Robert Matthews puts on his crampons for a thrilling climb in the highest maths Mathematical Mountaintops by John Casti, Oxford, £19.95, ISBN 0195141717
Reading about the exploits of mountaineers risking life and
limb to be the first up some perilous peak is one of life's great vicarious
pleasures. The thrills on offer in John Casti's account of assaults on the
five most famous peaks in mathematics are naturally rather more recherche'.
Even so, for those seeking something a bit more strenuous than the usual
trot around the foothills, Casti provides a bit of a treat. Not only does
he recount the stories of the many attempts-some doomed from the outset-on
these major mathematical challenges, but he also shows what vistas their
conquests have opened up. He gives us some inkling as to why mathematicians
try to scale such peaks, other than "because they're there.
Casti's first peak is the search for a method for showing
whether any given Diophantine equation has solutions involving only rational
numbers. What a singularly awful place to start his tour: an abstruse question
with an even more abstruse solution. By the time I'd staggered to the end
of this chapter, Casti had me feeling like a pensioner whose Alpine walking
holiday had begun with an ascent of the Eiger I won't bother to explain what
a Diophantine equation is,
except to say that the mathematicians who scrambled to the top of this peak
found that no, there isn't a general method - pretty scant reward for a hard
Fortunately, it's downhill a bit to Casti's second peak, the
proof of the famous 19th-century conjecture that four
colours suffice to colour a map so that no two
regions that share a boundary have the same colour. While the story of the
failed attempts and the ultimate
supercomputer-assisted success will be
familiar to many, Casti finds new angles on the story, such as what exactly
constitutes a "map", and philosophical aspects of proofs so computer-dependent
that no human can hope to check them. Next on Casti's
tour is the most thrilling climb of all:
Cantor's Continuum Hypothesis.
It states that there is no infinity between those of the natural numbers
(1, 2, 3 and so on) and the reals (numbers like 2/3 and
The very idea that there are varieties
of infinity is enough to bring on an attack of vertigo. Indeed, the eponymous
German mathematician seems to have succumbed to terminal altitude sickness
contemplating such possibilities. Casti is at his best when taking us through
the implication of the stunning result that the answer is yes and no.
Casti's fourth peak brings us back down to much more familiar
territory: stacking fruit. Roughly speaking,
Kepler's Conjecture states that fitting each layer of oranges into
the spaces in the layer below is the tightest way to
pack them-something all greengrocers know. Amazingly, it took almost
400 years and a big computer to prove this true. As Casti shows, still more
amazing is how a Hungarian mathematician found the tiny set of 150 footholds
up to the peak, out of the infinitude of nooks and crannies that led nowhere.
Finally, Casti leads us up
Fermat's Last Theorem (FLT).
As with Everest, guides of varying ability offer package tours up this peak
these days. Typically, Casti's route is more demanding, but offers captivating
views over landscapes usually ignored, such as the link between glamorous
FLT and dowdy old right-angle triangles. Seasoned mathematical ramblers will
also enjoy the tale of how one determined assault on FLT based on an entirely
natural-looking property of complex
numbers, worked fine for all integer powers up to 22 - then suddenly
failed. It seems even numbers can be cussed.