The Calculus Diaries - Jennifer Ouellette [14]
In one letter to Leibniz, Newton offered his “proof ” that he had invented calculus—but he couched it in a sort of anagram of a Latin sentence. He took all the individual letters and put them in alphabetical order: six a’s, two c’s, one d, thirteen e’s, two f ’s, and so forth. To Newton, it was perfectly obvious: Anyone could simply rearrange all of the letters and find the proof they sought that he, Isaac Newton, had prior knowledge of the key concepts. Very few people felt inclined to go to all that trouble, and frankly, even decoded, the “proof ” wasn’t especially clear. Roughly translated, the sentence read, “Having any given equation involving never so many flowing quantities, to find the fluxions, and vice versa.” That was his stab at summarizing derivatives; Newton would have been a lousy math teacher.
The Royal Society of England sided with Newton on the controversy, crediting him in 1715 with the discovery of calculus. Leibniz wasn’t given shared credit until after his death a year later. Today, the consensus seems to be that the two men represent two complementary approaches to the discipline they co-invented. Leibniz was the more abstract of the two, and it’s his system of notation that modern scientists still use today, while Newton focused on the more practical applications of calculus. Leibniz can also claim credit for coining the word calculus, named for a type of stone once used for counting purposes by the Romans.
Calculus did not find immediate acceptance within the scientific community; there was one final missing piece. The method worked, in that it gave the right answer, but mathematicians found the notion of the infinitesimal deeply troubling. Once again, the problem of infinity raised its ugly head. For instance, Newton relied on a bit of magical hand-waving to make his method work: He argued that since his fluxion units were so small—infinitely close to zero but not exactly equal to zero—they could be ignored for all practical purposes. In his equations, they effectively vanish for no reason. A rigorous explanation for what happens to those fluxion units when an equation is solved would not be found for another hundred years.
Leibniz adopted a symbolic notation—Δx, which stands for a tiny increment—that preserved the infinitesimals yet still enabled mathematicians to manipulate them as if they were actual numbers. (In modern notation, scientists often use dx to represent an infinitesimal.) Yet this approach seemed to many mathematicians to be a bit of a cheat. Chief among the naysayers was an Irish bishop named George Berkeley, who in 1734 (seven years after Newton’s death) criticized Newton and Leibniz for their fudging of the method, calling infinitesimals “ghosts of departed quantities” and observing that if they were comfortable with that sort of thing, they “need not, methinks, be squeamish about any point in divinity.”
TAKE IT TO THE LIMIT
A fictionalized Albert Einstein (portrayed by the late Walter Matthau) plays mischievous matchmaker between his egg-head niece, Catherine Boyd, and a good-hearted auto mechanic named Ed Walters in the charming 1994 romantic comedy I.Q. Some might object to the considerable liberties taken with historical fact and illustrious personages, but there’s a lot to admire in the film, if for no other reason than its inclusion of Einstein’s real-life cronies, Kurt Gödel, Boris Podolsky, and Nathan Liebknecht, as supporting characters. As Ed introduces Einstein to Frank, one of his co-workers at the garage, he declares, “This is Albert Einstein, the smartest man in the world!” Intones Frank in his best Joisey accent, “Hey, how they hangin’?”
There is a lovely scene in a diner, where Catherine tries to explain to Ed the gist of one of Zeno’s paradoxes. Zeno was a Greek philosopher living in the fifth century B.C. who thought a great deal about motion. Specifically, he speculated that all motion is illusory, and came up with a famous set of arguments to “prove” it. Catherine explains it thus: If she takes one step forward, and then halves the distance traveled