Criticism of Non-standard Analysis - Halmos' Remarks

Halmos' Remarks

Paul Halmos writes in "Invariant subspaces", American Mathematical Monthly 85 ('78) 182–183 as follows:

"the extension to polynomially compact operators was obtained by Bernstein and Robinson (1966). They presented their result in the metamathematical language called non-standard analysis, but, as it was realized very soon, that was a matter of personal preference, not necessity."

Halmos writes in (Halmos '85) as follows (p. 204):

The Bernstein–Robinson proof uses non-standard models of higher order predicate languages, and when sent me his reprint I really had to sweat to pinpoint and translate its mathematical insight.

While commenting on the "role of non-standard analysis in mathematics", Halmos writes (p. 204):

For some other, who are against it (for instance Errett Bishop), it's an equally emotional issue...

Halmos concludes his discussion of non-standard analysis as follows (p. 204):

it's a special tool, too special, and other tools can do everything it does. It's all a matter of taste.

Katz & Katz (2010) note that

Halmos's anxiousness to evaluate Robinson's theory may have involved a conflict of interests Halmos invested considerable emotional energy (and sweat, as he memorably puts it in his autobiography) into his translation of the Bernstein–Robinson result is blunt unflattering comments appear to retroactively justify his translationist attempt to deflect the impact of one of the first spectacular applications of Robinson's theory.