127,516 research outputs found
G\"odel Incompleteness and the Black Hole Information Paradox
Semiclassical reasoning suggests that the process by which an object
collapses into a black hole and then evaporates by emitting Hawking radiation
may destroy information, a problem often referred to as the black hole
information paradox. Further, there seems to be no unique prediction of where
the information about the collapsing body is localized. We propose that the
latter aspect of the paradox may be a manifestation of an inconsistent
self-reference in the semiclassical theory of black hole evolution. This
suggests the inadequacy of the semiclassical approach or, at worst, that
standard quantum mechanics and general relavity are fundamentally incompatible.
One option for the resolution for the paradox in the localization is to
identify the G\"odel-like incompleteness that corresponds to an imposition of
consistency, and introduce possibly new physics that supplies this
incompleteness. Another option is to modify the theory in such a way as to
prohibit self-reference. We discuss various possible scenarios to implement
these options, including eternally collapsing objects, black hole remnants,
black hole final states, and simple variants of semiclassical quantum gravity.Comment: 14 pages, 2 figures; revised according to journal requirement
Formal Arithmetic Before Grundgesetze
A speculative investigation of how Frege's logical views change between Begriffsschrift and Grundgesetze and how this might have affected the formal development of logicism
The logical anti-psychologism of Frege and Husserl
Frege and Husserl are both recognized for their significant contributions to the overthrowing of logical psychologism, at least in its 19th century forms. Between Frege's profound impact on modern logic that extended the influence of his anti-psychologism and Husserl's extensive attempts at the refutation of logical psychologism in the Prolegomena to Logical Investigations, these arguments are generally understood as successful. This paper attempts to account for the development of these two anti-psychologistic conceptions of logical objects and for some of the basic differences between them. It identifies some problems that are common to strongly anti-psychologistic conceptions of logic and compares the extent to which Frege's and Husserl's views are open to these problems. Accordingly, this paper is divided into two parts. Part I develops a conception of the problems of logical psychologism as they are distinctively understood by each philosopher, out of the explicit arguments and criticisms made against the view in the texts. This conception is in each case informed by the overall historical trajectories of each philosopher's philosophical development. Part II examines the two views in light of common problems of anti-psychologism
Abstracts of theses and related literature indicating current trends in arithmetic for the academically talented elementary school child written between 1957 and 1961
Thesis (Ed.M.)--Boston Universit
On an Irreducible Theory of Complex Systems
In the paper we present results to develop an irreducible theory of complex
systems in terms of self-organization processes of prime integer relations.
Based on the integers and controlled by arithmetic only the self-organization
processes can describe complex systems by information not requiring further
explanations. Important properties of the description are revealed. It points
to a special type of correlations that do not depend on the distances between
parts, local times and physical signals and thus proposes a perspective on
quantum entanglement. Through a concept of structural complexity the
description also computationally suggests the possibility of a general
optimality condition of complex systems. The computational experiments indicate
that the performance of a complex system may behave as a concave function of
the structural complexity. A connection between the optimality condition and
the majorization principle in quantum algorithms is identified. A global
symmetry of complex systems belonging to the system as a whole, but not
necessarily applying to its embedded parts is presented. As arithmetic fully
determines the breaking of the global symmetry, there is no further need to
explain why the resulting gauge forces exist the way they do and not even
slightly different.Comment: 8 pages, 3 figures, typos are corrected, some changes and additions
are mad
On Possible Implications of Self-Organization Processes through Transformation of Laws of Arithmetic into Laws of Space and Time
In the paper we present results based on the description of complex systems
in terms of self-organization processes of prime integer relations. Realized
through the unity of two equivalent forms, i.e., arithmetical and geometrical,
the description allows to transform the laws of a complex system in terms of
arithmetic into the laws of the system in terms of space and time. Possible
implications of the results are discussed.Comment: 26 pages, 4 figure
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