probability theory.pdf (17).pdf

c15b, 10/3/94
CHAPTER 15
P ARADO XES OF PR OBABILITY THEOR Y
\I protest against the use of innite magnitude as something plished, whic h
is nev er p ermissible in mathematics. Innit y is merely a gure of sp eec h, the
true meaning b eing a limit."
| C. F. Gauss
The term \parado x" app ears to ha v e sev eral dieren mon meanings. Sz  ek ely (1986)
denes a parado xasan ything whic h is true but surprising. By that denition, ev ery scien tic fact
and ev ery mathematical theorem qualies as a parado x for someone. W e use the term in almost
the opp osite sense; something whic h is absurd or logically con tradictory , but whic h app ears at rst
glance to b e the result of sound reasoning. Not only in probabilit y theory , but in all mathematics,
it is the careless use of innite sets, and of innite and innitesimal quan tities, that generates most
parado xes.
In our usage, there is no sharp distinction b et w een a parado x and an error. A parado xis
simply an error out of con trol; . one that has trapp ed so man yun w ary minds that it has gone
public, b ecome institutionalized in our literature, and taugh t as truth. It migh t seem incredible
that suc h a thing could happ en in an ostensibly mathematical eld; y et w e can understand the
psyc hological mec hanism b ehind it.
Ho wdoP arado xes Surviv e and Gro w?
As w e stress rep eatedly , from a false prop osition { or from a fallacious argumen t that leads to a
false prop osition { all prop ositions, true and false, ma y b e deduced. But this is just the danger; if
fallacious reasoning alw a ys led to absurd conclusions, it w ould b e found out at once and corrected.
But once an easy , short{cut mo de of reasoning has led to a few correct results, almost ev eryb o dy
accepts it; those who try to w arn against it are not listened to.
When a fallacy reac hes this stage it tak es on a life of its o wn, and dev elops v ery eectiv e
defenses for self{preser