An Ultimate Unit of Space and the Need for a New Calculus
Excerpt from “Why Knot?” Buy the book at Amazon
When I was reading The Fabric of the Cosmos by Brian Greene, which explains superstring theory for the masses, I was also reading Quicksilver by Neal Stephenson, an historical novel with Sir Isaac Newton as a character. On p. 670 of Quicksilver, one of the characters challenges a basic concept of calculus. He asks, “What happens then if we continue subdividing?… Is it the same all the way down? Or is it the case that something happens eventually, that we reach a place where no further subdivision is possible, where fundamental properties of Creation are brought into play?” The character is contrasting Newton’s notion of infinite subdivision, with other concepts of the world in which there is a natural limit to such subdivision.
There appears to be a contradiction between superstring theory, which postulates an ultimate unit of length, and the assumption of calculus that space is infinitely divisible.
I sent an email to Brian Greene, wondering if fundamental concepts and procedures of calculus need to be refined to take this ultimate unit of length into account.
He was kind enough to reply. “In fact,” he wrote, “that is just what we are working on today. The notion that the usual procedures of calculus are only relevant on length scales larger than some lower limit — we are trying to piece together the new procedures that take over.”
