[FOM] [FoM} Soft and hard routes to infinitesimals

[Mikhail Katz]

John Baez posted a note on infinitesimals at a physics blog here:

https://www.physicsforums.com/insights/struggles-continuum-part-1/

What I have to say is related more to FoM than physics so I decided to
post this to FoM with a copy to Baez.

Baez seems to favor the Smooth Infinitesimal Analysis (SIA) approach
to infinitesimals based on category theory.  That depends on one's
taste, I suppose.  One can either favor the "hard" route passing via a
reconsideration of foundations, or the "soft" route passing via more
traditional frameworks like the ZFC.

If one chooses the "soft" route, then there is no doubt that
Robinson's approach is preferable, since it takes place in ZFC in a
conservative fashion (in both senses) and requires no foundational
revolutions.

If one chooses the "hard" route, then certainly category theory is an
option for replacing the reliance on ZFC.  However, I would suggest
(borrowing Terry Tao's terminology) that a "cheaper" hard approach is
possible, and apparently at least as successful, as SIA.  Namely, one
can adopt the viewpoint that infinitesimals are already found within
the ordinary real line itself.  This is the approach of Edward Nelson
in his Internal Set Theory (IST).  Here non-associativity of
parentheses is significant, and the term should be parsed Internal
(Set Theory).  Namely, this is not a theory of internal sets, but
rather a new type of set theory called "internal".

I call this a "hard" approach because it also involves a foundational
revolution of sorts, but arguably a more easily digestible one.  What
this means is that ZFC is enriched by the introduction of a one-place
predicate called "standard".  Positive infinitesimals are then
positive numbers smaller than every positive standard real.  This
re-casting of Robinson's framework involves a foundational revolution
because the ZFC foundation is replaced by the IST foundation which is
a different axiomatisation.

On the other hand IST is a conservative extension of ZFC and merely
amounts to adding three additional axioms, arguably an easier lunch to
swallow than category theory.  The main advantage here is that one can
continue working in the ordinary real line, rather than any extension
thereof (as in Robinson's framework), and should be appealing to both
mathematicians and physicists (including Baez), if they are willing to
take the "hard" route.  As I mentioned above, those who prefer the
"soft" route will certainly choose Robinson's framework for
infinitesimals.