Paul Halmos

Paul Halmos (1916-2006) formulated a conjecture in operator theory called the Invariant Subspace Conjecture. The conjecture was solved by Bernstein and Robinson in 1964; the proof was published in 1966. The proof by Bernstein and Robinson relied on Nonstandard Analysis (NSA), a theory first developed by Robinson in 1961. Robinson sent an early draft of the paper to Halmos. The latter was able to develop a translation of the proof into traditional terms not using infinitesimals. Halmos's proof was published side-by-side with the Bernstein-Robinson proof in 1966. After Bernstein and Robinson solved Halmos's conjecture using NSA, Halmos made it a bit of a pastime attacking NSA in particular and nonstandard models more generally in various publications, even into the 2000's shortly before Halmos's death.

Halmos made various comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is "certainty" and "architecture" yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lession, namely that the castle is floating in midair. Halmos' realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied.

Halmos often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians' concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson's framework is "unnecessary" but Henson and Keisler argue that Robinson's framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos' criticisms. See the article 16b.
See also this MO post: https://mathoverflow.net/q/225455