Call for Registration: 31st Novembertagung on the History and Philosophy of Mathematics

[Mikhail Katz]

As I recall the comment that "the axiomatic view on mathematics may be
harmful in that it omits fundamental aspects of mathematical practice
and idealizes mathematical reasoning in an unfaithful way" turned out
to be controversial at FoM and led to a flurry of postings.  I would
like to point out one sense in which the claim that "the axiomatic
view idealizes mathematical reasoning in an unfaithful way" is
arguably correct.  If one takes the "axiomatic view" to refer to ZF,
an "mathematical reasoning" to refer to mathematical analysis, then
the theory ZF is arguably an unfaithful idealisation in the sense that
it omits an aspect of mathematical analysis that was considered its
central feature, namely the infinitesimal, or more precisely the
distinction that goes back to Leibniz between assignable and
inassignable quantities.  This aspect of classical analysis is
formalized more successfully in a theory SPOT, an axiomatic approach
to analysis (conservative over ZF) based on a pair of relations "st"
and "\in" rather than a single relation "\in".  The single-place
standardness predicate "st" is a formalisation of the Leibnizian
assignable/inassignable distinction.  For a more detailed introduction
see https://u.math.biu.ac.il/~katzmik/spot.html

M. Katz