Physicist Albert Bartlett is quoted as saying that “the greatest shortcoming of the human race is our inability to understand the exponential function”.
Starting with a thought experiment in which two competitors are challenged to come up with the bigger finite number, Scott Aaronson has written an accessible and fact-filled essay about large numbers, touching on topics such as AI, NP problems, the computational power of our brains, and much more besides.
Place value, exponentials, stacked exponentials: each can express boundlessly big numbers, and in this sense they’re all equivalent. But the notational systems differ dramatically in the numbers they can express concisely. […] It takes the same amount of time to write 9999, 9999, and 9999—yet the first number is quotidian, the second astronomical, and the third hyper-mega astronomical. The key to the biggest number contest is not swift penmanship, but rather a potent paradigm for concisely capturing the gargantuan.
Such paradigms are historical rarities. We find a flurry in antiquity, another flurry in the twentieth century, and nothing much in between. But when a new way to express big numbers concisely does emerge, it’s often a by-product of a major scientific revolution: systematized mathematics, formal logic, computer science. Revolutions this momentous, as any Kuhnian could tell you, only happen under the right social conditions. Thus is the story of big numbers a story of human progress.
This essay inspired the 2007 Big Number Duel at MIT.