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 M Sherry

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)1-13
Number of pages13
JournalLogica Universalis
DOIs
StateAccepted/In press - Jul 6 2016

Fingerprint

Nonstandard Analysis
Logic
Category Theory
Tend
Incompleteness
Syntactics
First-order Logic
Algebra
Eliminate
Semantics
Color
Uncertainty
Imply
Resources
Term
Theorem
Architecture
Framework
Teaching

ASJC Scopus subject areas

  • Logic
  • Applied Mathematics

Cite this

Błaszczyk, P., Borovik, A., Kanovei, V., Katz, M. G., Kudryk, T., Kutateladze, S. S., & Sherry, D. M. (Accepted/In press). A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos. Logica Universalis, 1-13. https://doi.org/10.1007/s11787-016-0153-0

A Non-Standard Analysis of a Cultural Icon : The Case of Paul Halmos. / Błaszczyk, Piotr; Borovik, Alexandre; Kanovei, Vladimir; Katz, Mikhail G.; Kudryk, Taras; Kutateladze, Semen S.; Sherry, David M.

In: Logica Universalis, 06.07.2016, p. 1-13.

Research output: Contribution to journalArticle

Błaszczyk, P, Borovik, A, Kanovei, V, Katz, MG, Kudryk, T, Kutateladze, SS & Sherry, DM 2016, 'A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos', Logica Universalis, pp. 1-13. https://doi.org/10.1007/s11787-016-0153-0
Błaszczyk P, Borovik A, Kanovei V, Katz MG, Kudryk T, Kutateladze SS et al. A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos. Logica Universalis. 2016 Jul 6;1-13. https://doi.org/10.1007/s11787-016-0153-0
Błaszczyk, Piotr ; Borovik, Alexandre ; Kanovei, Vladimir ; Katz, Mikhail G. ; Kudryk, Taras ; Kutateladze, Semen S. ; Sherry, David M. / A Non-Standard Analysis of a Cultural Icon : The Case of Paul Halmos. In: Logica Universalis. 2016 ; pp. 1-13.
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