James Joseph Sylvester at a glance:

J. J. Sylvester and Arthur Cayley were/are often referred to as the "invariant twins" for their important contributions to the theory of invariants. Although the two mathematicians held similar interests, their temperments and mathematical style were quite the opposite. Cayley was calm, methodical, and refined, and incomparably sweet to his colleagues and students. Sylvester, on the other hand, was known for violent outbursts, rambling lectures, and horrible poetry. Oh yes, poetry. Sylvester loved poetry, and kept that hobby for much of his life. An amusing bit from a textbook:

"One evening, at the Peabody Institute in Baltimore, (Sylvester) read his Rosalind poem, which consists of 400 lines all rhyming with the heroine's name 'Rosalind.' So as not to interrupt the poem, he first spent an hour and a half reading his explanatory footnotes, many of which led to further extemporaneous elaborations. Then, to the remnant of his audience that was left, he read the poem itself."1

Sylvester could barely keep his temper under control. When he was 14, he attempted to stab a fellow student with a table knife over a minor dispute. In 1841, he resigned his position at the University of Virginia after only a few months because of a dispute with two of his students. Other personality quirks include forgetfulness (he often could not even remember his own findings) and a complete lack of interest in other mathematicians' work.

A strange man, for sure. But brilliant, as well. Sylvester is commonly credited for coining terms such as "matrix" (1848). He also showed a great interest in formal proofs, and spent a good deal of time working formal proofs on work Cayley discovered and accepted on a quasi-empirical basis. For example, Cayley simply accepted the Cayley-Hamilton theorem as true after proving it for matrices of rank 2 and 3. Sylvester rigorously proved the theorem later for matrices of rank n. I suppose you could say Cayley was a realist which Sylvester was a formalist.

1Eves, Howard. An Introduction to the History of Mathematics, 5th ed. 1983: CBS College Publishing.
2Cullen, Charles G. Matrices and Linear Transformations, 2nd ed. 1990: Dover Publications, Inc.
3Burton, David M. The History of Mathematics: an Introduction, 4th ed. 1999: McGraw Hill Companies, Inc.

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