FOM:Geometry & weird arithmetics

[Robert Tragesser]

        One standard view of PRA (or in any case finitist mathematics) is
that it be fundamental for nature/physics (Hilbert;  Simpson in Partial
Realizations of HP  especially).  In this connections notice "non-standard"
"finitary mathematics"-- 
        The writings of the Danish geometer J.Hjelmslev circa 1919 [
pointed out to me by Kreisel] develop a purely phenomenal geometry,   e.g.,
 straight lines are tangent to circlse for a piece,  two mutually external
but osculating but noncoinciding circles wiil have an arc in common -- so
that, e.g.,  the triangle inequality breaks down.
        This suggests to Hjelmslev a kind of short,  approximation
arithmetic.
        It is worth mentioning Rashevskii's classic little paper "The dogma
of the natural numbers" in which he proposes that a natural number series
that decays as it lengthens may be best for the foundations of the
mathematics of quantum mechanics.
        It is also worth mentioning that Brouwer's natural numbers are very
very strange -- he has them straddle the discrete and the con tinuous (and
contrary to Heyting,  he maintains this to the end,  I think). 
Unfortunately,  this is obscured by post-Brouwer intuitionists.
        As Hjelmslev already noticed,  it is very likely that in all of
these ideas -- and I think even in Brouwer if his "intuition" is taken very
seriously -- lead to a break-down of logics whose structure are directly
codable by standard natural numbers.  That is,  e.g.,  as logical
implications lengthen,  their reliability quickly decays.  This makes these
arithmetics and geometries incaspable (perhaps) of deductive science.  Such
arithmewtics and geometries would,  that is,  have a definite empirical
content,  perhaps with moments of free ad hoc decisions having to be made. 
(I haven't been able to see any way around this. . .)

Robert Tragesser
West(Running)Brook, Connecticut