the climate between periods of benign temperature and ice ages. Climate
f lipping is one of the reasons that many scientists are concerned about
the phenomenon of greenhouse warming. The climate record contains
periods where there have been relatively abrupt transformations, and it is
not at all clear what causes the climate to seesaw from one temperature
regime to another. Those who feel that humans must take measures to
prevent global warming point to the fact that it is impossible to know
whether the relative fraction of carbon dioxide in the atmosphere is the
trigger of chaotic behavior, but until we are more knowledgeable, it seems
prudent to err on the side of caution. On the other side of the argument,
the climate seems to have had its own strange attractors for millions of
years before man began using fossil fuels as a power source, so we’re just
Johnny-come-latelies in cycles that have been going on for millions of
years without us.
There’s a lot to gain by being able to model chaotic systems. Imagine
how valuable it would be to be able to predict cardiac arrhythmia before it
actually shows up. We’ve actually taken an important step by knowing
that cardiac arrhythmia is a chaotic phenomenon rather than a random
one. If it were random, there would be no hope of doing anything about
individual cases; the best we could do is to know what percentage of peo-
ple displaying certain patterns would be liable to suffer heart attacks.
With chaotic behavior, there is the possibility that we can do things in
individual situations. This probably lies some distance in the future,
though, as chaos is a very young discipline.^9 But at least it’s not a disci-
pline characterized by extreme confusion and disorder.
NOTES- C. Sagan, Cont act (New York: Simon & Schuster, 1985).
- See http://mathworld.wolfram.com/NormalNumber.html. Like many of the ref-
erences in Mathworld, you have to be a pro to take full advantage of the informa-
tion, but the basics are reasonably comprehensible. - See http:// mathworld .wolfram .com/ AbsolutelyNormal .html.
- Borel’s normal number theorem states that the set of numbers that are not nor-
mal in every base is a set of Lebesgue measure zero. You need an upper-division
math course to be really comfortable with Lebesgue measure, but it attaches
numbers to sets that generalize the idea of length. The Lebesgue measure of the
unit interval, all real numbers between 0 and 1, is 1, as you would expect. How-
ever, the Lebesgue measure of all the rational numbers in that interval is 0. The
proof of Borel’s normal number theorem uses the axiom of choice. Probability
for sets of real numbers is closely tied up with Lebesgue measure, so when
we say that a randomly selected number is almost certain to be normal, that is
simply a restatement of Borel’s normal number theorem in the more-intuitive
The Disor ga nized Universe 183