[FOM] pathology
Tom Dunion
tom.dunion at gmail.com
Tue Aug 25 13:05:39 EDT 2015
Martin Davis said:
>It is certainly worthwhile to study mathematical entities whose existence
can be proved but for which it can be proved that explicit examples (using
acceptable mathematical language) can not be given. But I don't like using
a pejorative term like "pathology" to describe them. We don't know what
role such things will play in future developments, and calling them
pathological simply validates what may turn out to be simply prejudice of
our day.<
Agreed. One might wish that more end-users of f.o.m. themes (such as
theoreticians of physics like J. Bub and I. Pitowsky) would point out the
desirability of thinking outside the box of current regularity properties.
The classic example of such is non-Lebesgue measurable sets, which perhaps
are best thought of as sets which don't "play nice" with all other sets (as
in the Caratheodory characterization of measurability for example). In
this sense, non-measurable sets can be somewhat analogous to elements
outside the center of a non-abelian group.
Sure, it's good to know what contexts give lots of such regularity
properties -- Determinacy, supercompacts, L(R) and so forth -- but who can
tell at this point if future progress in quantum physics for example will
incorporate an expanded concept of probability beyond today's tidy minded
axiomatization in the tradition of Borel and Kolmogorov, and under what
set-theoretic assumptions.
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