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

[Monroe Eskew]

Of course, nonstandard analysis is perfectly compatible with ZF, and it was through set-theoretic methods that Robinson gave nonstandard analysis a firm foundation and ignited a renaissance of infinitesimal reasoning.  Are you suggesting that it is preferable to leave infinitesimals as unformalized or as logically primitive?

Best,
Monroe

> On Nov 10, 2020, at 1:29 AM, Mikhail Katz <katzmik at macs.biu.ac.il> wrote:
> 
> 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
> 
> 
>