The Library Paradox
Page 31
‘Done it unconsciously, or meant to do it?’ I said. ‘Do you know, Jonathan, you never told me why you were going to the library that day. You never told anyone. But I have wondered about it a great deal. What were you doing there?’
He squirmed about a little.
‘Well, I will tell you the truth,’ he said at length. ‘I do think you have deserved to know, and I can trust you. But please, never tell anyone what I am about to tell you. I don’t want my family to know, above all not my uncle. I don’t know what he would think of me. You see, I was becoming obsessed with the fact that he was about to be released, and that he would be free, and that he talked and thought about nothing but the professor – the black dog, as we called him. I was terrified that he would immediately go and slaughter him and then perhaps commit suicide, or go back to prison forever. Not only my uncle, but my mother would have been destroyed by it. I have to admit that I felt that I couldn’t let it happen. I went through a very difficult period trying to come to a decision.’
‘Yes?’ I said encouragingly, as he came to a standstill. ‘I suppose that after all that has been said, you are not going to tell me that you decided to kill him in your uncle’s stead, after all!’
‘No, of course not. I’m no murderer,’ he said. ‘I was going to warn him, actually. I was going to tell him that my uncle was about to come out of prison, and that he was in mortal danger. I imagine he knew that my uncle was due to be released, and he knew about my uncle’s letter to the judge, but he did not know that my uncle and I knew his identity, so I assumed that he must feel quite secure.’
‘So you meant to warn him for his own safety?’ I said.
‘Well, yes, I meant to warn him, but to be completely honest, I also intended to reveal my knowledge of his role in my uncle’s trial in a dramatic kind of way – to frighten him, to put him in the position of being accused. I wasn’t just going to say, “My uncle wants to kill you, so go somewhere where you’ll be safe.” If I had said that, he would probably just have had my uncle put under police surveillance or even rearrested. I meant to make him feel frightened and threatened. I wanted him to fear for his life. I wanted him to wake up at night, hearing noises, terrified and not knowing what he was afraid of or where the threat was coming from. I really hated him. God knows how it would have turned out if I had done what I meant to. I can tell you that when I saw him lying dead, I was terrified. I literally wondered for a moment if my thoughts couldn’t have had such an effect. But then I remembered that he must have had any number of enemies who mortally hated him.’
‘He did,’ I said. ‘Isn’t it strange that he himself was his own worst enemy in the end?’
MATHEMATICAL HISTORY IN THE LIBRARY PARADOX
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.
About the Author
CATHERINE SHAW is a professional mathematician and academic living in France. The Library Paradox is her third mystery novel.
By Catherine Shaw
The Three-Body Problem
Flowers Stained with Moonlight
The Library Paradox
The Riddle of the River
Fatal Inheritance
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First published in Great Britain by Allison & Busby in 2006.
This ebook edition published by Allison & Busby in 2013.
Copyright © 2006 by CATHERINE SHAW
The moral right of the author is hereby asserted in accordance with the Copyright, Designs and Patents Act 1988.
All characters and events in this publication other than those clearly in the public domain are fictitious and any resemblance to actual persons, living or dead, is purely coincidental.
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A CIP catalogue record for this book is available from the British Library.
ISBN 978–0–7490–1454–4