The Library Paradox (38 page)

The famous library paradox of Bertrand Russell (1872–1970) is generally thought to have been discovered in 1901. Russell was a young man of excellent family; his paternal grandfather, Lord John Russell, had been a Prime Minister. He completed his studies and left Cambridge in 1894, turning to philosophy.

At about this time, probably in 1896, Cesare Burali-Forti (1861–1931), an assistant to Giuseppe Peano – and a mathematician whose interests, being ahead of his time, prevented him from ever obtaining a university position – discovered the first version of what became later known as Russell’s paradox. As described in the book, his observation was that since the set of ordinals is a well-ordered set, it must have an ordinal; however, this ordinal must be both an element of the set of all ordinals and yet greater than every ordinal: a contradiction. Russell, a recent graduate from Cambridge, must have learnt of Burali-Forti’s work,
which would naturally have influenced him in the discovery of his own version, which he made public in 1901.

The content of Russell’s paradox is essentially identical to Burali-Forti’s; however, it has the advantage of being expressed only in terms of common notions rather than specifically mathematical ones. The version of librarians and catalogues is one of the commonest methods used by mathematicians for explaining the paradox to laymen. In terms of simple set theory, one considers the problem of sets which are, or are not, members of themselves (for instance, a catalogue of books in a library which lists, or does not list, itself as one of the books in the library). The paradox arises when trying to define the set of all sets that
are not
members of themselves. A moment’s reflection will make it clear that if such a set is a member of itself, then it cannot be a member of itself, and vice versa: hence the paradox.

Russell’s paradox is considered to have been of fundamental importance in the development of modern axiomatic set theory and logic. Gottlob Frege (1848–1925) was finishing up an enormous treatise on logic, the
Grundgesetze der Arithmetik,
when he received a letter from Russell containing the paradox. The paradox showed that Frege’s work was based on axioms that were inconsistent and caused a sort of revolution in the way mathematicians considered logic and set theory. Any collection of objects was no longer considered worthy of the name ‘set’. A set now had to be formulated as a collection satisfying certain basic axioms, and the principles of set theory apply in a
consistent way only to these sets and not to collections in general. As is well known, Bertrand Russell went on to do more foundational work in logic, and then became deeply involved in affairs of global peace, winning the Nobel Prize in Literature in 1950.

On the subject of taking degrees at Cambridge, the Jewish mathematician J.J. Sylvester was Second Wrangler in 1837, and was refused a degree. Another Jew, Numa Hartog, was Senior Wrangler in 1869; he participated actively in an effort to change the rules, testifying before the House of Lords. These efforts culminated with the passing of the Universities Tests Act in 1871, allowing students of all religions to graduate from Oxford and Cambridge by abolishing the requirement that they sign the Thirty-Nine Articles of the Church of England. Note that the act was passed just two years after a Jewish student became Senior Wrangler. However, although the brilliant young mathematics student Philippa Fawcett was classed
above the Senior Wrangler
in 1890, it took several more decades before women obtained the right to take degrees at Cambridge.

Incidentally, the fascinating history of the Dreyfus affair is exactly as recounted in the book, except that this was only the beginning. The full story, filled with spies and secrets, stunning reversals, sublime nobility and treacherous villainy, lasted until 1906, and ended with Dreyfus being awarded the Légion d’Honneur. It makes a more extraordinary tale than many a novel, and constitutes one of the most important and profound political events of fin-de-siècle Europe.

I would like to extend my warmest thanks to Peter Kenyon, a retired history lecturer whose main interest is in late nineteenth- and early twentieth-century social history, for his invaluable advice, suggestions and corrections on dozens of historical details throughout the manuscript. They ranged from questions of language to matters of law to details of carriages, trains, post offices, newspapers, and even chimneys and coals.

C
ATHERINE
S
HAW
is a professional mathematician and academic living in France.
The Library Paradox
is her third mystery novel.

The Three-Body Problem

Flowers Stained with Moonlight

The Library Paradox

The Riddle of the River

Fatal Inheritance

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