Logica Universalis

, Volume 10, Issue 4, pp 393–405 | Cite as

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

  • Piotr Błaszczyk
  • Alexandre Borovik
  • Vladimir Kanovei
  • Mikhail G. Katz
  • Taras Kudryk
  • Semen S. Kutateladze
  • David Sherry
Article

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 lesson, 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.

Mathematics Subject Classification

01A60 26E35 47A15 

Keywords

Archimedean axiom Bridge between discrete and continuousmathematics Hyperreals Incomparable quantities Indispensability infinity Mathematical realism Robinson 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Piotr Błaszczyk
    • 1
  • Alexandre Borovik
    • 2
  • Vladimir Kanovei
    • 3
  • Mikhail G. Katz
    • 4
  • Taras Kudryk
    • 5
  • Semen S. Kutateladze
    • 6
  • David Sherry
    • 7
  1. 1.Institute of MathematicsPedagogical University of CracowCracowPoland
  2. 2.School of MathematicsUniversity of ManchesterManchesterUnited Kingdom
  3. 3.IPPI, Moscow, and MIITMoscowRussia
  4. 4.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  5. 5.Department of MathematicsLviv National UniversityLvivUkraine
  6. 6.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  7. 7.Department of PhilosophyNorthern Arizona UniversityFlagstaffUSA

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