- Lukacs, John.
Five Days in London.
New Haven, CT: Yale University Press, 1999.
ISBN 0-300-08466-8.
-
Winston Churchill titled the fourth volume of his
memoirs of The Second World War,
describing the events of 1942,
The Hinge of Fate.
Certainly, in the military sense, it was in that year
that the tide turned in favour of the allies—the
entry of the United States into the war and the Japanese
defeat in the Battle of Midway, Germany's failure at Stalingrad
and the beginning of the disastrous consequences for the
German army, and British defeat of Rommel's army at El Alamein
together marked what Churchill described as, “…not the
end, nor is it even the beginning of the end, but, it is perhaps, the
end of the beginning.”
But in this book, distinguished historian John Lukacs argues
that the true “hinge of fate” not only of World War
II, but for Western civilisation against Nazi tyranny, occurred
in the five days of 24–28 May of 1940, not on the
battlefields in France, but in London, around conference
tables, in lunch and dinner meetings, and walks in the garden.
This was a period of unmitigated, accelerating disaster for
the French army and the British Expeditionary Force in
France: the channel ports of Boulogne and Calais fell to the
Germans, the King of Belgium capitulated to the Nazis, and
more than three hundred thousand British and French troops
were surrounded at Dunkirk, the last channel port still in
Allied hands. Despite plans for an evacuation, as late as
May 28, Churchill estimated that at most about 50,000 could
be evacuated, with all the rest taken prisoner and all the
military equipment lost. In his statement in the House of
Commons that day, he said, “Meanwhile, the House
should prepare itself for hard and heavy tidings.”
It was only in the subsequent days that the near-miraculous evacuation
was accomplished, with a total of 338,226 soldiers rescued by June
3rd.
And yet it was in these darkest of days that Churchill vowed that
Britain would fight on, alone if necessary (which seemed increasingly
probable), to the very end, whatever the cost or consequences. On May
31st, he told French premier Paul Reynaud, “It would be better
far that the civilisation of Western Europe with all of its
achievements should come to a tragic but splendid end than that the
two great democracies should linger on, stripped of all that made life
worth living.” (p. 217).
From Churchill's memoirs and those of other senior British
officials, contemporary newspapers, and most historical
accounts of the period, one gains the impression of a
Britain unified in grim resolve behind Churchill to fight on
until ultimate victory or annihilation. But what actually
happened in those crucial War Cabinet meetings as the disaster
in France was unfolding? Oddly, the memoirs and collected
papers of the participants are nearly silent on the period,
with the author describing the latter as having been
“weeded” after the fact. It was not until the
minutes of the crucial cabinet meetings were declassified
in 1970 (thanks to a decision by the British government to reduce
the “closed period” of such records from fifty
to thirty years), that it became possible to reconstruct what
transpired there. This book recounts a dramatic
and fateful struggle of which the public and earlier historians
of the period were completely unaware—a moment when
Hitler may have come closer to winning the war than at any
other.
The War Cabinet was, in fact, deeply divided. Churchill, who had only
been Prime Minister for two weeks, was in a precarious position, with
his predecessor Neville Chamberlain and the Foreign Secretary Lord
Halifax, who King George VI had preferred to Churchill for Prime
Minister as members, along with Labour leaders Clement Attlee and
Arthur Greenwood. Halifax did not believe that Britain could resist
alone, and that fighting on would surely result in the loss of the
Empire and perhaps independence and liberty in Britain as well. He
argued vehemently for an approach, either by Britain and France
together or Britain alone, to Mussolini, with the goal of keeping
Italy out of the war and making some kind of deal with Hitler which
would preserve independence and the Empire, and he met on several
occasions with the Italian ambassador in London to explore such
possibilities.
Churchill opposed any effort to seek mediation, either by
Mussolini or Roosevelt, both because he thought the chances
of obtaining acceptable terms from Hitler were
“a thousand to one against” (May 28, p. 183)
and because any approach would put Britain on a
“slippery slope” (Churchill's words in the same
meeting) from which it would be impossible to restore the
resolution to fight rather than make catastrophic concessions.
But this was a pragmatic decision, not a Churchillian declaration of
“never,
never, never, never”. In the May 26 War Cabinet
meeting (p. 113), Churchill made the rather astonishing
statement that he “would be thankful to get out of our
present difficulties on such terms, provided we retained
the essentials and the elements of our vital strength,
even at the cost of some territory”.
One can understand why the personal papers of the
principals were so carefully weeded.
Speaking of another conflict where the destiny of Europe hung in the
balance, the Duke of Wellington said of Waterloo that it was
“the nearest run thing you ever saw in your life”.
This account makes it clear that this moment in history
was much the same. It is, of course, impossible to forecast
what the consequences would have been had Halifax prevailed
and Britain approached Mussolini to broker a deal with Hitler.
The author argues forcefully that nothing less than
the fate of Western civilisation was at stake. With
so many “what ifs”, one can never know. (For
example, it appears that Mussolini had already decided by
this date to enter the war and he might have simply rejected a
British approach.) But in any case this fascinating, thoroughly
documented, and lucidly written account of a little-known but
crucial moment in history makes for compelling reading.
- Roberts, Siobhan.
King of Infinite Space.
New York: Walker and Company, 2006.
ISBN 0-8027-1499-4.
-
Mathematics is often said to be a game for the young. The
Fields
Medal, the most prestigious prize in mathematics, is restricted
to candidates 40 years or younger. While many older mathematicians
continue to make important contributions in writing books,
teaching, administration, and organising and systematising
topics, most work on the cutting edge is done by those in
their twenties and thirties. The life and career of
Donald Coxeter (1907–2003), the subject of
this superb biography, is a stunning and inspiring counter-example.
Coxeter's publications (all of which are
listed in an appendix to this book) span a period of eighty
years, with the last, a novel proof of
Beecroft's
theorem, completed just a few days before his death.
Coxeter was one of the last generation to be trained in
classical geometry, and he continued to do original work and
make striking discoveries in that field for decades after
most other mathematicians had abandoned it as mined out
or insufficiently rigorous, and it had disappeared from the
curriculum not only at the university level but, to a
great extent, in secondary schools as well. Coxeter worked
in an intuitive, visual style, frequently making models,
kaleidoscopes, and enriching his publications with numerous
diagrams. Over the many decades his career spanned, mathematical
research (at least in the West) seemed to be climbing an endless
stairway toward ever greater abstraction and formalism,
epitomised in the work of the
Bourbaki group.
(When the unthinkable happened and a diagram
was included in a
Bourbaki book,
fittingly it was a
Coxeter
diagram.)
Coxeter inspired an increasingly fervent group of followers
who preferred to discover new structures and symmetry using
the mind's powers of visualisation. Some, including Douglas Hofstadter
(who contributed the foreword to this work) and John Horton
Conway (who figures prominently in the text) were inspired
by Coxeter to carry on his legacy. Coxeter's interactions with
M. C. Escher
and
Buckminster Fuller
are explored in two chapters,
and illustrate how the purest of mathematics can both inspire and
be enriched by art and architecture (or whatever it was that Fuller
did, which Coxeter himself wasn't too sure about—on one occasion
he walked out of a new-agey Fuller lecture, noting in his diary
“Out, disgusted, after ¾ hour” [p. 178]).
When the “new math” craze took hold in the 1960s, Coxeter
immediately saw it for the disaster it was to be become and involved
himself in efforts to preserve the intuitive and visual in mathematics
education. Unfortunately, the power of a fad promoted by purists is
difficult to counter, and a generation and more paid the price of
which Coxeter warned. There is an excellent discussion at the end of
chapter 9 of the interplay between the intuitive and formalist
approaches to mathematics. Many modern mathematicians seem to have
forgotten that one proves theorems in order to demonstrate that the
insights obtained by intuition are correct. Intuition without rigour
can lead to error, but rigour without intuition can blind one to
beautiful discoveries in the mathematical objects which stand behind
the austere symbols on paper.
The main text of this 400 page book is only 257 pages.
Eight appendices expand upon technical topics ranging
from phyllotaxis to the quilting of toilet paper and
include a complete bibliography of Coxeter's publications.
(If you're intrigued by “Morley's Miracle”,
a novel discovery in the plane geometry of triangles
made as late as 1899, check out this
page
and Java applet which lets you play with it interactively.
Curiously, a diagram of Morley's theorem appears on the
cover of Coxeter's and Greitzer's
Geometry Revisited, but
is misdrawn—the trisectors are inexact and the
inner triangle is therefore not equilateral.)
Almost 90 pages of endnotes provide both source citations
(including Web links to
MathWorld for
technical terms and
the
University
of St. Andrews biographical archive for
mathematicians named in the text) and detailed
amplification of numerous details. There are a few typos and
factual errors (for example, on p. 101 the planets
Uranus and Pluto are said to have been discovered in
the nineteenth century when, in fact, neither was: Herschel
discovered Uranus in 1781 and Tombaugh Pluto in 1930), but none
are central to the topic nor detract from this rewarding
biography of an admirable and important mathematician.
- Kauffman, Stuart A.
Investigations.
New York: Oxford University Press, 2000.
ISBN 0-19-512105-8.
-
Few people have thought as long and as hard about the origin
of life and the emergence of complexity in a biosphere as
Stuart Kauffman. Medical doctor, geneticist, professor of
biochemistry and biophysics, MacArthur Fellow, and member of
the faculty of the Santa Fe Institute for a decade, he has
sought to discover the principles which might underlie
a “general biology”—the laws which
would govern any biosphere, whether terrestrial, extraterrestrial, or
simulated within a computer, regardless of its physical
substrate.
This book, which he describes on occasion as “protoscience”,
provides an overview of the principles he suspects, but cannot
prove, may underlie all forms of life, and beyond that systems
in general which are far from equilibrium such as a modern
technological economy and the universe itself. Most of science
before the middle of the twentieth century studied complex
systems at or near equilibrium; only at such states could the
simplifying assumptions of statistical mechanics be applied to
render the problem tractable. With computers, however, we can now
begin to explore open systems (albeit far smaller than those in nature)
which are far from equilibrium, have dynamic flows of energy and
material, and do not necessarily evolve toward a state of maximum
entropy.
Kauffman believes there may be what amounts to a fourth law of
thermodynamics which applies to such systems and, although we don't
know enough to state it precisely, he suspects it may be that these
open, extremely nonergodic, systems evolve as rapidly as possible to
expand and fill their state space and that unlike, say, a gas in a
closed volume or the stars in a galaxy, where the complete state space
can be specified in advance (that is, the dimensionality of the space,
not the precise position and momentum values of every object within
it), the state space of a non-equilibrium system cannot be prestated
because its very evolution expands the state space. The presence of
autonomous agents introduces another level of complexity and
creativity, as evolution drives the agents to greater and greater
diversity and complexity to better adapt to the ever-shifting fitness
landscape.
These are complicated and deep issues, and this is a very difficult
book, although appearing, at first glance, to be written for a popular
audience. I seriously doubt whether somebody who was not previously
acquainted with these topics and thought about them at some length
will make it to the end and, even if they do, take much away from the
book. Those who are comfortable with the laws of thermodynamics,
the genetic code, protein chemistry, catalysis, autocatalytic
networks, Carnot cycles, fitness landscapes, hill-climbing strategies,
the no-go theorem, error catastrophes, self-organisation, percolation
phase transitions in graphs, and other technical issues raised in the
arguments must still confront the author's prose style. It seems
like Kauffman aspires to be a prose stylist conveying a sense of
wonder to his readers along the lines of Carl Sagan and
Stephen Jay Gould. Unfortunately, he doesn't pull it off as well,
and the reader must wade through numerous paragraphs like the following
from pp. 97–98:
Does it always take work to construct constraints? No, as we will soon
see. Does it often take work to construct constraints? Yes. In
those cases, the work done to construct constraints is, in fact,
another coupling of spontaneous and nonspontaneous processes. But
this is just what we are suggesting must occur in autonomous
agents. In the universe as a whole, exploding from the big bang into
this vast diversity, are many of the constraints on the release
of energy that have formed due to a linking of spontaneous and
nonspontaneous processes? Yes. What might this be about? I'll say
it again. The universe is full of sources of energy. Nonequilibrium
processes and structures of increasing diversity and complexity arise
that constitute sources of energy that measure, detect, and capture those
sources of energy, build new structures that constitute constraints on
the release of energy, and hence drive nonspontaneous processes to
create more such diversifying and novel processes, structures, and
energy sources.
I have not cherry-picked this passage; there are hundreds of others
like it. Given the complexity of the technical material and the
difficulty of the concepts being explained, it seems to me that the
straightforward, unaffected Point A to Point B style of explanation
which Isaac Asimov employed would work much better. Pardon my
audacity, but allow me to rewrite the above paragraph.
Autonomous agents require energy, and the universe is full
of sources of energy. But in order to do work, they require
energy to be released under constraints. Some constraints
are natural, but others are constructed by autonomous agents
which must do work to build novel constraints. A new constraint,
once built, provides access to new sources of energy, which
can be exploited by new agents, contributing to an ever
growing diversity and complexity of agents, constraints, and
sources of energy.
Which is better? I rewrite; you decide. The tone of the prose is
all over the place. In one paragraph he's talking about
Tomasina the trilobite (p. 129) and Gertrude the ugly squirrel
(p. 131), then the next thing you know it's “Here, the
hexamer is simplified to 3'CCCGGG5', and the two complementary
trimers are 5'GGG3' + 5'CCC3'. Left to its own devices,
this reaction is exergonic and, in the presence of excess trimers
compared to the equilibrium ratio of hexamer to trimers, will flow
exergonically toward equilibrium by synthesizing the hexamer.”
(p. 64). This flipping back and forth between colloquial
and scholarly voices leads to a kind of comprehensional
kinetosis. There are a few typographical errors, none serious, but
I have to share this delightful one-sentence paragraph from
p. 254 (ellipsis in the original):
By iteration, we can construct a graph connecting the founder
spin network with its 1-Pachner move “descendants,”
2-Pachner move descendints…N-Pachner move
descendents.
Good grief—is Oxford University Press outsourcing their
copy editing to Slashdot?
For the reasons given above, I found this a difficult read. But it is an
important book, bristling with ideas which will get you looking at the
big questions in a different way, and speculating, along with the
author, that there may be some profound scientific insights which
science has overlooked to date sitting right before our eyes—in
the biosphere, the economy, and this fantastically complicated
universe which seems to have emerged somehow from a near-thermalised
big bang. While Kauffman is the first to admit that these are
hypotheses and speculations, not science, they are eminently testable
by straightforward scientific investigation, and there is every reason
to believe that if there are, indeed, general laws that govern these
phenomena, we will begin to glimpse them in the next few decades. If
you're interested in these matters, this is a book you shouldn't miss,
but be aware what you're getting into when you undertake to read it.