Written: Oct. 27, 2006.
I just finished reading the wonderful biography, by Karen Parshall, of my great hero James Joseph Sylvester. Parshall concludes her masterpiece with quoting, with agreement, MacMahon's verdict that Sylvester, while definitely one of the greatest mathematicians of his time, was even more definitely, not among the greatest mathematicians of all time.
First, what's so great about being ``great'', or even ``greatest''? It is more important to be interesting, and James Joseph was surely more interesting than Euler and Gauss combined. Just read Parshall's biography, or better still, browse through his Collected Works.
But even if it is not so great being the "greatest", if I had to name a mathematician who, all things considered (constructing a measure that is more concentrated on the things that really count, like vision, originality and foresight) then Sylvester has no rivals.
He was way ahead of his time. He was also way ahead of our time, witness Parshall's agreement with the verdict of his contemporaries: `pretty great but no way amongst the greatest', and Karen Parshall ends with a conciliatory note:
"In his time and his place, he was both a leader and a pathbreaker."[my emphasis] .
Why was he so great? First, he knew that algebra is more important than analysis. He tried to do everything algebraically. He also knew that algebra was just combinatorics in disguise, and his Constructive Theory of Partitions, helped by his brilliant Johns Hopkins students [Fabian Franklin and William Pitt Durfee], is a masterpiece that is still not fully appreciated today. All these so-called analytical theta-function identities proved via one picture!
He was also a great algorithmitican, way before the word existed. Towards the end of his life, there was a young Turk named David Hilbert who proved existence, and didn't care much about construction. Hilbert ruled for the next 100 years, and that was one of the reasons Sylvester and his style was looked down upon as "old hat".
But the best reason why James Joseph Sylvester was the greatest was his vision and realization that mathematics is not a deductive science but an experimental science. In a public speech before the Mathematical and Physical Section of the British Association, delivered in 1869, he responded, in very strong terms, to the conventional wisdom of Thomas Huxley- who had his own agenda to promote the education of the empirical sciences at the expense of fossilized mathematics- delivered at an after-dinner speech, who claimed that
"Mathematics is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation.''
Sylvester admitted that the way mathematics was taught may give that false impression (unfortunately for us, this is still true to a large extent today). But the way mathematics is discovered is purely experimental, and he went on to present many convincing examples from his own and other mathematicians' work.
Sylvester was so fond of his speech that he included it as an appendix to his poetry treatise "Laws of Verse". His approach to poetry was experimental as well, and he must have seen the connection.
Sylvester was already right yesterday , way back in 1869. He is even more right today, and tomorrow, his vision will seem so obvious, that once again he would be in danger of not being considered so great, since all he said were platitudes known to everyone.