[FOM] Infinitesimals

[Alasdair Urquhart]

A short comment on Ayan Mahalanobis's posting.  

Alain Connes has objected in
print to the non-constructive character of
Robinsonian infinitesimals, and has advocated
instead his own non-commutative geometry,
where explicitly definable infinitesimals
are available.

These kind of objections seem to me a red
herring.  There is no reason to think that
there should be a unique theory of infinitesimals.
We already have two quite distinct theories of
infinitesimals in logic, namely the original
Robinsonian theory, and the theory of smooth
analysis, that has recently been given a nice
introductory exposition by John Bell.  
The informal practice of the 17th and 18th centuries
provides support for both explications.

Robinsonian infinitesimals provide a very compact
and efficient notation for talking about limit
phenomena, and are increasingly used for their
heuristic power.  Non-constructive methods are
ubiquitous in modern mathematics.  Why should
the theory of infinitesimals be different?