Is nonstandard analysis relevant for the philosophy of mathematics?

Abstract

It is a common belief of mathematics that the geometric line is a pointset which can be coordinatized by the set of real numbers, and that the reals in turn can be constructed "from below" by either Dedekind cuts, Cauchy sequences, or some similar device. Thus the creed goes: In the beginning there was the empty set, the rest is an exercise in axiomatic set theory. Nonstandard analysis suggests a different point of view. The geometric line is not a pointset, but acts as a support of pointsets which could be different from, even richer, than the set of standard reals. To elaborate a bit, take the usual axiomatization of the affine plane. There are two basic categories of objects, lines and points. Any point lies on a line, but a line is not a set of points. Axioms with a "true" geometric content allow us to introduce coordinates from a field. But we need the less elementary Archimedian axiom to tell us that the field of an ordered geometry is isomorphic to a subfield of the reals. Is the Archimedian axiom a "true" geometric fact? What is given in our immediate experience is a limited part of the geometric line with at most a finite number of points marked on it, representing, e.g., the result of some physical measurement. The rest is an extension, ideal or real. In the orthodox view the real numbers are created out of the rationals by adjoining points representing certain equivalence classes of con vergent sequences. And the claim is made that the "ideal elements" thus created fill up the line. And, as has often been argued, this has a certain plausibility seen from the point of view of recorded measure ments. But this point of view is static, we do not, e.g., pay any attention to the rate of convergence, i.e., to how we arrive at the recorded measurement. If we do this, we are led to richer pointsets on the line. And if we in addition also care about asymptotic behavior of sequences, we are almost inevitably pushed to the full notion of hyperreals or nonstandard reals. This point of view gives a certain "constructive" justification for the