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

PIOTR B LASZCZYK, Alexandre Borovik, VLADIMIR KANOVEI, Mikhail G. Katz, TARAS KUDRYK, SEMEN S. KUTATELADZE, DAVID SHERRY

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    Abstract

    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.
    Original languageEnglish
    JournalLogica Universalis
    Volume10
    Issue number4
    Early online date6 Jul 2016
    DOIs
    Publication statusPublished - Dec 2016

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