Moby Dick Meet Theory

Sarah Hart holds a chair that dates back to 1597—Gresham Professor of Geometry at Birkbeck, University of London. She is a British mathematician specialising in group theory. The British modifier is redundant since I spelled specialising with a “s” not a “z”:

Today’s British English spellings mostly follow Samuel Johnson’s A Dictionary of the English Language (1755), while many American English spellings follow Webster’s An American Dictionary of the English Language (1828).

Today we will discuss history, making history, and more.

Hart is an excellent writer about math with a special interest in history and in non-standard applications of math to things like Moby-Dick. See the recent article in the science section of the New York Times about Hart:

For the mathematician Sarah Hart, a close reading of “Moby-Dick” reveals not merely “one of the strangest and most wonderful books in the world” and “the greatest book of the sea ever written,” but also a work awash in mathematical metaphors.

Hart’s paper on Moby-Dick is titled Ahab’s Arithmetic. Here is an updated link to her article: new version as of June 2021.

I bet few articles on the arxiv website are on fictional works like Moby-Dick:

Herman Melville’s novel Moby-Dick contains a surprising number of mathematical allusions. In this article we explore some of these, as well as discussing the questions that naturally follow: why did Melville choose to use so much mathematical imagery? How did Melville come to acquire the level of mathematical knowledge shown in the novel? And is it commensurate with the general level of mathematical literacy at that time?

Hart also has given many talks on math from a unique point of view.

Group Theory History

Hart is the current president of the British Society for the History of Mathematics:

Mathematics has been part of human culture since the beginnings of civilization. Its study and practice has gone hand in hand with the evolution and development of commerce, architecture, legal theory, cosmology, astrology, and countless other activities. The history of mathematics today addresses a rich cultural heritage across continents, peoples, and ages, taking in the practical traditions of merchants’ accounts and surveying just as it does the elevated traditions of learned scholars at court or the modern day university professor.

Hart adds:

In this article we explore mathematical allusions in Herman Melville’s novel Moby-Dick. We argue that both the quantity and sophistication of these allusions are evidence for Melville’s high level of mathematical knowledge and ability. We discuss some of the most compelling mathematical imagery, as well as giving background on the several mathematicians and mathematics books mentioned in the novel. We also include some biographical details supporting the assertion that Melville had an unusually good mathematical education.

Another view of history is given by Alma Steingart of the history department of Columbia. Steingart has written, for example, an article on the history of the classification of all simple groups:

Over a period of more than 30 years, more than 100 mathematicians worked on a project to classify mathematical objects known as finite simple groups. The Classification, when officially declared completed in 1981, ranged between 300 and 500 articles and ran somewhere between 5,000 and 10,000 journal pages. Mathematicians have hailed the project as one of the greatest mathematical achievements of the 20th century, and it surpasses, both in scale and scope, any other mathematical proof of the 20th century.

Decades ago as soon as we started to think about the group isomorphism problem we realized there is an application of the classification theorem:

We can use it to prove that isomorphism for finite simple groups is in polynomial time. The proof is simple—bad pun: Just use the fact that all simple groups are generated by at most elements. So just try all possible pairs. See GLL for some more comments.

Making Group Theory History

Hart may be interested in the history of math, but proving new theorems is perhaps the best way to affect history. The paper titled Small maximal sum-free sets by Michael Giudici and Hart solves a previously open problem. The site Theorem of day highlights this paper here. Also Tim Gowers proved some related results.

Hart’s main theorem is:

Theorem 1 Let be a finite group and let be the subgroup generated by a subset of the elements of . Then unless , cannot be both maximally sum-free and minimal with respect to generating .

Consider a set of elements in a finite group . There are two properties that can have, and these properties act in opposite directions.

• Force to be smaller: The set is sum-free. This means that for in ,

  

• Force to be larger: The set is minimal generator for . This means that it is not possible to generate with less elements than . Intuitively these properties work against each other. Adding an element to could violate sum-free; deleting an element from could violate generation.

Open Problems

Here is an interview with Hart—take a look.