Logica Universalis 10 (4):393-405 (2016)

Authors
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.
Keywords Archimedean axiom  Bridge between discrete and continuousmathematics  Hyperreals  Incomparable quantities  Indispensability  infinity  Mathematical realism  Robinson
Categories (categorize this paper)
DOI 10.1007/s11787-016-0153-0
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 53,514
Through your library

References found in this work BETA

Non-Standard Analysis.A. Robinson - 1961 - North-Holland Publishing Co..
Applied Nonstandard Analysis.Martin Davis - 1978 - Journal of Symbolic Logic 43 (2):383-384.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Review: Paul R. Halmos, Naive Set Theory. [REVIEW]Alfons Borgers - 1969 - Journal of Symbolic Logic 34 (2):308-308.
An Autobiography of Polyadic Algebras.Paul R. Halmos - 2000 - Logic Journal of the IGPL 8 (4):383-392.
Algebraic Logic IV. Equality in Polyadic Algebras.Paul R. Halmos - 1959 - Journal of Symbolic Logic 24 (3):252-252.
Review: Paul R. Halmos, Polyadic Boolean Algebras. [REVIEW]Roland Fraïssé - 1958 - Journal of Symbolic Logic 23 (2):223-223.

Analytics

Added to PP index
2016-07-06

Total views
21 ( #472,289 of 2,348,301 )

Recent downloads (6 months)
2 ( #329,111 of 2,348,301 )

How can I increase my downloads?

Downloads

My notes