A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We 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 his
philosophical position of naive set-theoretic realism. In the words
of an MAA biography, Halmos thought that mathematics is “cer-
tainty” and “architecture” yet 20th century logic teaches us is that
mathematics is full of uncertainty or more precisely incomplete-
ness. If the term architecture meant to imply that mathematics is
one 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 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. Hal-
mos claimed that Robinson’s framework is “unnecessary” but Hen-
son 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; in-
dispensability; infinity; mathematical realism; Robinson.