FOM: rigidity

[Randall Holmes]

Dear FOMers,

Friedman objects to infinitesimals on the grounds that one cannot
give an example.  

Note that it is impossible to give an example of a point in classical
Euclidean geometry.  All the points are the same.  Geometry (as
originally understood) is not rigid in the way that arithmetic is, and
it is not self-evident that this is a disadvantage.  (Of course, this
fact is disguised by the way we now implement geometry using R^2 or
R^n).

To take an example from my own experience, if one works in NFU
(Quine's NF as modified by Jensen to allow urelements) plus the axiom
of choice one can prove that there are urelements, but one cannot
produce one -- because they are indistinguishable.  It is important
that there are urelements, but there is no particular important
urelement.  The indiscernibility of the urelements is not a practical
disadvantage for NFU as a theory of sets, at any rate.

Infinitesimals can be very useful without our having or even wanting
an example of an infinitesimal.  Similarly, the fact that the reals
can be well-ordered (which, to be used, requires occasional appeal to
an "arbitrary" well-ordering of the reals) can be very useful without
anyone being especially interested in producing an example (which is
fortunate, since we can't produce one -- unless we are willing to
assume V=L, of course, in which case we can give an example of an
infinitesimal in a nonstandard model of the reals as well).

                                  Sincerely,
                                  Randall Holmes