### Why not use another theory
containing infinitesimals?

There are two main trends of nonstandard analysis in
which some form of infinitesimal quantities are defined: the hyperreal
number system (established by A. Robinson and formalised also by K.D.
Stroyan and W.A.J. Luxemburg) in which the real numbers are extended to
contain infinitesimals which are not real numbers and IST (established by
E. Nelson) an axiomatic extension in which it is the language that is
extended to enable defining infinitely small real numbers.

Both approaches are interesting in their own right and
share a common principle: there are standard numbers and nonstandard
numbers. Both approaches have a difficulty with the derivative: there is a
definition for the derivative of a standard function at a standard value
and that formula can be made very intuitive. But to extend the definition
to nonstandard values and nonstandard functions, the level of technicality
required to do it rigorously seems to defeat the purpose.

If the derivative is not easily defined at nonstandard
points, the derivative function becomes problematic. If nonstandard
functions are not easily defined, the tangent function becomes
problematic.

The distinction between internal and external properties
is crucial in both approaches but it is technically sophisticated and requires a good understanding of mathematical logic.

One of the authors of the book faced these problems when
a first attempt at teaching using infinitesimals was used in 2000, and in
2002 the meeting with K. Hrbacek lead to the present approach. It is an
axiomatic extension similar to Nelson’s IST but not limited to two levels.
The words used (ultrasmall and ultralarge) avoid confusion with other
similar but not identical concepts.

In our theory (analysis with ultrasmall numbers), the definition of the derivative is
valid for all numbers and properties that satisfy a specific syntactic
rule are automatically internal. External properties are altogether
avoided. It can be made intuitively acceptable to students and there is a certain number of teachers who have been using this approach in two
Geneva High Schools for the last 15 years. The integration of these
students in university courses (or polytechnical school EPFL) has been
good.

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