The conversation around climate change can sometimes seem riddled with magic numbers and dark thresholds. Researchers talk about the dangers of surpassing 2 degrees of warming. Headlines blare about the importance of reducing emissions by 2030. Even your friendly neighborhood climate reporter (hi!) gets tripped up by the complex calculations involved.

But all this math didn’t get pulled out of a hat; it’s the product of years of careful research by scientists and painstaking negotiations among diplomats. So it’s worth understanding how the world comes up with these targets — and what might happen if we fail to meet them.

Way back in 1992, world leaders signed a treaty promising to stabilize greenhouse gas concentrations in the atmosphere “at a level that would prevent dangerous anthropogenic” — human-caused — “interference with the climate system.” This set up the United Nations Framework Convention on Climate Change (UNFCCC), and member countries have been trying to figure out how to achieve its goals ever since.

As Texas Tech University climate scientist Katharine Hayhoe has pointed out, “dangerous” is a pretty subjective standard. If humans are going to hold ourselves accountable to the terms of the UNFCCC, we need to understand the consequences of warming and decide what we’re willing to live with.

Scientists have done a great deal of work to help answer the first half of that question. As early as the 1970s, Yale economist William Nordhaus suggested that humans would want to avoid a global average temperature increase that went beyond the natural variations seen in the last few hundred thousand years (as long as our species has been around). This would mean limiting warming to roughly 2 degrees Celsius (3.6 degrees Fahrenheit) above preindustrial levels.

(Side note: Even though most Americans — and The Washington Post’s copy editors — prefer to talk about temperature in Fahrenheit, metric is the chosen system of scientists and pretty much everyone else in the world. That’s why you’ll almost always see climate targets discussed in degrees Celsius.)

Nordhaus was an economist, not a climate scientist, and he referred to that suggestion of 2 degrees as “a first approximation.” But a growing body of work by agriculture experts, meteorologists, ice scientists, ecologists and other researchers revealed that warming beyond 2 degrees would result in all kinds of horrible consequences.

In 2001, the Intergovernmental Panel on Climate Change produced what is known as the “burning embers diagram,” a barometer of risks associated with various levels of warming. As temperatures rose from 0 to 5 degrees above preindustrial levels, the risk of extreme weather events, the likelihood of damage to the economy and harm to humans was illustrated with bars that graduated from pale yellow to fire engine red. The measure of most risks appeared to tip into the red around 2C.

“There isn’t a line on those burning embers that says, ‘Here the world ends,” said Richard Alley, a geoscientist at Pennsylvania State University and contributor to multiple IPCC reports. But at 2 degrees, the diagram suggested, “you are having impacts on most people, impacts on the market, that make it hard for everyone to live.”

At the same time, diplomats from countries that signed the UNFCCC were gathering annually to consider the science and come up with a plan for addressing climate change. But it wasn’t until their 2015 meeting in Paris that they finally agreed on a definition of “dangerous.”

The result was the Paris Climate Accord, the first treaty in which countries committed to limit warming to “well below 2 degrees Celsius.” From this target, scientists calculated a carbon budget — how much more greenhouse gas could spew into the atmosphere without surpassing that level of warming. In 2015, most estimates put the global carbon budget somewhere between 500 and 1,000 gigatons. That’s about as much as the world emits in a decade or two.

There was a problem, though: The Paris accord couldn’t force nations to accept specific limits on emissions. No one would have agreed to it, said Yale University lecturer Susan Biniaz, who from 1989 to 2017 was the top climate lawyer at the State Department.

Instead, participants crafted voluntary plans to lower emissions. But most of those plans — particularly the ones from the biggest emitters, like the United States and China — weren’t ambitious enough to achieve the UN’s overall temperature goal. A recent analysis found that the current commitments would still allow the world to warm 3.2 degrees Celsius (5.8 degrees Fahrenheit) by the end of the century. Few countries are on track to meet even their own modest targets, and the United States plans to pull out of the agreement entirely.

Meanwhile, as climate models have improved, researchers have realized that 2 degrees of warming might be more dangerous than originally thought. The United Nations asked the IPCC to produce a special report on an even lower target: 1.5 degrees C (2.7 degrees Fahrenheit).

The report, which came out in 2018, “paints a pretty dramatic difference between a 1.5 [degree] world and a 2 degree world,” Biniaz said. “It’s been alarming to a lot of people and countries and groups.”

At two degrees of warming, more than a third of the world’s population will be subjected to repeated severe heat waves, the report found. Coral reefs will “mostly disappear,” and ice-free summers in the Arctic become ten times more likely. The report also warned of the looming possibility of triggering unpredictable and uncontrollable feedbacks, which would further speed warming — such as the disintegration of the Antarctic ice sheet or catastrophic burning of major forests.

If 1.5 degrees Celsius is to be the new goal, a U.N. report said, greenhouse gas emissions need to start falling 7.6 percent annually, starting this year. The world should aim to curb emissions 45 percent by the end of the decade, and to reach zero emissions by 2050. Achieving that goal will require a dramatic about-face: In 2019, emissions hit a record high.

Climate change is not pass/fail, scientists like to say. Crossing warming thresholds doesn’t mean that Earth collapses into climate oblivion. And meeting these targets doesn’t guarantee humanity will avoid climate change’s dangerous consequences.

It’s too late for that. Already, many people around the world have gotten a glimpse of what we can expect as the planet becomes hotter. A Washington Post analysis found that 20 percent of the globe has warmed by at least 1.5 degrees Celsius. In those places, the ground is collapsing from melting permafrost. Fish species that people depend on for food are vanishing from the seas. Summers have gotten so hot that cities are air conditioning the outdoors. The bush fires currently raging across Australia come after a year that was 3.2 degrees Celsius (5.8 degrees Fahrenheit) hotter than average.

“You want to know how to define ‘dangerous?” said Veerabhadran Ramanathan, a climate scientist at the Scripps Institution of Oceanography. “I think nature is answering that question. We know we have already reached a dangerous spot.”