My Time in Space, page 15
The consequences of there being nothing! This took me back to a passage from which I had averted my attention in Laws of Form:
There is a tendency, especially today, to regard existence as the source of reality, and thus as a central concept. But as soon as it is formally examined, existence is seen to be highly peripheral and, as such, especially corrupt (in the formal sense) and vulnerable…. It is the intellectual block which most of us come up against at the points where, to experience the world clearly, we must abandon existence to truth, truth to indication, indication to form, and form to void, that has so long held up the development of logic and its mathematics.
That the world has produced itself out of nothing, I do believe, utter mystery though it be, and that we can trace the self-creation of its elements back to the very seedgrain of time – but there falls the cliff edge, the ultimate distinction; was Spencer-Brown sacrificing the existent to the formal structures of its possibility? This did not suit the naïve-realist side of my temperament, my sense of the clamorous demand of every blade of grass on the clifftop for recognition. I began to find references to him on the Internet and elsewhere that were disquieting in other ways too. In the Scientific American of February 1980 Martin Gardner had disparaged Spencer-Brown in his well-known column ‘Mathematical Games’:
In December of 1976 G. Spencer-Brown, the maverick British mathematician, startled his colleagues by announcing he had a proof of the four-color theorem that did not require computer checking. Spencer-Brown’s supreme confidence and his reputation as a mathematician brought him an invitation to give a seminar on his proof at Stanford University. At the end of three months all the experts who attended the seminar agreed that the proof’s logic was laced with holes. But Spencer-Brown returned to England still sure of its validity. The ‘proof has not yet been published. Spencer-Brown is the author of a curious little book called Laws of Form, which is essentially a reconstruction of the propositional calculus by means of an eccentric notation. The book, which the British mathematician John Horton Conway once described as beautifully written but ‘content-free’, has a large circle of counter-cultural devotees.
In the mouth of Gardner (whom the anarchic philosopher of science Paul Feyerabend has called ‘the pit-bull of scientism’), ‘counter-cultural’ would be a term with bite. What had urged him to this attack? To my amazement I found a letter of Gardner’s, of slightly earlier date, quoted on two web sites including the Math Forum discussion group, saying that he had proposed to write a column on Spencer-Brown but had been dissuaded by another mathematician, Donald Knuth, on the grounds that to do so would give publicity to a charlatan! What could this work be, that seemed to have provoked a move to suppress it? Although I had little confidence in my ability to judge it, I was determined to look into it.
The only available account by Spencer-Brown of his controversial proof of the four-colour theorem, I gathered from the LoF site, was in an English appendix to a German edition of Laws of Form, to be had from a company called Astro. When I located Astro’s website I was discouraged to find it largely concerned with astrology and sexual magic. In any case, they failed to respond to my e-mails, and so, with some trepidation (one Internet source having described him as ‘a dreadful curmudgeon but at the same time one of the most charming, humorous, and delightful people I have ever met’), I wrote to the author himself. I was hoping for some closure to the story of high intellectual endeavour I wanted to tell and of which I had already written the earlier part; it would have been satisfying to see the new proof victorious over all doubters, so that the conclusion of my essay could carry it in procession as Cimabue’s new Madonna was carried by the admiring townsfolk of Florence. Instead, I have become, briefly, the Fool of pelting storms, too fraught with personal matters to be recounted here, far out on a conceptual heath. How simple it would be to round off this adventure if Spencer-Brown and his proof were, ‘not in lone splendour hung aloft the night’, but glittering away up there as unquestionable constellations with Pascal and his mystic hexagram! But I have discovered, with pain, that the realm of pure ideas is one of the battlefields of human affairs, of which we will never have a colourable map.
A week or so after the date of my letter, the telephone rang: George Spencer-Brown himself, from his new address in the West Country. He began with a long complaint that the telephone number on my letterheading did not include the international dialling code, continued with some disobliging remarks on a little book I had sent him on prime numbers by a friend of mine, and half an hour later he was still telling me about his own astounding discoveries in the theory of prime numbers, including an unpublished proof of Goldbach’s conjecture. I arranged to send him a cheque for the price of the German edition of Laws of Form, which arrived a few days later together with a photocopy of a long and rather bizarre-looking handwritten paper on prime numbers, and a twenty-page letter in which he dealt at length with the dialling-code question again, again abused my friend’s book, went into a rant against G.H. Hardy and his book, (‘a [mediocre] mathematician’s apology [for being mediocre]’), broke off to include a poem ‘In Praise of Lying’ by Richard de Vere, presumably a pseudonym of Spencer-Brown himself, got into his stride by remarking that if he is let loose in any mathematical discipline, he will either abolish it or transform it, and ended with a rich account of his dealings with God, Sir Stanley Unwin and Bertrand Russell over the publication of Laws of Form.
Several parts of this inordinate communication infuriated me, and I wrote a riposte calculated to raise blisters on a battleship, but did not put it in the post because at the same time I was reading the new work on the four-colour theorem and finding it full of wonders. Instead, I devised a challenge, in a logical format that recurs in Spencer-Brown’s own writing:
Dear George – It was generous of you to send the ms on primes between squares with the copy of Gesetze der Form. I’m nibbling my way through it slowly line by line with pleasure and excitement, as I am the four-colour proof. I hope shortly to be able to write something about my wonderment over the pure silk of your mathematics, even if I fail to follow the threads all the way to the conclusions.
However there were several passages in your letter that angered me. I have written a sharp retort but refrained from posting it. What to do? I propose that you choose whether to receive my expression of wonderment, or my retort, or both, or neither. If you choose not to choose, I will choose which if any to send you. But I think you should choose. Yours sincerely …
His reply completely ignored this provocation; without even a preliminary ‘Dear Tim’ it opened with the claim that the only two great mathematicians of the twentieth century were Ramanujan and himself, and that while Ramanujan was probably the cleverer I would already know that Spencer-Brown was the deeper. What was I to make of this? If false, the boast was absurd; if true, then it is to be expected that he would know it to be so, and it would be legitimate, if unconventional, to proclaim it. The question added to the interest of the journey I made to visit him, as a gesture of friendship and concern, in the summer of 2000.
I found Spencer-Brown to be a vigorous seventy-odd-year-old, with the fine domed head one associates with the conventional image of genius, and combative bushy eyebrows. He is still extremely productive – the floor of his cottage was ankle-deep in calculations – but in cramped circumstances, having held no academic post since the early 1980s when he was Visiting Professor of Mathematics at the University of Maryland, because, he claims, of the sort of calumny I had come across on the Internet. He has been denied outlet for his papers other than as appendices to successive editions of Laws of Form – a shame, this, because it is a classic text, and in its current form suffers as a Palladian villa would from disproportionate extensions and ad hoc lean-to sheds. His conversation was challenging (we disagreed on the fundamental nature of reality, because, he said, I was ‘unenlightened’ – which remark enlightened me instantly to the fact that Buddhism can be as oppressive as any other system of belief), amusing (‘Most cats think the way to play chess is to knock the pieces off the board and chase them around the floor; but my kitten, having spent up to two hours sitting on my shoulder while I analyze a chess position, knows how to put out its paw and gently push a bishop from one square to another’), sometimes profound (‘To write a great book one must love one’s reader’), but unremittingly self-centred, so that after a few hours my own ego was struggling for oxygen. When I picked up the phone to call a taxi, I found that the spiral flex between handset and base was so intricately twisted that I could only prise them a few inches apart, which made speaking difficult. During our subsequent telephone conversations I have often thought of that flex, for they have been extremely tangled and contorted. Calls would go on for an hour or more, leaving me sometimes with a valuable insight into Spencer-Brown’s mathematics and sometimes with a burden of history, too personal in relation to himself and libellous in relation to his fellow professionals (‘pipsqueaks’, the most of them) to be anything other than an obstacle to my writing of this account, and on several occasions ending in as much physical violence as can be transmitted by telephone.
He had some unfair advantages, I felt, in these bouts: an extraordinarily quick intelligence; an unshakeable belief in the correctness of whatever he was saying, even if, or especially if, it was the exact opposite of what he had said last time; and an utter ruth-lessness as to my amour-propre. Sometimes I tried to hang up on him, but he was always quicker on the draw and while I was still aiming a parthian shot I would hear the decisive click of his receiver. Latterly I undertook such exchanges with the enthusiasm I might have for running before the bulls of Pamplona, in expectation of an exhilarating scamper ending with a high probability of being trampled and gored. The relationship terminated, as M had predicted, with my dismissal into the category of pipsqueak. I had written asking for elucidation of certain steps that left me doubtful in two of his proofs of the four-colour theorem; there was no reply, and I accepted the break in this overheating correspondence with some relief, but eventually, since I needed guidance in order to progress with the mathematics, I telephoned him. A useful little tutorial followed, but when I showed a reluctance to follow him in a death-defying leap from a result in Laws of Form to the obvious truth of the four-colour theorem he told me that if I couldn’t see the connection, after the thousands of poundsworth of free tuition he’d given me, I was an idiot. Like all the mathematicians who couldn’t or wouldn’t read his work, I just wasn’t good enough. And he rang off. I rang back instantly to remind him that despite his rudeness I would be writing about him, and was finally annihilated by another roar of ‘You’re not good enough!’
So, the message of all that sound and fury is: Caveat lector, any idiocies in the following sketch of Spencer-Brown’s reformulation of the theory of maps are my own. In any case I can hardly go beyond the opening moves here, but I hope to convey something of the finesse of his approach. I begin by quoting the introduction to his own presentation:
Once I had constructed the primary arithmetic in Laws of Form, I became aware that I had a technique that, suitably applied, would solve the map-colouring problem. I did not immediately apply it to the problem, because I felt that to make what would almost certainly be a difficult proof, in a completely novel system of mathematics, would occasion the hostility and disbelief of the more superficial members of the mathematical profession…. This subsequently turned out to be the case.
In 1961 my brother came to visit me and I showed him the problem. He went away contemptuously, saying,
‘Soon prove that!’
After a week he returned with the news that it was turning out to be more difficult than he had at first anticipated. But he had found an astonishing algebraic colouring algorithm [i.e. a rule-bound procedure for colouring any given map] … In 1975 my father died, leaving an estate worth half-a-million pounds (a substantial sum in those days) that had been in the family for more than two centuries. My mother who, for vindictive reasons, maintained that my brother and I ‘did not deserve it’, forged documents and, with the connivance of her lawyers, succeeded in stealing it all and bequeathing it to my cousins, leaving my brother and me destitute. The unpleasantness and the distress of it killed my brother and nearly killed me.
It was in these circumstances of despair and bereavement that I decided things could not get worse, so I might as well prove the four-colour theorem. My brother’s algorithm was now lost, so I set about it the only way I knew how, using two elementary marks to set up the special case.
A map, of the abstract sort we are considering, consists of a number of regions separated by borders, drawn on a flat surface; we count the area one would normally think of as outside the map (as the sea is outside a map of Ireland) as one of its regions. The points where borders meet are called nodes. Adjacent regions are those that share a border (not just a node). The shapes and sizes of the regions and borders are irrelevant; we can stretch and bend them as convenient. The first step is to standardize the map so that just three borders meet at any one node, as in Fig. 19. If the four-colour theorem is true for standard maps it must be true for all maps; so from here on we need only consider standard maps.
Fig. 19. Left: Eliminating a node at which more than three regions meet by expanding one of them. Right: If the map containing this new configuration can be four-coloured, so can an otherwise identical map containing the old configuration.
Suppose we have a map that can be coloured (so that no two adjacent regions are of the same colour – and from now on we will take this clause to be understood) with just four different colours. An economizing printer looking at this as a production problem might hit on the idea of using a red and a blue ink, say, plus a purple made by overprinting red with blue (or vice versa), leaving the white paper to represent the fourth colour; this is the format: r, or b, or both, or neither. Thus the four-coloured map could be printed from two plates, one of a number of red shapes and the other of a number of blue shapes – i.e. the map can be factored into two nodeless maps each of two colours (red and white, or blue and white), rather as the composite number 15 can be factored into the prime numbers 3 and 5 (Fig. 20). Spencer-Brown’s strategy is to prove that any standard map can be so factored – from which it follows that any map can be coloured with just four colours.
Fig. 20. A four-coloured map and its two-coloured factors.
Fig. 21. The three types of border in a four-coloured map, and the alternation of types of borders around one of its regions.
Fig. 22. A symmetrical map (derived from a regular solid, the dodecahedron), colourable with some puzzling, and the petersen, an uncolourable network.
Without going too far into the lengthy reasoning necessary to capture this proof, I can give the willing reader a glimpse of the striking techniques for manipulating the colourings of maps that Spencer-Brown has devised on the basis of the above simple ideas. First we shift attention from the regions of the map to its network of borders. Fig. 21 shows how the borders of the four-coloured map shown in Fig. 20 are made up from the borders of its two factors. Note that at each node borders of three types meet. We can think of them as having three different colours. A little experimentation will convince the reader of the theorem (first published by P.G. Tait in 1880) that if a map can be four-coloured, its network of borders can be three-coloured (i.e. coloured so that the three borders meeting at a node are all of different colours), and vice-versa. So the problem of four-colouring maps has been reduced to one of three-colouring (standard) networks. Just to show that this is a real problem, here are two little labyrinths, in one of which lurks the beast, uncolourability (Fig. 22). By trial and error the ingenious reader will be able to colour the first, but the other, a fascinating entity known as the petersen, cannot be three-coloured; in fact it is the smallest of all the uncolourable networks. It is three-dimensional, and since its links pass under and over each other they cannot be the borders of regions in a map; so it is not a counter-example to the four-colour map theorem. (One of Spencer-Brown’s proofs of the theorem proceeds by showing that all uncolourable networks must be three-dimensional, from which it would follow that all planar networks are colourable.) It is instructive to try colouring the petersen, starting from any node and working outwards, labelling the links 1, 2 or 3, adhering to the rule that the three links meeting at any node must have different labels. Whatever choices one makes in this process, one soon finds oneself trapped in a corner where one cannot avoid breaking the rule. The sensation is like that of coming up against a contradiction in a chain of reasoning, or of trying to work out whether the statement ‘This statement is false’ is true or false. Paradoxicity, not uncolourability, is the relevant paradigm in this mathematical situation, according to Spencer-Brown; the network is colourable – but only with imaginary colours analogous to the imaginary values he introduces as solutions of self-contradictory equations in Laws of Form. (This is the great leap from Laws of Form to the four-colour theorem that I jibbed at, in that last apocalyptic telephone conversation.)
I shall try to convey the fascination and simplicity of Spencer-Brown’s methods and follow him some way into the theory of map colourations. Returning to Fig. 21, we can take the three colours of the network to be blue, red, and blue-red, i.e. purple (b, r, and br). Of course at this stage we do not know if all maps can be four-coloured, or, equivalently, if all planar networks can be three-coloured; so we have to be able to represent a link we cannot colour or which we choose to leave uncoloured. Fortunately we have the fourth alternative in Spencer-Brown’s logical pattern, neither b nor r, for such a link. Note that the links involving r, for instance, form closed circuits, around which they alternate: r, br, r, br etc. (These circuits are just the boundaries of the colour-patches in the r-factor of the map.) Suppose we have succeeded in three-colouring a network in which there is a circuit as shown in Fig. 23. Clearly we can interchange the b-links with the br-links without having to change any of the r-links and so upsetting the colouration of the rest of the map. This is the sort of operation one often needs to perform in map-theory, and Spencer-Brown has invented a brilliant way of thinking about it. Imagine one could take a closed band of colour r and superimpose it on the circuit. What are the rules for superimposing one colour on another? (Instead of ‘superimpose’ we should say ‘idempose’, as the two colours will not just be one on top of the other but will occupy the identical space.) The answer goes to the heart of what Spencer-Brown calls ‘formal mathematics’, that is, the mathematics of elements whose order, size and shape are undefined or irrelevant. Consider a region in a map, and another region of the same colour idemposed on the map so that part of its border coincides with part of the border of the first region (Fig. 24). Since we have the same colour on either side of it, we can omit the doubled part of the border; the two regions fuse like raindrops, eliminating the surface films between them. This suggests that the rule should be that idemposition of two identical elements cancels both of them:
