TY - JOUR

T1 - A Non-Standard Analysis of a Cultural Icon

T2 - The Case of Paul Halmos

AU - Błaszczyk, Piotr

AU - Borovik, Alexandre

AU - Kanovei, Vladimir

AU - Katz, Mikhail G.

AU - Kudryk, Taras

AU - Kutateladze, Semen S.

AU - Sherry, David

N1 - Publisher Copyright:
© 2016, Springer International Publishing.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - 01A60

KW - 26E35

KW - 47A15

UR - http://www.scopus.com/inward/record.url?scp=84978036267&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978036267&partnerID=8YFLogxK

U2 - 10.1007/s11787-016-0153-0

DO - 10.1007/s11787-016-0153-0

M3 - Article

AN - SCOPUS:84978036267

SN - 1661-8297

VL - 10

SP - 393

EP - 405

JO - Logica Universalis

JF - Logica Universalis

IS - 4

ER -