From Nautil.us, August 22, 2013:
If two
statisticians were to lose each other in an infinite forest, the first
thing they would do is get drunk. That way, they would walk more or less
randomly, which would give them the best chance of finding each other.
However, the statisticians should stay sober if they want to pick
mushrooms. Stumbling around drunk and without purpose would reduce the
area of exploration, and make it more likely that the seekers would
return to the same spot, where the mushrooms are already gone.
Such considerations belong to the
statistical theory of “random walk” or “drunkard’s walk,” in which the
future depends only on the present and not the past. Today, random walk
is used to model share prices, molecular diffusion, neural activity, and
population dynamics, among other processes. It is also thought to
describe how “genetic drift” can result in a particular gene—say, for
blue eye color—becoming prevalent in a population. Ironically, this
theory, which ignores the past, has a rather rich history of its own. It
is one of the many intellectual innovations dreamed up by Andrei
Kolmogorov, a mathematician of startling breadth and ability who
revolutionized the role of the unlikely in mathematics, while carefully
negotiating the shifting probabilities of political and academic life in
Soviet Russia.
As a
young man, Kolmogorov was nourished by the intellectual ferment of
post-revolutionary Moscow, where literary experimentation, the artistic
avant-garde, and radical new scientific ideas were in the air. In the
early 1920s, as a 17-year-old history student, he presented a paper to a
group of his peers at Moscow University, offering an unconventional
statistical analysis of the lives of medieval Russians. It found, for
example, that the tax levied on villages was usually a whole number,
while taxes on individual households were often expressed as fractions.
The paper concluded, controversially for the time, that taxes were
imposed on whole villages and then split among the households, rather
than imposed on households and accumulated by village. “You have found
only one proof,” was his professor’s acid observation. “That is not
enough for a historian. You need at least five proofs.” At that moment,
Kolmogorov decided to change his concentration to mathematics, where one
proof would suffice.
It
is oddly appropriate that a chance event drove Kolmogorov into the arms
of probability theory, which at the time was a maligned sub-discipline
of mathematics. Pre-modern societies often viewed chance as an
expression of the gods’ will; in ancient Egypt and classical Greece,
throwing dice was seen as a reliable method of divination and fortune
telling. By the early 19th century, European mathematicians had
developed techniques for calculating odds, and distilled probability to
the ratio of the number of favorable cases to the number of all equally
probable cases. But this approach suffered from circularity—probability
was defined in terms of equally probable cases—and only worked for
systems with a finite number of possible outcomes. It could not handle
countable infinity (such as a game of dice with infinitely many faces)
or a continuum (such as a game with a spherical die, where each point on
the sphere represents a possible outcome). Attempts to grapple with
such situations produced contradictory results, and earned probability a
bad reputation.
Reputation and renown were qualities
that Kolmogorov prized. After switching his major, Kolmogorov was
initially drawn into the devoted mathematical circle surrounding Nikolai
Luzin, a charismatic teacher at Moscow University. Luzin’s disciples
nicknamed the group “Luzitania,” a pun on their professor’s name and the
famous British ship that had sunk in the First World War. They were
united by a “joint beating of hearts,” as Kolmogorov described it,
gathering after class to exalt or eviscerate new mathematical
innovations. They mocked partial differential equations as “partial
irreverential equations” and finite differences as “fine night
differences.” The theory of probability, lacking solid theoretical
foundations and burdened with paradoxes, was jokingly called the “theory
of misfortune.”...MORE
HT for the reminder we had a Nautil.us story primed, The Big Picture
