Doom has always been fashionable. "The Idea of Decline in Western History" by Arthur Herman shows this. The doomsayers have always been right (disasters happen all the time) and always wrong - every disaster (so far) has made us stronger. The reason for this is natural selection. Competence tends to survive more often than incompetence. Competence tends to accumulate. Disaster creates terrible pain, and kills the incompetent. Among the compassionate survivors, it encourages defiance - never again. And that leads to a faster spread of competence, as we are shaken out of our complacency.
Economic growth is nothing more than the accumulation of competence. It isn't the accumulation of gold, or the production of iron, or the number of buildings on a college campus, though all those things can be opportunities to manifest competence. Economic competence is produced by human minds, and humans invent new ways to be competent faster than ever before. Some fools invent new ways to be incompetent, of course, but again (and very importantly), incompetence is selected against.
In modern economies, competence is a network effect. Globally connected nations thrive, isolated nations wither. South Koreans builds ocean fleets, North Koreans starve. They have similar genetics, similar resources, and similar histories before the 1940s. The South Koreans are better networked.
My paper book library is bigger than Thomas Jefferson's, which was the best library in North America 200 years ago. Far larger public and private libraries are close to my house, closer than the University of Virginia is to Monticello. The internet, only a couple of decades old, connects me to far more. I have access to billions of different kinds of artifacts made by billions of people. I can design an object, send the specification to a factory in China, and get finished goods in a few weeks. That can potentially become a few hours, and for information objects, a few seconds. Thomas Jefferson was way smarter than I am, but my network makes me more competent than he was. How many Chinese or Central American or South African people did Jefferson work with?
I can imagine far better connectivity between humans over time. Together, we can cure old and new diseases, explore the solar system, probe the deep past and the far future, create attractive food and entertainment that is superbly healthy and mind enhancing. As we learn to build larger coalitions and operate them more efficiently, our capabilities expand exponentially. So if the growth rate of physical inputs tapers off, or even declines, what we can do with those inputs will continue to grow.
We've barely tapped the resources of the physical universe. Many of us do not even understand what some of those resources are, and destroy them in the pursuit of distraction and incompetence. Since incompetence is selected against, those who refuse to learn competence from the natural world will be replaced by those who can learn. As long as knowledge accumulates, those who learn will have increasing advantage over those who do not.
So yes, you can extrapolate exponential curves from very short trend lines, and show that those curves cannot continue in a highly restricted scenario. Energy growth is an example. It is easy to show that manufactured energy between 1700 to 1970 follows a somewhat straight line on a semilog graph with a 2.9% per year slope. Of course, that is not a measurement of total human energy use, just the components the graph maker chose to include in the sum. The actual plot of that cherry-picked data bumps above and below the line by half an order of magnitude, as old energy sources are replaced by new ones, and new populations join the party.
According to the graph, manufactured power has increased by a factor of (1.029)270 or 2250 in 270 years. If we extrapolate that 2.9% line backwards from 2010AD to 3500BC - the invention of writing and the beginning of rapid competence accumulation - then we go from a 14TW world to a 7E-56 watt world. Ridiculous. If that kind of extrapolation is ridiculous going backwards, what makes it valid going forwards?
If a brilliant physicist in 3500BC Uruk extrapolated from a few thousand humans dissipating an average of 50 watts each at a density of 200 humans per square kilometer (WAG), he would conclude that at a 2.9% increase rate, in a mere 10 thousand years Uruk would be 1.6E33 degrees Kelvin, hotter than the Planck temperature of the big bang. And his extrapolation would be as invalid as those of his modern straight-line colleagues.
Right now, in 2012, we are correcting global energy inequalities, with high growth in developing nations. Growth is rapid (too rapid for some) because the poverty of existing humans outweighs the hypothetical suffering of the future earth. Those of us who care about both the earth and human suffering have an opportunity to develop new energy sources which increase the current growth rate, while reducing the ecological burden. This is a difficult problem, unprecedented in human history. On the other hand, never before in human history have we had so much knowledge and capability, and never before have we been blessed with the emergence of so many educated and empowered human minds. Few of us are Thomas Jeffersons, but together we can think and create exponentially faster than all historic intellectuals combined. We are more than equal to the task, if we don't fall prey to doom mongering and fashionable groupthink.
What we can learn from the past is not where a particular aggregate number must go in the future, but the value of courage and the worthlessness of despair. If you don't like the choices we are faced with, invent some new ones. If you are not competent to do so, learn. Don't assume that the other 7 billion of us are all incompetent, too.