Re: A paradox? Why not?

[John Denker <jsd@MONMOUTH.COM>]

At 04:41 PM 7/15/99 +0100, Ludwik Kowalski wrote:
> Paradoxes are worth fishing for; they are great in promoting
> critical thinking

The subject line seems to ask whether it is a paradox.  I'd say yes.  It
even has a name;  in the statistics literature it is known as Richardson's
Paradox.

IMHO it is high on the list of good paradoxes.  I rank it below the truly
historic paradoxes (e.g. the ultraviolet catastrophe) but well above most
of the hokey "paradoxes" we see in relativity texts.

Searching for paradoxes and counterexamples is good sport, but it requires
judgement.  I am reminded of the preface to the book "Counterexamples in
Topology" which explains that counterexamples are good when they disprove
an *interesting* or *important* conjecture.

Richardson's paradox is interesting and important because
 1) many people are surprised by the result
 2) it and its corollaries disprove several dangerous notions e.g.:
    a) "the" observed probability of past events is a good predictor of
"the" probability of future events, and
    b) there is a well-defined basis for saying what "the" probability is.

In my experience, physicists are quite commonly taken in by fallacy (2b) --
the idea that there is only one possible probability measure and the first
and/or simplest probability measure we come across must be "the" one that
describes the world.  This is anagous to fallaciously assuming that
Euclidean geometry must be "the" geometry of the real world.  I recommend
that we adopt the mathematicians' point of view, namely that there are
innmerably many probability measures (just as there are innumerably many
non-Euclidean geometries) and it is a task for observation (not just for
axioms or assumptions or omphaloskepsis) to decide which if any of these
describes the real world.

As for fallacy (2a)... Richardson's Paradox is a graphic demonstration of
why a reliable statistical analysis requires things like random sampling.
It also shows the difference between common sense and the equally common
nonsense.  Extrapolation of historical data, without randomization or other
controls, is an example of the latter.

Cheers --- jsd