## A Brain on Fire: James Joseph Sylvester

**September 24, 2006**

**Book ReviewPhilip J. Davis**

**James Joseph Sylvester: Jewish Mathematician in a Victorian World.** *By Karen Hunger Parshall, Johns Hopkins University Press, Baltimore, 2006, 461 pages, $69.95. *

When Karen Parshall's biography of Sylvester came across my desk, I was initially torn, preferring in general to review books that have an applied mathematical angle. On the one hand, I wanted to write about a man long known to me as one of the more colorful characters in the mathematical pantheon. On the other hand, I think of Sylvester as primarily an algebraist who forged a path to what has been called "modern algebra." I couldn't then think of a way of pulling him into the applied orbit. A bit more thought and the way became clear. So here goes.

"This book aims to tell, for the first time, the complex story of Sylvester's life as fully as possible within the political, religious, mathematical and social current of nineteenth-century England."

Parshall, aided by a copious---one might even say a glutted---paper trail, devoted many years to her project (she edited a collection of Sylvester's letters that appeared in 1998). She has produced a masterpiece that amply and admirably fulfills her announced goal. She gives us a complete picture of Sylvester the man and his activities: his constant mathematizing, lecturing; his long and deep collaboration with Arthur Cayley; his exchanges of letters with the mathematicians of the world; his frequent writing about mathematical education; his traveling, complaining, arguing, bargaining. Sylvester was in love with and obsessed by his subject, a "brain on fire," as he wrote of himself in 1870.

James Joseph Sylvester (1814–1897) was a brilliant, inductive, exuberant, euphoric, prolific, passionate, egocentric, touchy, contentious bachelor. He was born in London to Jewish parents. The family name was Joseph, but James, following the lead of his older brothers, tacked on the "Sylvester" very early. Parshall has done a splendid job in researching and describing the Anglo–Jewish roots and background of the Joseph family, going back to the mid-18th century. In the 19th century, England was relatively liberal with respect to religious minorities; Sylvester fought constantly and stubbornly against the prejudice and exclusions that still existed and that, in the course of his lifetime, lessened significantly.

In 1828, at the age of 14, Sylvester was a student of Augustus de Morgan at the University of London. By 1831 he was studying at St. John's College, Cambridge. His studies were then interrupted by a prolonged illness. In 1837 he sat the Cambridge Math Tripos and came in second. A year later, still lacking a degree because of the religious exclusionary policy of Cambridge, he became a professor of natural philosophy at University College, London. Trinity College in Dublin, operating under more liberal rules, awarded him the BA and the MA in 1841.

In that same year, at the age of 27, he sailed across the Atlantic and took a professorship of mathematics at the relatively new University of Virginia in Charlottesville. Very rapidly there ensued one of the first of the many contretemps and spats that peppered his life. Sylvester found himself in the middle of a student culture in which, Parshall writes, "if not outright xenophobia, then at least a pervasive sense of localism had worked against foreign faculty members, making them obvious targets of hostility." One student, William H. Ballard of New Orleans, plagued and threatened Sylvester almost from the start, claiming ultimately that Sylvester had violated his "Southern gentlemen's code of honor." When push came to shove, Sylvester, who had bought a sword cane for self-protection, "drew the sword cane and struck [Ballard] a glancing blow off a rib with no damage done." I had heard this story many years ago but was amazed at the wealth of detail preserved in the university archives, which Parshall was able to dig out and present.

Sylvester concluded wisely that though scarcely four months had passed since his arrival, he'd better get out of town. He traveled to New York, where his older brother, Sylvester Joseph Sylvester, was in the brokerage and lottery business. Letters flew back and forth and visits and acquaintances were made as Sylvester tried to secure for himself a mathematics professorship in the U.S., but all attempts proved futile. Late in November 1843, he sailed back to England.

There he studied law and worked as an actuary. He met Arthur Cayley (1821–1895), a lawyer and mathematician with whom he was to collaborate fruitfully for years. Eric Temple Bell, in his 1937 *Men of Mathematics*, called Cayley and Sylvester "The Invariant Twins." From 1855 to 1869, Sylvester was a professor of mathematics at the Royal Military Academy, Woolwich. More spats arose during this 14-year hitch. The military authorities, who knew little mathematics and probably cared less, kept a tight grip on all aspects of the operation of the academy. Sylvester, particularly during the years between 1861 and 1863, was at loggerheads with them over curricular reform, his job description, questions of authority, and numerous other issues. His tendency to go over the heads of the local administration in his quarrels could hardly have endeared him to them.

At the military academy, mandatory retirement came for Sylvester in 1869, but some apparatchik, operating according to rule, terminated his job six weeks before he would have been entitled to a full civil service pension. Sylvester went public with a complaint and won. Prime Minister Glad-stone "rectif[ied] . . . a lamentable departmental error."

Fifteen years ago, while working on my book *Spirals: From Theodorus to Chaos*, I came across a spiral that Sylvester had studied in connection with the design of gun-carriages. For more details I was led to his *Collected Mathematical Papers**. A benefit of my trip to the library was that I read first-hand about a great talk Sylvester gave in Exeter in 1869 as the president of Section A of the British Association for the Advancement of Science. In this talk, Sylvester refuted views of mathematics put forward by Thomas Henry Huxley, a prominent and vocal advocate of Darwin's theory of evolution. Here is the story: In 1868 Huxley delivered a speech in which he "contrasted the inductive quality of science with what he understood to be the deductive quality of mathematics." Mathematics, Huxley asserted,

"is that which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation."

Shortly thereafter, taking on Huxley, who was then very much in the limelight, Sylvester rebutted that mathematics

"unceasingly calls forth the faculties of observation and comparison; one of its principal weapons is induction: it has frequent recourse to trial and verification; and it affords a boundless scope for the exercise of the highest efforts of imagination and invention."

Because the philosophy that Sylvester expressed was one that I found friendly, I extrapolated that his Ghost might nod in approval of my own philosophy of mathematics. I was delighted to see that Parshall tells this story in full detail.

In 1877, Daniel Coit Gilman, first president of the newly founded Johns Hopkins University in Baltimore, offered Sylvester, on his retirement from Woolwich, the "Inaugural Professorship of Mathematics." Sylvester accepted, crossed the Atlantic once again, and there, at the age of 63, experienced a phoenix-like restoration of his mathematical interests and powers. In 1878 he founded the *American Journal of Mathematics*, the first substantial journal in the United States devoted to research. (My first published math paper was accepted by Aurel Wintner for this journal.)

Though touching on many areas of mathematics, including number theory, mechanics, linkages, geometry, Sylvester's main productions were in invariant theory and matrix theory. Although never having warmed to the former, I do have a few words to say about the latter.

Matrix theory grew out of determinants, whose origins have been traced to the Japanese mathematician Takakazu Seki in the 1670s and, almost simultaneously and civilizations away, to Leibniz in Germany. Cayley was the first to regard the square arrays of numbers (the shape of determinants) as producing an algebra, and in 1850 Sylvester, who had a strong penchant for coining strange mathematical neologisms, dubbed them "matrices." In an often repeated description, Sylvester wrote that a matrix is

"a rectangular array of terms out of which different systems of determinants may be engendered, as from the womb of a common parent."

Notice the emphasis on determinants; today, they play only a supporting role in matrix theory, but it was from the study of determinants that numerous matrix results originally emerged. As late as 1930, Sir Thomas Muir was writing large volumes about determinants. Out of his considerable corpus of writings about matrices, the following results currently bear Sylvester's name (unless they have now been so well absorbed into matrix theory that ascriptions are bypassed): Sylvester's identity (for subdeterminants), his Laws of Rank, and his Law of Inertia. Incidentally, for all the work Sylvester did on invariants, the text of Peter Olver's 1999 book *Classical Invariant Theory* uses only Sylvester's Law of Inertia.

This brings me to the hostility and the spat over priority in a matrix discovery between the brilliant, egotistical, cranky mathematician/philosopher Charles Sanders Peirce and the equally cranky Sylvester. It unfolded in two acts.

*Act One.* In 1878, George Bruce Halstead, a PhD student of Sylvester and an egotist in his own right, scheduled a talk at Johns Hopkins in which he originally planned to mention some results of Peirce. He later changed his mind, writing a very brusque letter to Peirce claiming that "your writings exhibited a tendency to undervalue everybody and everything mentioned. Besides this I was particularly discouraged by Prof. Sylvester's adding that your articles in the *Popular Science Monthly* were pretentious without being at all profound."

*Act Two.* In 1879, Sylvester, who had come up with two 3 × 3 matrices that form the generators of a nine-dimensional algebra that he called the *nonions* (in imitation of Hamilton's *quaternions*), wrote it up as a note and submitted it for publication in *The Johns Hopkins University Circulars*, which was under the editorship of Peirce. Peirce added a note that Sylvester's forms can be derived from his (Peirce's) 1870 *Logic of Relatives*. Sylvester, who rarely read anyone else's work, saw red. The ensuing brouhaha continued through 1883 and pulled in President Gilman. Sylvester, eager for priority and fame, was in the wrong. Nonetheless, his discovery is now known as the Sylvester–'t Hooft generators (Geradus 't Hooft, 1999 Nobelist in physics) and plays a role in the theory of quantum chromodynamics.

Matrices are now everywhere in applications---think, for example, of MATLAB---and the word "matrix" has taken on a life of its own, independent of mathematics, especially in the movies. The word "matrix" has the cachet of the cutting edge and forceful technological objectivity.

In 1883, Sylvester, replete with honors, was offered the Savilian Professorship of Geometry at Oxford. He snapped it up as the ultimate, if delayed, vindication of his British existence. Incredible. His contemporary Alfred Tennyson wrote, "For men may come, and men may go/ But I go on forever." Sylvester's second and final retirement came in 1892.

To wrap up. As judged by mathematician Percy MacMahon (1854–1929), Sylvester is not among the all-time top dogs in the math biz. Still, there's a crater on the Moon named after him. So there we are.

*Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at **philip_davis@brown.edu**.*

*In 1973, Chelsea reprinted in a "corrected" edition the Cambridge University Press version of 1904–1912. In 1997, the AMS bought Chelsea's list and has just announced the republication of Volume I.