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.