My Time in Space (19 page)

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:

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:

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:

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:

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:

The consequences of there being nothing! This took me back to a passage from which I had averted my attention in
Laws
of
Form:

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’: