They further add that both the economic and population growth must be exponential, but they don't consider the results of exponential growth.
Exponentials grow fast. In less than the time from the Roman Empire until now--that is, within a few thousand years--even modest exponential growth would run into physical absurdities, as illustrated in these posts in the Do The Math blog, by Caltech-trained UC San Diego physics professor Tom Murphy:
(My PhD is in physics, so I understand Tom's approach. I hope readers don't misunderstand how a physicist describes things like energy, growth, and exponentials. It requires understanding conservation of energy and thermodynamics, in this case.)
If we had infinite resources, infinite growth would work. We don't, though, so trying to grow forever will result in unintended consequences.
Some economists have studied steady-state economies, which differ from stagnant economies. Culture and societies would differ from today's growth-based economies, but aren't impossible.
Among the economists that imagined steady-state economies is Adam Smith, pictured above on the twenty pound note.
Societies have lived for long times in steady states. Think of islands that could support tens of thousands of people, where people knew much of the rest of the population.
They knew that if a couple had three children, another couple had to have fewer. They could see that resources would run out otherwise.
Maybe because of the Earth's size relative to a person or because of the distribution of resources, we seem unable to see the limits of resources like an island nation can. The Earth's resources are still limited even if we can't see the limitations.
Growth-based economies have also had their shares of problems of depressions, overproduction, and famine. They haven't been able to solve the Tragedy of the Commons, which wasn't a problem when we could always expand to new areas with new resources.
What does economics mean, anyway?
Economics studies the distribution of resources. We need to distribute resources necessary for life to everyone. We do so pretty well now, but not perfectly.
To get food, shelter, energy, and other necessities for life to everyone doesn't require exponential growth. People just believe it does.
A transition to steady state from growth would create a challenge. People suggest that we'd have a period of an again population where feeding unproductive retired people would be difficult, but given productivity gains we've seen, such as the United States going from a nation of 25% farmers to 2%, this redistribution doesn't have to be untenable.
Science means theory must conform to observation, not the other way around
People believing exponential economic and population growth can continue forever are denying the foundation of science: that theories have to conform to observation, not the other way around.
If you stick to your beliefs in conflict with nature, you're dreaming.
If you accept that nature has the last word, you change your theories to conform with nature. Only then can you prepare for the conflicts that arise when you try to overcome natural limits. Nature will win that conflict.
If you believed the population would rise forever, it won't. If we overshoot natural limits--which we can temporarily by using up stored resources, which we're doing with fossil fuels, top soil, fish, etc--populations will collapse, bringing cultures down with them.
Alternatively, we can prepare for steady-state economies and lowering birth rates without collapse.