Laws of Form

Contents:Preface to the fifth English Edition Preface to the first American edition Preface Introduction A note on the mathematical approach1 The form2 Forms taken out of the form 3 The conception of calculation4 The primary arithmetic5 A calculus taken out of the calculus6 The primary algebra 7 Theorems of the second order 8 Re-uniting the two orders 9 Completeness10 Independence11 Equations of the second degree12 Re-entry into the form NotesAppendix 1. Proofs of Sheffer's postulatesAppendix 2. The calculus interpreted for logic Index of references Index of formsAppendix 3. Bertrand Russell and the Laws of FormIntroduction to Appendices 4 & 5Appendix 4. An algebra for the natural numbers Appendix 5. Two proofs of the four-colour map theorem My simplest proof of the four-colour map theoremAppendix 6. Last wordAppendix 7. The prime limit theoremAppendix 8. Primes between squaresAppendix 9. A proof of Riemann's hypothesis via Denjoy's equivalent theoremClosing remarks

"Preface to the fifth English edition: As is now well known, Laws of Form took ten years from its inception to its publication, four years to write it and six years of political intrigue to get it published. Typically of all unheralded best sellers from relatively obscure authors, it was turned down by six publishers, including Mark Longman who published my earlier work on probability. Even Sir Stanley Unwin refused to publish it until his best author, Bertrand Russell, told him he must. This crucial recommendation was not achieved without intrigue, and required me (not unwillingly) to sleep with one of Russell's granddaughters, who asked me in the morning, 'What exactly do you want from Bertie?' 'To endorse what he said about the book when he first read it in typescript,' I told her. 'He never will!' she exclaimed. 'You'll have to twist his arm, you'll have to blackmail him. How can I help?' The next few years were spent in vigorous arm-twisting and incessant blackmail from us both. One of her threats was to invite me to Plas Penrhyn as her guest while Bertie and Edith were away in London. This sent Bertie into a paroxysm of terror of what the neighbours might think. He also had an irrational fear of spoiling his reputation as a mathematician, which was not good anyway, by recommending a book that had not yet been tried by the critics. He seemed totally unaware that any book he recommended, however ridiculous, would have no effect whatever on this. When we finally got him cornered, in my next visit to Plas Penrhyn, he carefully avoided mentioning the subject during the whole of my stay, and I considered it too dangerous to mention it myself. The next morning I was due to depart while Bertie and Edith were still in bed, and I thought I had failed miserably. But no! I missed my train because they had not ordered me a taxi to the station, which was their way of telling me that my visit was to be prolonged by another day. The evening of this extra day came, and still nothing was mentioned. Ten o' clock bedtime arrived, and I thought I had failed again, when Bertie suddenly said, 'What exactly do you want of me?' 'To endorse what you said about the book three years ago,' I told him. 'You must remind me what it was,' he said. I produced a verbatim report of his remarks, neatly typed out, and thrust it in his face. 'Are you sure this is all you want?' he said. 'Don't you want me to write a detailed introduction to the work, as I did for Wittgenstein?' I told him that that would be very nice, but that this was all I needed just now. He contemplated the page of typescript for a moment, and then a wicked gleam lit up his face, and he rubbed his hands. 'Supposing I don't?' he grinned. 'Then,' I heard myself saying, 'it might delay the publication for a year or so, but the book will still be published in the end, and you won't be associated with it.' 'Oh,' he said. 'I never thought of that. How would you like me to sign it?' There is no stronger mathematical law than the law of complementarity. A thing is defined by its complement, i.e. by what it is not. And its complement is defined by its uncomplement, i.e. by the thing itself, but this time thought of differently, as having got outside of itself to view itself as an object, i.e. 'objectively', and then gone back into itself to see itself as the subject of its object, i.e. 'subjectively' again. Thus we are what we see, although what we see looks like (and is) what we are not. This incessant crossing of the thing boundary, to look at it from one side and then the other, is called scrutiny, which as a small child was I told is not polite, because by scrutinizing a person or thing we shall notice uncomplimentary (same sound, different word) qualities of the person or thing that it is rude to mention or think about ..."