College Publications logo   College Publications title  
View Basket
Homepage Contact page
Academia Brasileira de Filosofia
Cadernos de Lógica e Computação
Cadernos de Lógica e Filosofia
Cahiers de Logique et d'Epistemologie
Communication, Mind and Language
Cuadernos de lógica, Epistemología y Lenguaje
Encyclopaedia of Logic
Historia Logicae
IfColog series in Computational Logic
IfColog Lecture series
IfColog Proceedings
Journal of Applied Logics - IfCoLog Journal
Editorial Board
Scope of the Journal
Forthcoming papers
Logics for New-Generation AI
Logic and Law
Logic and Semiotics
Logic PhDs
Logic, Methodology and Philosophy of Science
The Logica Yearbook
Neural Computing and Artificial Intelligence
The SILFS series
Studies in Logic
Student Publications
Texts in Logic and Reasoning
Texts in Mathematics
Digital Downloads
Information for authors
About us
Search for Books

Forthcoming papers


Groups of Worldview Transformations Implied by Isotropy of Spac

Judit X. Madarasz, Mike Stanett and Gergely Szekely

Given any Euclidean ordered field, $Q$, and any `reasonable' group,
$G$, of (1+3)-dimensional spacetime symmetries, we show how to
construct a model $M_{G}$ of kinematics for which the set $W$ of
worldview transformations between inertial observers satisfies $W =
G$. This holds in particular for all relevant subgroups of $gal$,
$cpoi$, and $ceucl$ (the groups of Galilean, Poincar'e and
Euclidean transformations, respectively, where $c in Q$ is a
model-specific parameter corresponding to the speed of light in the
case of Poincar'e transformations).

In doing so, by an elementary geometrical proof, we demonstrate our
main contribution: spatial isotropy is enough to entail that the set
$W$ of worldview transformations satisfies either $W subseteq gal$, $W
subseteq cpoi$, or $W subseteq ceucl$ for some $c > 0$. So
assuming spatial isotropy is enough to prove that there are only 3
possible cases: either the world is classical (the worldview
transformations between inertial observers are Galilean
transformations); the world is relativistic (the worldview
transformations are Poincar'e transformations); or the world is
Euclidean (which gives a nonstandard kinematical interpretation to
Euclidean geometry). This result considerably extends previous
results in this field, which assume a priori the (strictly stronger)
special principle of relativity, while also restricting the choice of
$Q$ to the field $mathbb{R}$ of reals.

As part of this work, we also prove the rather surprising result that, for any $G$ containing translations and rotations fixing the time-axis $taxis$, the requirement that $G$ be a subgroup of one of the
groups $gal$, $cpoi$ or $ceucl$ is logically equivalent to the
somewhat simpler requirement that, for all $g in G$: $g[taxis]$ is
a line, and if $g[taxis] = taxis$ then $g$ is a trivial
transformation (ie $g$ is a linear transformation that preserves
Euclidean length and fixes the time-axis setwise).

December 2020

© 2005–2022 College Publications / VFH webmaster