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A Non-Standard Analysis of a Cultural Icon:The Case of Paul Halmos
No full textWe examine Paul Halmos’ comments on category the-ory, Dedekind cuts, devil worship, logic, and Robinson’s infinites-imals. Halmos’ scepticism about category theory derives from hisphilosophical position of naive set-theoretic realism. In the wordsof an MAA biography, Halmos thought that mathematics is “cer-tainty” and “architecture” yet 20th century logic teaches us is thatmathematics is full of uncertainty or more precisely incomplete-ness. If the term architecture meant to imply that mathematics isone great solid castle, then modern logic tends to teach us the op-posite lession, namely that the castle is floating in midair. Halmos’realism tends to color his judgment of purely scientific aspects oflogic and the way it is practiced and applied. He often expresseddistaste for nonstandard models, and made a sustained effort toeliminate first-order logic, the logicians’ concept of interpretation,and the syntactic vs semantic distinction. He felt that these werevague, and sought to replace them all by his polyadic algebra. Hal-mos claimed that Robinson’s framework is “unnecessary” but Hen-son and Keisler argue that Robinson’s framework allows one to digdeeper into set-theoretic resources than is common in Archimedeanmathematics. This can potentially prove theorems not accessibleby standard methods, undermining Halmos’ criticisms.Keywords: Archimedean axiom; bridge between discrete andcontinuous mathematics; hyperreals; incomparable quantities; in-dispensability; infinity; mathematical realism; Robinson
