amusing a story he wrote in which he described the logical consequences
surrounding the discovery of thiotimoline, a substance that dissolved in
water 1.2 seconds before the water was added.
Asimov’s best-known major work of science fiction is the Foundation
trilogy.^5 In it, a future interstellar empire is collapsing, a collapse foreseen
by Hari Selden, a mathematical sociologist who uses statistics to predict
the future history of the empire. Selden’s computations are upset by the
arrival of the Mule, a mutant whose special psychic abilities enable him
to seize and hold power. Others, such as Georg Hegel, had emphasized
that history is shaped by specific individuals, whom he called “world-his-
torical individuals.” Asimov’s contribution was to note that the propensity
of civilizations to produce such individuals may invalidate any hope of
applying techniques, such as those used in statistical mechanics, to his-
tory. After all, no single air molecule has the capability to alter the behav-
ior of large quantities of other air molecules the way a single individual
can alter the course of history.
Does this mean it is impossible to come up with a mathematical scheme
for assessing or predicting history? It’s an interesting question, and I am
unaware of any Hari Seldens, past or present, who have attempted any
sort of scheme with any success. Possibly, when the mathematics of chaos
is sufficiently well developed, it may be possible to set some sort of limits
on what might be accomplished in this area.
In the 1960s, the French mathematician René Thom inaugurated a
branch of mathematics now called catastrophe theory.^6 This was an at-
tempt to analyze dramatic changes in behavior of phenomena arising
from small changes in the parameters describing the phenomena. Sound
familiar? It certainly has a good deal of the f lavor of chaos theory. Addi-
tionally, catastrophe theory looks at nonlinear phenomena, as does much
of chaos theory. A major difference, however, is that catastrophe theory
views dramatic shifts in behavior of the underlying parameters as mani-
festations of standardized geometrical behavior in a larger parameter
space. From a practical standpoint in actually predicting impending ca-
tastrophes, this isn’t much help. It might be nice to know that the next
stock market collapse is simply the expected behavior of a well-defined
geometrical structure in a higher-dimensional space, but unless we
can get a handle on exactly what the parameters governing that higher-
dimensional space measure, and do so in some a priori manner, it’s not
very useful.
There are, of course, numerous applications of mathematics in the so-
cial sciences; courses in these applications are offered at almost every
institution above the secondary level. Some have proved remarkably
Through a Glass Darkly 243