely streaks of cirrus or a dense, billowing cumulonimbus. How low cooling clouds might form or if such clouds might disappear all together as the climate warms have major impacts on overall global warming.

Among all the numbers commonly bandied about global warming, the most important one in climate change is not 400 (parts-per-million CO2 in the atmosphere), two degrees Celsius (average warming of global temperatures), one trillion tons (of carbon budget), or even $100 billion (in climate adaptation funding per year). It's not even a single number, it's a range: 1.5 degree C to 4.5 degree C, according to the most recent effort of the United Nations Intergovernmental Panel on Climate Change, the Nobel Peace Prize-winning IPCC.

That's the expected global warming over centuries from a doubling of atmospheric CO2 based on the output of around 40 planetary-scale simulations of the ocean and atmosphere known as global climate models. Each model, like the Community Earth System Model at the National Center for Atmospheric Research in Boulder, Colo., is run on massive supercomputers. The most important number's formal name is "equilibrium climate sensitivity," and it's meant to represent the new equilibrium in surface temperatures after a change in the amount of the Sun's energy trapped here on Earth rather than radiated back to space. The number is an estimate of a warming range because neither scientists nor computer models can agree on just how sensitive Earth's biogeochemical cycles are to a thickening blanket of invisible greenhouse gas trapping more heat.

In fact, despite decades of better observations and simulations, this range of climate sensitivity hasn't changed much since 1979 when a National Research Council report on climate change led by meteorologist Jule Gregory Charney assessed climate sensitivity in the range of two to four degrees C of warming from a doubling of atmospheric CO2. "We may be just as unsure as before," says Gavin Schmidt, a climate modeler and director of the NASA Goddard Institute for Space Studies. "But we are unsure on a much more solid footing."

The Earth is a complex system, an irreducible complexity that defies simplification into computer models. As a result, real world effects of increasing concentrations of atmospheric CO2, like the meltdown of Arctic land and sea ice, appear to be happening faster than massive computer simulations have predicted. But the range of sensitivity is not the simple product of computers running simulations of how the atmosphere and ocean—two great roiling fluids in congress—react to more heat trapped by more CO2. It is also based on ancient air entombed in Antarctic ice, the steady decay of radioactive elements in stone and other observations of the planet's remote past. In terms of climate, the deep past isn't merely the past, it's a preview of what the world might experience again in the future.

Take for example the Eocene, some 30 million years ago, the warmest period in recent Earth history where atmospheric concentrations of CO2 rose above 700 ppm and palm trees and crocodile-like animals thrived in near Arctic latitudes. Climate models have a hard time explaining how the Eocene could be so warm at the poles even with CO2 concentrations much higher than today. "We're either looking at new feedbacks that kick in in the polar regions at high temperatures, possibly associated with vegetation and aerosols or hazes," Schmidt says. Or "it could be exotic physics that happens."

On the other side is the last glacial maximum roughly 26,000 years ago, which saw global average temperatures lower by four degrees Celsius.* That proved enough to encase large swaths of the northern hemisphere in kilometer-thick ice sheets that stretched south as far as New York City. Tracking temperatures over the last 420 million years suggests that climate change will exceed 2 degrees C if CO2 concentrations double—but how much temperatures will exceed that number is unknown.

It's not just clouds making that prediction complicated, of course. Figuring out the energy balance of our home planet is complex. Volcanic eruptions appear to have a bigger influence on global climate than previously thought (giving yet more hope to would-be geoengineers) as does the amount of heat swallowed into the oceans, both of which help to cool the climate, at least in the short run. Merely understanding the circulation of the Southern Ocean that swirls around Antarctica and keeps the continent locked in deep freeze could improve climate models. That's a key bit of observational input easier asked for than acquired given the realities of the Roaring Forties and Furious Fifties, powerful winds and attendant waves that make observation by ship or even robot glider difficult. And it's not just the Southern Ocean: how the Pacific and Atlantic circulate, absorb and disgorge heat remains a reality that models fail to capture. "The evolving warming depends also on how quickly the ocean warms," notes Gabriele Hegerl, a climate scientist at the University of Edinburgh.

Climate change also is not just driven by CO2 and other greenhouse gases from fossil fuel burning and other human activities. It's also affected by clearing forests for farmland or allowing trees to regrow, the vagaries of the Sun's waxing and waning strength, and choking vapors from China and other uncontrolled aerosol pollution sullying the atmosphere for a spell. "It's a bit like watching traffic at a busy crossroad for an hour," says Reto Knutti, a climate scientist at ETH Zurich. "You can learn a lot about the rules of traffic by doing that, but it would be very difficult to predict how traffic will change over the next decade or century."

As the saying goes: all models are wrong, but some are useful. The impact of all of these real world uncertainties can perhaps be synthesized and made into one constraint to rule them all, a kind of master ring for climate change models based on Bayesian statistical techniques that estimates an overall uncertainty based on the probability associated with each contributing factor. "We haven't quite designed that experiment yet," Schmidt says. But "the path toward that synthesis is quite clear."

The physics of clouds and their seemingly capricious behavior—the kind of activity that gave rise to humanity's penchant for mercurial sky gods—bedevil modern climate scientists and meteorologists, though the eyes in the sky provided by satellites should help as should more powerful computers on which to run models that get down to grids of just one square kilometer and finally capture those thunderheads. Then there's the all too human failings of a company like Volkswagen, whose employees lied about emissions from the cars they manufacture, even enough to throw off the pollution factored into a model run, and thus the predictions of the future generated by that model. Feedbacks have unknown unknowns in other words.

Still, the computer models—even the hand calculations of Guy Callendar in 1938 or the guesswork of the Charney report in 1979—have done a good job of predicting climate change as it has happened, suggesting that climate sensitivity over the long term may not matter as much as how the climate will respond in the next several decades. "The whole point is to predict what's going to happen before it happens," Schmidt says. "To know what happens in 2050, it's not climate sensitivity that's the issue, it's what's the emissions path we're on."

Still, most climate scientists agree with all their predecessors: a doubling of atmospheric CO2 will mean a warming of global average temperatures of about 3 degrees C, plus or minus a degree or so. One way to see if that consensus is right is to continue on civilization's present course of burning fossil fuels, cutting down forests and other greenhouse gas-emitting activities.

"We could pass through two degree C of warming by 2065 even with no acceleration, and be over 2.5 C by end of century," says Kevin Trenberth, a climate scientist at the National Center for Atmospheric Research. "There could be acceleration because of melting ice and darker surfaces. But just maybe something good comes out of Paris [climate change talks] and the rates begin to drop down somewhat. Let's hope so."

*Correction 12/4/15: This sentence has been updated to reflect the correct cooling during the last glacial maximum.