6 research outputs found
Vienna Circle and Logical Analysis of Relativity Theory
In this paper we present some of our school's results in the area of building
up relativity theory (RT) as a hierarchy of theories in the sense of logic. We
use plain first-order logic (FOL) as in the foundation of mathematics (FOM) and
we build on experience gained in FOM.
The main aims of our school are the following: We want to base the theory on
simple, unambiguous axioms with clear meanings. It should be absolutely
understandable for any reader what the axioms say and the reader can decide
about each axiom whether he likes it. The theory should be built up from these
axioms in a straightforward, logical manner. We want to provide an analysis of
the logical structure of the theory. We investigate which axioms are needed for
which predictions of RT. We want to make RT more transparent logically, easier
to understand, easier to change, modular, and easier to teach. We want to
obtain deeper understanding of RT.
Our work can be considered as a case-study showing that the Vienna Circle's
(VC) approach to doing science is workable and fruitful when performed with
using the insights and tools of mathematical logic acquired since its formation
years at the very time of the VC activity. We think that logical positivism was
based on the insight and anticipation of what mathematical logic is capable
when elaborated to some depth. Logical positivism, in great part represented by
VC, influenced and took part in the birth of modern mathematical logic. The
members of VC were brave forerunners and pioneers.Comment: 25 pages, 1 firgure
Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory
There are several first-order logic (FOL) axiomatizations of special
relativity theory in the literature, all looking essentially different but
claiming to axiomatize the same physical theory. In this paper, we elaborate a
comparison, in the framework of mathematical logic, between these FOL theories
for special relativity. For this comparison, we use a version of mathematical
definability theory in which new entities can also be defined besides new
relations over already available entities. In particular, we build an
interpretation of the reference-frame oriented theory SpecRel into the
observationally oriented Signalling theory of James Ax. This interpretation
provides SpecRel with an operational/experimental semantics. Then we make
precise, "quantitative" comparisons between these two theories via using the
notion of definitional equivalence. This is an application of logic to the
philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in
Logi
Towards Formal Verification of Computations and Hypercomputations in Relativistic Physics
It is now more than 15 years since Copeland and Proudfoot introduced the term hypercomputation. Although no hypercomputer has yet been built (and perhaps never will be), it is instructive to consider what properties any such device should possess, and whether these requirements could ever be met. Aside from the potential benefits that would accrue from a positive outcome, the issues raised are sufficiently disruptive that they force us to re-evaluate existing computability theory. From a foundational viewpoint the questions driving hypercomputation theory remain the same as those addressed since the earliest days of computer science, viz. what is computation? and what can be computed? Early theoreticians developed models of computation that are independent of both their implementation and their physical location, but it has become clear in recent decades that these aspects of computation cannot always be neglected. In particular, the computational power of a distributed system can be expected to vary according to the spacetime geometry in which the machines on which it is running are located. The power of a computing system therefore depends on its physical environment and cannot be specified in absolute terms. Even Turing machines are capable of super-Turing behaviour, given the right environment