On 22 February, 1877, James Joseph Sylvester gave an “Address on Commemoration day at Johns Hopkins University”.

Sylvester, the very excellent English mathematician, worked in areas of what we would today call algebra, number theory, and combinatorics. He is known for his algebraic work in invariant theory; he is known for his work in combinatorics, such as Sylvester’s Problem in discrete geometry; and for much else. He invented several terms which are commonplace in mathematics today — “matrix”, “graph” (in the sense of graph theory) and “discriminant”. He was also well known for his love of poetry, and indeed his poetic style. (He in fact published a book, The Laws of Verse, attempting to reduce “versification” to a set of axioms.)

I came across this address of Sylvester, not through mathematical investigations or in the references of a mathematical book, but rather in the footnotes of the book “Awakenings”, in which the late neurologist Oliver Sacks discusses, in affectionate and literary detail, the case histories of a number of survivors of the 1920s encephalitis lethargica (“sleeping sickness”) epidemic — an interesting and mysterious event in itself — as those patients are treated in the 1960s with the then-new drug L-DOPA and experience wondrous “awakenings”, often after decades of catatonia, although often followed by severe tribulations. (These awakenings were the subject of the 1990 Oscar-nominated movie of the same name.) These tribulations, in each patient, form an odyssey through the depths of human ontology, in which the effects of personality, character, physiology, environment, and social context are all present and deeply intertwined.

Sacks comes to the conclusion that a reductionist approach to medicine, focusing on the cellular and the chemical, is wholly deficient:

What we do see, first and last, is the utter inadequacy of mechanical medicine, the utter inadequacy of a mechanical world-view. These patients are living disproofs of mechanical thinking, as they are living exemplars of biological thinking. Expressed in their sickenss, their health, their reactions, is the living imagination of Nature itself, the imagination we must match in our picturing of Nature. They show us that Nature is everywhere real and alive and that our thinking about Nature must be real and alive. They remind us that we are over-developed in mechanical awareness; and that it is this, above all, that we need to regain, not only in medicine, but in all science.

Indeed, Sacks quotes from W H Auden’s “The Art of Healing”:

Papa would tell me,
‘is not a science,
but the intuitive art
of wooing Nature.

In an accompanying footnote, Sacks notes that mathematical thinking is real and alive, in just the same way. He quotes the aforementioned address of Sylvester.

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.

Sylvester is right, and if anything his argument is not forceful enough. Mathematics has always been limitless — and even more limitless than the seemingly (to Sylvester, at least) infinite possibilities of astronomy and biology — for, unlike the experimental or observational sciences, it requires no substrate in reality beyond the imagination of those who think it. Liberated from the necessity to study only this world, mathematics studies all the worlds it can imagine, which include our own but go far beyond our own one. (It is perhaps surprising, and even “unreasonable”, as Wigner argued, that we can count our own world as among those which are mathematical; but it is not surprising that its worlds transcend ours.)

The progress of science has displayed, in an absolute sense, how mathematics outstrips the limitlessness of other sciences.

However many may be the worlds of the astronomer — now teeming also with exoplanets and gravitational waves — they are still finite; the observable universe has a finite radius.

Sylvester’s panpsychism (everything has consciousness) is now out of fashion, but seems focused on biology — and we now know that biological life is constrained by genetics, and at the molecular level by DNA and related biochemistry. Mathematics knows no such constraint.

Taking panpsychism more generally, there is an argument — and a strong one, in my view — that understanding consciousness will eventually require a radical revision of our understanding of physics. But even then, I very much doubt any such radical revision would completely transcend mathematics — and I very much doubt that mathematics would not encompass infinitely more.

It is worth noting, though, that mathematics is, in a certain sense, reductionism par excellence. Even accepting what we know about incompleteness theorems and the like, mathematics, theoretically at least, can be reduced to sets of axioms and logical arguments, in the end consisting only of formal logic, modus ponens and the like. That is not how mathematicians do mathematics in practice, but that is the orthodox view on what mathematics formally is. Even the standard theorems that mathematics “knows no bounds” — the Godel incompleteness theorems, the Cantor diagonalisation argument, the set-theoretic paradoxes like Russell’s, for instance — can themselves be expressed, reductionistically, in this formal way.

All the infinite possibilities, the unboundedness, of mathematics, then, can be expressed in a very finite, very discrete, very reductionistic way. This is not surprising — even with finitely many letters one can construct an infinity of sentences, one can burst all brazen clasps, one can empty all veins and lodes, one can exhaust all soils, there is no end to the harvest, however dizzying and rarefied the altitude at which it is sown.

And as for definitions of permanent validity? At least in terms of the experience of learning, doing and discovering mathematics, I cannot go past Ada Lovelace’s definition of the “poetical science” as “the language of the unseen relations between things”.

There is much else of interest — and not just historical interest — in Sylvester’s address. Mathematics impedes public speaking; university study and research ought to avoid monetary reward and public recognition; students should avoid “disorder or levity”; all researchers should simultaneously engage in teaching; anecdotes of arithmetic in the French revolution; every science improves as it becomes more mathematical; and the taste for mathematics is much broader than one might think. So argues Sylvester, poet, mathematician; perhaps I will return to these arguments one day.

Limitless as that space too narrow for its inspirations
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