FOM: Response to Torkel Franzen

Moshe' Machover moshe.machover at
Tue Nov 11 10:13:01 EST 1997

TF writes:

>  Probably from most points of view, non-standard analysis is an
>interesting development in mathematics that is not "only" technical.
>Just to mention two striking aspects, it yields a justification of
>something very like the traditional manipulation of infinitesimals,
>and it exploits, surprisingly, the limitations of first order logic to
>turn pathologies into new and useful mathematical structure.
>  So is this a foundational development? That would depend on how we
>interpret "foundational". In the sense of foundations put forward by
>Steve Simpson, it's not foundational, because it is a mathematical
>theory that comes late in the chain of explanation and justification
>in mathematics. The basics of non-standard analysis use a lot of stuff
>about the real numbers and some logical theory. (Nor is there in
>non-standard analysis any uniquely characterized extended domain of
>numbers.) We don't set out to explain the real numbers to anybody in
>terms of non-standard analysis, and it's far from obvious that this
>would be possible.

Perhaps. And it is certainly true that NSA is applied model theory *par
excellence*. Robinson actually insisted on this: that was his style of
model theory. (And remember: he was also a first-rate applied
mathematician, an expert on the aerodynamics of supersonic flight.)

*But* NSA can be said to explain notions such as continuity and limit,
which were once thought to be foundational, and perhaps still are. And it
is unequalled in the light it throws on the notion of compactness. My
feeling is that once you have shown these and similar notions worked out in
NSA, you will see them in a completely new light. So I am not sure that TF
is right in claiming:

> But non-standard analysis as it now exists is not a foundational subject.

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