My Time in Space, page 14
This extraordinary construction (epoch-making, indeed, if it accomplishes what it claims to have done) was written in a style of Pascalian grace, if not with irreproachable clarity; its air of stark rigour contrasted strangely with the somewhat shamanistic view of the mathematician it implied, as the one who brought back almost incommunicable truths from a distant and dangerous Otherworld:
To any person prepared to enter with respect into the realm of his great and universal ignorance, the secrets of being will eventually unfold, and they will do so in a measure according to his freedom from natural and indoctrinated shame in his respect of their revelation.
To arrive at the simplest truth … requires years of contemplation. Not activity. Not reasoning. Not calculating. Not busy behaviour of any kind…. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set about it in secret, pretending meanwhile to be engaged in the frantic diversions and to conform with the deadening personal opinions which are being continually thrust upon them. In these circumstances the discoveries that any person is able to undertake represent the places where, in the face of induced psychosis, he has, by his own faltering and unaided efforts, returned to sanity. Painfully, and even dangerously, maybe. But nonetheless returned, however furtively.
What appealed to my temperament most deeply in Laws of Form was its conceptual knife-play. ‘The theme of this book is that a universe comes into existence when a space is severed or taken apart,’ says Spencer-Brown in an introductory note. As a beginning, this move has its precedent: ‘The earth was without form, and void…. and God said, Let there be light: and there was light … and God divided the light from the darkness.’ That the basis of existence is difference, that the first step in understanding is discrimination, is for me an axiom that feels as if it were innate. As it happened, while I was dealing with Spencer-Brown’s diagrams I was also writing a catalogue note for an exhibition of abstract paintings by my closest friend, Peter Joseph, and I borrowed Spencer-Brown’s dictum as an epigraph:
‘… a universe comes into being when a space is severed …’
… a world in which the divisions of night from day, the shores, the horizons, are sharp enough to cut you. In the paintings of ’67–8 Peter Joseph reduces landscape to the clearest oppositions of the simplest elements, each work being composed of just three or four rectangles of pure colour…. One of these paintings, a long rectangle divided once from top to bottom and once from side to side to give four identical areas, conveys the sensations one would hope for from a trans-polar orbiting: three of its quadrants, in brilliant but somehow sombre primaries, have an arctic, auroral, chill; the fourth contains six months of night.
What is gained by such a severe process of reduction? In these condensed transcriptions of experience, all ambiguities of twilight, poetry of misty horizons, flux and riddle of coast-lines, are suppressed in favour of the immediacy of the great elements. One comes out of a cave, and faces the sky; one crosses the brow of a hill, and there is the Atlantic….
I sent a draft of this to Spencer-Brown, looking for permission to quote from his then unpublished book, and later phoned him. He sounded mildly interested in my poetic application of his words, and I arranged to call on him at his flat in Richmond. I found him to be a middle-aged man with a dark and intellectually hungry look, a shadow of D.H. Lawrence projected into abstract realms. He was very ready to expound his discoveries, though a little short with my mathematical ignorance; as a student, his IQ had been ‘off the clock’, he told me. He scribbled in the flyleaf of my proof copy of Laws of Form a little diagram of how i is re-entered into its defining equation, as mentioned above. He identified his ‘marked and unmarked states’ with the Yang and Yin of Taoism – concepts I had thought long worn away to nothing by mindless overapplication. But his counting device, a stack of circuits like the one described above that counts to two, each feeding its output into the next, and so registering a number in the binary system, was a very concrete application of his theories; it was patented, he told me, and used in the real world by British Rail for keeping track of the number of railway wagons entering and leaving tunnels. Spencer-Brown had developed this device in collaboration with his brother, whom he mentioned also as the discoverer of some abstruse equations for which he ‘had to go very far’. The impression I came away with of this brother was slightly mysterious; he seemed to combine an uncanny degree of intelligence with a certain mythical remoteness from the actual, like Holmes’s Mycroft.
Mathematics cohabited with mysticism in Spencer-Brown’s bookshelves. I noticed J.W. Dunne’s An Experiment With Time, which had caused a commotion in the 1920s by purporting to make precognitive dreams scientifically respectable. Dunne argues that since time passes, there must be another time with which to measure how fast it passes, and yet another to measure the passing of that, and so on; also, we have a consciousness, a consciousness of that consciousness, etc., corresponding to this infinite number of time-dimensions, and that these higher-dimensional consciousnesses are (naturally!) able to see at least some way into the future of ordinary everyday time. H.G. Wells had been impressed by Dunne at first but later said his thesis was ‘an entertaining paradox expanded into a humourless obsession’. I had early learned to sneer at it, but now Spencer-Brown put me right; Dunne had grasped the importance of ‘recursion’ (the sort of snake-swallowing-its-own-tail process exemplified by Spencer-Brown’s expressions that re-enter themselves), and had merely lacked the mathematics to develop it.
Peter Joseph’s exhibition was due to open at the Lisson Gallery soon thereafter, and I invited Spencer-Brown to the opening. M and I arrived a little late, and found him looking rather adrift among the gallery-going crowd. We stayed him with conversation. I asked – it was out of order as party chitchat, but I did want to know, having been fiddling with the question inconclusively – if his arithmetic of indications was isomorphic with the arithmetic of the null set and the universal set. (‘Isomorphic’ is jargon for ‘structurally identical’.) He concentrated inwardly for a few seconds before answering, ‘Yes.’ He seemed a little put out, so I hastened to add, ‘I suppose no one ever thought to write down the arithmetic of the null set and the universal set.’ ‘Quite!’ he replied. Later an exotic girl, prime ornament of the art-gallery set of that season, black with a bloom of gold-dust, came in, and we all turned to admire her. M said she could see purple smoke rising from her, and Spencer-Brown explained that that was because she was seeing her directly; if the light-level had been lower the girl’s ‘essential body’ would have been visible hovering above her head.
During one or other of my two encounters with Spencer-Brown we talked of the four-colour hypothesis, which was then still unproven. This is one of the most famous problems in mathematics, quite simple to state but doggedly resistant to solution. It arose as follows. In 1852 a Francis Guthrie, colouring a map of the counties of England, found that he needed only four colours to avoid having adjoining counties (i.e. counties sharing a length of border, not just a point) of the same colour. Would this be the case for any map, however complicated? he asked himself, and being unable to answer the question he put it to William De Morgan, his former mathematics teacher at University College London. De Morgan immediately passed it on to his colleague Alexander Hamilton in Dublin. Neither of them could prove that four colours are always sufficient, nor disprove it by concocting a map for which they are insufficient. Later the American logician, Charles Peirce, and Arthur Cayley, Professor of Mathematics at Cambridge, tried and failed with it. In 1879 Cayley’s former student Alfred Kempe published what was thought to be a proof of the hypothesis, and became Sir Arthur Kempe FRS partly on the strength of it. But its strength was not sufficient; in 1890 P.J. Heawood, a lecturer at Durham, found a subtle mistake in Kempe’s reasoning. Much more penetrating but very laborious techniques were invented to attack the problem as an understanding of its difficulties deepened over the next eighty years. It was realized that a proof might depend on consideration of a number of special configurations of map-regions, the obstacle being that this number was dauntingly large and some of the configurations intractably extensive. Such was the state of play at the time I was occupied with Laws of Form.
In this book Spencer-Brown had claimed that his use of imaginary values would lead to the proof of certain unspecified theorems beyond the powers of a mathematics based only on real values; in fact, he wrote, ‘I may say that I believe that at least one such theorem will shortly be decided by the methods outlined in the text. In other words, I believe that I have reduced their decision to a technical problem which is well within the capacity of an ordinary mathematician who is prepared, and who has the patronage or other means, to undertake the labour.’ This theorem, he now told me, was to be the four-colour theorem. Furthermore, he had an idea that its solution was deeply connected with one of the most famous outstanding problems of the theory of numbers, Goldbach’s conjecture, that any even number bigger than 2 can be represented as the sum of two prime numbers (for example, 36 = 31 + 5) – a simple statement that has no known exceptions, having been tested for numbers up into the quadrillions, but that has not yet been proved to be true of all numbers however vast. I was intrigued by this coming dragon-hunt: would my strange acquaintance march out, armed with imaginary values, and triumph on the shadowy horizon of the known?
In the following years, having moved to Aran, I did not keep in contact with Spencer-Brown and heard no more of him. Then in 1976 came an announcement that left the mathematical world both dumbfounded and disappointed. Two mathematicians, Kenneth Appel and Wolfgang Haken of the University of Illinois, claimed to have proved the four-colour theorem, having got the number of special configurations down to about 1500 and carried out elaborate calculations on each. But as well as their own ingenuity and skill their proof had called for 1200 hours of computer time. Few seriously doubted that they were right, that the theorem was true – but could one call this a proof, if the nub of it was miles of computer print-out that no human could ever check, and, more fundamentally, if it provided no insight into why the result was true?
A few months later a friend sent me a newspaper cutting of an interview with Spencer-Brown, described as a maverick Cambridge mathematician. The announcement of Appel and Haken’s proof had spurred him on to produce a proper, readily comprehensible, proof using his own methods, which involved thinking ‘in a way that almost blows the mind’. It took him, he said, just two weeks; then he thought that if there was one proof there must be another, and proved it again by a different route. He was to present his proof in a lecture, and leave his diagrams and tapes of the lecture with London University.
It was strange to me that I never heard another word about this proof. Books I picked up now and then on contemporary mathematics retold the saga of the computer proof, but Spencer-Brown was unmentioned. Was he an exploded myth, a forgotten eccentric? However, when I began to explore the Internet in the late ’90s I discovered the existence of a Spencer-Brown cult. It was also clear that much highly professional work was being done in the field established by him, even if the accounts of some of it emanated from institutions with unreassuring names. From Jack Engstrom of the Maharishi University of Management I learned that:
Louis H. Kauffman … has applied Laws of Form to topology, natural numbers, electronic circuits, imaginary values in logic, and other areas. Francisco Varela extended Laws of Form into biology, autopoiesis, three-valued logic, and cognitive systems. William Bricken has used it for logical calculations on computers and has applied it to natural numbers. Jeffrey James has applied it to real and imaginary numbers. I have applied it to natural numbers, to set theory, to a philosophy of transcendence, and to a philosophy of wholes and parts. Nathaniel Hellerstein has applied Laws of Form to a logic of paradox. Laws of Form has also been applied to neural processing, automata, semiotics, and more.
Eventually I tracked down an Internet mailing list entitled ‘Spencer-Brown in America’, to which other searchers as puzzled as myself had posted notices. One of them read:
Dear list members; Now I wonder whether anybody knows anything about Spencer Brown. The more autopoietic systems theory I read, the more central does the concept of form seem to be, which allegedly origins from Spencer Brown. Spencer Brown himself, however, seems to be both a key-and a shadow-figure at the same time. What is Spencer Brown up to at the time being? For some time I assumed him dead. Wrongly, it seems….
An eminent Princeton mathematician, John Conway (known to followers of the theory of artificial intelligence as the inventor of ‘The Game of Life’) had replied to such queries:
In his ‘Laws of Form’ he recasts some of logic in a very elegant new way, but it can’t really be said that this removes the paradoxes from formal logic. I don’t believe his ‘proof’ of the 4-color theorem (and don’t know any other professional mathematician who does). When he first made this claim, I bet him 10 pounds that his proof wasn’t valid. At that time, it wasn’t written down, but he spent a good few hours describing it. I told him that I certainly wasn’t going to pay up without having seen a written copy of the proof, and I’m still waiting to do so!
I posted a query myself to this mailing list, calling it ‘À la recherche de Spencer-Brown’, and over a year later my query was answered by a journalist who had entertained Spencer-Brown in California. His headlong e-mails brought to my desk a slipstream of excitement off the western world’s leading-edge of innovation:
tim i saw your message about g. spencer-brown, via John conway, dated 3 jan1998, only recently on the internet; you can find much information about spencer-brown at the laws of form web pages, http://user.aol. com/lawsofform/lof.html…. perhaps you’ve found him by now. a couple of people are currently working on applications of laws of form: dick shoup of interval research in palo alto (shoup@interval.com) and william bricken (bricken@halcyon.com). shoup is developing logic systems that calculate using imaginary values, eventually he wants to build field-programmable gate arrays that can be programmed on the fly, in microseconds, so that a chip can be a graphics accelerator on one cycle and a cpu on another. he thinks this is essential for when silicon hits the wall and moore’s law works no more. shoup’s systems are written in a laws of form computer language called losp, which is a variant of lisp written by bricken…. i’m sure either he or shoup would be glad to help you out – email them and mention that i gave you their addresses, they’re great folks, (interval, you may know, is paul alien’s private think tank in palo alto.) as for the four-color theorem, you may have seen spencer-brown’s letter to nature in 1976 in which he announced that he had a solution, it was just after the publication of this letter that brown came to california for a couple of months, a riotous experience, i introduced him to some people who hired him to lecture at xerox pare on his proof, and apparently it was a disaster – no one could follow it…. i did visit your website and was fascinated by the maps. i thought your illustrations for laws of form were just perfect….
all the best, cliff barney
Gratified to learn that my humble hackwork was appreciated in this world of great folks with private think-tanks, I dashed to the ‘LoF web site’, and found a cornucopia of curious references. A Vita of our hero suggested a personality driven by the imperative of excelling in every field. For instance, in the Royal Navy he qualified as a Radio Mechanic with the highest mark of all candidates in the practical examination, and later undertook successful trials of hypnosis for dentistry and the rehabilitation of wounded personnel; at Cambridge he captained the University chess team when it beat Oxford, qualified without dual instruction for the Silver C Badge in gliding in the world-record time of 78 days, joined the Cambridge University Air Squadron and became the first ab initio member of any university squadron to qualify for Instrument Rating, led the Formation Aerobatic Team to victory in the Cambridge Squadrons Formation Flying Contest, learned racing driving with Gavin Maxwell, worked with Wittgenstein on the Foundations of Philosophy and appeared with the University Actors in a Shakespearean production. As a professional psychotherapist he used hypnosis and sleep-learning techniques to enhance performance in sporting and other competitive activities, and specialized in the education of superintelligent children. As adviser to the Federal Naval Research Laboratory in Washington, DC, he made discoveries in optics, coding and code-breaking. He has worked with Lord Cherwell on Goldbach’s conjecture and with Bertrand Russell on the Foundations of Mathematics, been Soccer Correspondent to the Daily Express and Bridge Correspondent to Parson’s Pleasure. Apart from some Visiting Professorships he has worked mainly as an engineer, and he runs his own publishing house. His recreations are as various as his professional activities; they range from shooting to Mozart, from writing and singing ballads to constructing ingenious machines and inventing games.
Laws of Form, I learned from the LoF site, had evolved from edition to edition, and in 1994:
a fourth preface was added in which he talks about ‘triunions’ or ‘triple identities’ such as of reality, appearance and awareness, or imaginability, possibility and actuality, or what a thing is, what it isn’t and the boundary between them. He claims/acknowledges that Sakyamuni (the Buddha) is ‘the only other author who evidently discovered these laws.’ He invites the reader to join a siblinghood and help found a school of his methods for intuitively feeling and naturally acting upon the consequences of there being nothing. He … asks for money and volunteers to help him found schools for superintelligent children such as he was.
