Big Numbers February 2015
An amazing story by Scott Aaronson on Big Numbers.
Keywords: exponential growth, NP-complete problems, Ackermann sequence, Turing Machine, Halting Problem, Busy Beaver,
some quotes :)
In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces "Eighty-three!" The second, mightily impressed, replies "You win."
Consider, for example, the oft-repeated legend of the Grand Vizier in Persia who invented chess. The King, so the legend goes, was delighted with the new game, and invited the Vizier to name his own reward. The Vizier replied that, being a modest man, he desired only one grain of wheat on the first square of a chessboard, two grains on the second, four on the third, and so on, with twice as many grains on each square as on the last. The innumerate King agreed, not realizing that the total number of grains on all 64 squares would be 264-1, or 18.6 quintillion—equivalent to the world’s present wheat production for 150 years.
Rado called this maximum the Nth "Busy Beaver" number. (Ah yes, the early 1960’s were a more innocent age.)
To solve the Halting Problem for super machines, we’d need an even more powerful machine: a ‘super duper machine.’ And to solve the Halting Problem for super duper machines, we’d need a ‘super duper pooper machine.’
If we could run at 280,000,000 meters per second, there’d be no need for a special theory of relativity: it’d be obvious to everyone that the faster we go, the heavier and squatter we get, and the faster time elapses in the rest of the world. If we could live for 70,000,000 years, there’d be no theory of evolution, and certainly no creationism: we could watch speciation and adaptation with our eyes, instead of painstakingly reconstructing events from fossils and DNA. If we could bake bread at 20,000,000 degrees Kelvin, nuclear fusion would be not the esoteric domain of physicists but ordinary household knowledge.
But do people fear big numbers? Certainly they do. I’ve met people who don’t know the difference between a million and a billion, and don’t care. We play a lottery with ‘six ways to win!,’ overlooking the twenty million ways to lose. We yawn at six billion tons of carbon dioxide released into the atmosphere each year, and speak of ‘sustainable development’ in the jaws of exponential growth.