[FOM] Infinitesimals

[Alexander M Lemberg]

It is true that NSA has provided a framework for efficient and intuitive
proofs of certain results in analysis. However, I don't regard the system
itself, burdened as it is with its galaxies and so on as compact or
efficient. 

More significantly, does it reflect the original heuristic motivations
for infinitesimals and origins of calculus? I believe that  the theory of
 "smooth analysis" does so to a far greater extent. 

Calculus and infinitesimals were developed to address our apprehensions
of space, time and motion. At small distances, in the limit, it is in
principle impossible to say whether a quantity is zero or nonzero
infinitesimal (or even a small finite quatnity). This state of affairs is
captured by the intuitionistic framework of smooth analysis.

The Robinson theory was a historic breakthrough in that it demonstrated
possibility of providing a rigorous framework for infinitesimal arguments
within ZFC. But I believe that the rival  the theory of smooth analysis
goes more deeply into the heart of the matter.

I would be interested in what others think about this.

Sandy

On Mon, 28 Jul 2003 10:35:10 -0400 Alasdair Urquhart
<urquhart at cs.toronto.edu> writes:

"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."