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[my emphasis] .histime andhisplace, he was both a leader and a pathbreaker."

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.

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