Dr. Kurzweil has other graphs showing a century of exponential growth in the number of patents issued, the spread of telephones, the money spent on education. One graph of technological changes goes back millions of years, starting with stone tools and accelerating through the development of agriculture, writing, the Industrial Revolution and computers. (For details, see nytimes.com/tierneylab.)

Now, he sees biology, medicine, energy and other fields being revolutionized by information technology. His graphs already show the beginning of exponential progress in nanotechnology, in the ease of gene sequencing, in the resolution of brain scans. With these new tools, he says, by the 2020s we’ll be adding computers to our brains and building machines as smart as ourselves.

This serene confidence is not shared by neuroscientists like Vilayanur S. Ramachandran, who discussed future brains with Dr. Kurzweil at the festival. It might be possible to create a thinking, empathetic machine, Dr. Ramachandran said, but it might prove too difficult to reverse-engineer the brain’s circuitry because it evolved so haphazardly.

“My colleague Francis Crick used to say that God is a hacker, not an engineer,” Dr. Ramachandran said. “You can do reverse engineering, but you can’t do reverse hacking.”

Dr. Kurzweil’s predictions come under intense scrutiny in the engineering magazine IEEE Spectrum, which devotes its current issue to the Singularity. Some of the experts writing in the issue endorse Dr. Kurzweil’s belief that conscious, intelligent beings can be created, but most think it will take more than a few decades.

He is accustomed to this sort of pessimism and readily acknowledges how complicated the brain is. But if experts in neurology and artificial intelligence (or solar energy or medicine) don’t buy his optimistic predictions, he says, that’s because exponential upward curves are so deceptively gradual at first.

“Scientists imagine they’ll keep working at the present pace,” he told me after his speech. “They make linear extrapolations from the past. When it took years to sequence the first 1 percent of the human genome, they worried they’d never finish, but they were right on schedule for an exponential curve. If you reach 1 percent and keep doubling your growth every year, you’ll hit 100 percent in just seven years.”