Sets and Functions in Theoretical Physics

Adonai S. Sant’Anna, Otavio Bueno

Research output: Contribution to journalArticle

Abstract

It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue.

Original languageEnglish (US)
Pages (from-to)257-281
Number of pages25
JournalErkenntnis
Volume79
Issue number2
DOIs
StatePublished - Apr 1 2014

Fingerprint

Physics
Set Theory
Classical Field Theory
Group Theory
Reformulation
Mechanics
Theoretical Physics
Formulation
Concepts
Physical
Standards
Language

ASJC Scopus subject areas

  • Philosophy
  • Logic

Cite this

Sets and Functions in Theoretical Physics. / Sant’Anna, Adonai S.; Bueno, Otavio.

In: Erkenntnis, Vol. 79, No. 2, 01.04.2014, p. 257-281.

Research output: Contribution to journalArticle

Sant’Anna, Adonai S. ; Bueno, Otavio. / Sets and Functions in Theoretical Physics. In: Erkenntnis. 2014 ; Vol. 79, No. 2. pp. 257-281.
@article{164f532bbd9e4d749e71d39d8caffd94,
title = "Sets and Functions in Theoretical Physics",
abstract = "It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue.",
author = "Sant’Anna, {Adonai S.} and Otavio Bueno",
year = "2014",
month = "4",
day = "1",
doi = "10.1007/s10670-013-9491-y",
language = "English (US)",
volume = "79",
pages = "257--281",
journal = "Erkenntnis",
issn = "0165-0106",
publisher = "Springer Netherlands",
number = "2",

}

TY - JOUR

T1 - Sets and Functions in Theoretical Physics

AU - Sant’Anna, Adonai S.

AU - Bueno, Otavio

PY - 2014/4/1

Y1 - 2014/4/1

N2 - It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue.

AB - It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue.

UR - http://www.scopus.com/inward/record.url?scp=84956801132&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956801132&partnerID=8YFLogxK

U2 - 10.1007/s10670-013-9491-y

DO - 10.1007/s10670-013-9491-y

M3 - Article

AN - SCOPUS:84956801132

VL - 79

SP - 257

EP - 281

JO - Erkenntnis

JF - Erkenntnis

SN - 0165-0106

IS - 2

ER -