A non-standard analysis of a cultural icon: The case of Paul Halmos

TL;DR

This paper critically examines Paul Halmos' views on various mathematical concepts, highlighting his philosophical stance and contrasting it with Robinson's nonstandard analysis, revealing the depth of set-theoretic and logical debates.

Contribution

It offers a detailed analysis of Halmos' philosophical perspective on logic and nonstandard models, contrasting it with Robinson's framework, and discusses implications for mathematical realism.

Findings

Halmos was skeptical of nonstandard models and Robinson's infinitesimals.

Robinson's framework enables deeper set-theoretic exploration than Halmos acknowledged.

The paper highlights philosophical differences impacting the understanding of mathematical foundations.

Abstract

We examine Paul Halmos' 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. He 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. Keywords: Archimedean axiom; bridge between discrete and continuous mathematics; hyperreals; incomparable quantities; indispensability; infinity; mathematical realism; Robinson.