Alumnus co-writes article, selected as one of 2014’s best on mathematics
Brandeis alumnus Joseph Shipman Ph.D.’91 co-wrote an article that was chosen to be in the 2014 edition of The Best Writing on Mathematics. He wrote the article, titled “Extreme Proofs I: The Irrationality of √2” with John H. Conway, an established mathematician who teaches at Princeton University and the University of Cambridge.
Shipman, who recieved his Ph.D. in Mathematical Logic from Brandeis, originally published his article with Conway in Vol. 35, Number 3 of The Mathematical Intelligencer.
The article tackles the irrationality of the square root of two. The so-called “extreme” proofs seek to explain one of math’s simplest theorems: the rational root theorem, often attributed to the Pythagorean school, write Shipman and Conway in the article.
The article begins by exploring the very notion of proofs, claiming from the start that “the ordering of proofs cannot be a total order,” as an individual might relate to one proof more than another based off of personal experience or interests.
Still, the authors wrote, exploring even the most basic proofs can be a worthwhile use of time.
“It is enjoyable and instructive to ﬁnd proofs that are optimal with respect to one or more such value functions, not only because they tend to be beautiful, but because they are more likely to point to possible generalizations and applications,” Shipman and Conway wrote in the article. “It can be difﬁcult to decide whether two proofs are ‘really the same.’”
The writers acknowledge in the article that its logic assumes many of the mathematical basics to be fact, and yet, it is unique in it being one of few works to tackle a seemingly simple concept.
“This proof uses a hammer to crack a nut. It is not self-contained, because we have not proven the Fundamental Theorem of Arithmetic [which states that every number greater than one is either prime or the product of prime numbers],” the authors wrote.
“How is this proof ‘extreme?’ Well, it’s the shortest proof and the most transparent proof if the Fundamental Theorem of Arithmetic ‘comes for free,’ and it generalizes to arbitrary integers in both the base and the exponent,” they continued.
The article’s inclusion in The Best Writing on Mathematics 2014 is especially noteworthy, as the collection features writing from leading mathematicians from a diverse range of backgrounds and fields of study, such as artist and mathematician Sarah-Marie Belcastro’s “Adventures in Mathematical Knitting” and scientific author Brian Hayes’ “Crinkly Curves.”
Each year, the Princeton University Press publishes a new edition of the collection. According to the Press’ blog, in preparation for the collection’s publication, editor Mircea Pitici reads several journals and magazines—ranging in topic from scientific and mathematical to general-interest and interdisciplinary—and selects the best writings he comes across.
“Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2014 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them,” according to the Princeton University Press website regarding the edition.
Additionally, notes the Princeton Press, the collection—including Shipman’s article—is relevant to a variety of subjects and disciplines.
“These writings offer surprising insights into the nature, meaning, and practice of mathematics today,” Princeton University Press wrote. “They delve into history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today’s hottest mathematical debates.”