43,521 research outputs found
Some Varieties of Superparadox. The implications of dynamic contradiction, the characteristic form of breakdown of breakdown of sense to which self-reference is prone
The Problem of the Paradoxes came to the fore in philosophy and mathematics with the discovery of Russell's Paradox in 1901. It is the "forgotten" intellectual-scientific problem of the Twentieth Century, because for more than sixty years a pretence was maintained, by a consensus of logicians, that the problem had been "solved"
The ultimate tactics of self-referential systems
Mathematics is usually regarded as a kind of language. The essential behavior
of physical phenomena can be expressed by mathematical laws, providing
descriptions and predictions. In the present essay I argue that, although
mathematics can be seen, in a first approach, as a language, it goes beyond
this concept. I conjecture that mathematics presents two extreme features,
denoted here by {\sl irreducibility} and {\sl insaturation}, representing
delimiters for self-referentiality. These features are then related to physical
laws by realizing that nature is a self-referential system obeying bounds
similar to those respected by mathematics. Self-referential systems can only be
autonomous entities by a kind of metabolism that provides and sustains such an
autonomy. A rational mind, able of consciousness, is a manifestation of the
self-referentiality of the Universe. Hence mathematics is here proposed to go
beyond language by actually representing the most fundamental existence
condition for self-referentiality. This idea is synthesized in the form of a
principle, namely, that {\sl mathematics is the ultimate tactics of
self-referential systems to mimic themselves}. That is, well beyond an
effective language to express the physical world, mathematics uncovers a deep
manifestation of the autonomous nature of the Universe, wherein the human brain
is but an instance.Comment: 9 pages. This essay received the 4th. Prize in the 2015 FQXi essay
contest: "Trick or Truth: the Mysterious Connection Between Physics and
Mathematics
A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points
Following F. William Lawvere, we show that many self-referential paradoxes,
incompleteness theorems and fixed point theorems fall out of the same simple
scheme. We demonstrate these similarities by showing how this simple scheme
encompasses the semantic paradoxes, and how they arise as diagonal arguments
and fixed point theorems in logic, computability theory, complexity theory and
formal language theory
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
Mathematical Foundations of Consciousness
We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical
primitives. The Anti-foundation Axiom plays a significant role in our
development, since among other of its features, its replacement for the Axiom
of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic
interpretations. These interpretations also depend on such allied notions for
sets as pictures, graphs, decorations, labelings and various mappings that we
use. A syntax and semantics of operators acting on sets is developed. Such
features enable construction of a theory of non-well-founded sets that we use
to frame mathematical foundations of consciousness. To do this we introduce a
supplementary axiomatic system that characterizes experience and consciousness
as primitives. The new axioms proceed through characterization of so- called
consciousness operators. The Russell operator plays a central role and is shown
to be one example of a consciousness operator. Neural networks supply striking
examples of non-well-founded graphs the decorations of which generate
associated sets, each with a Platonic aspect. Employing our foundations, we
show how the supervening of consciousness on its neural correlates in the brain
enables the framing of a theory of consciousness by applying appropriate
consciousness operators to the generated sets in question
Hypermedia Learning Objects System - On the Way to a Semantic Educational Web
While eLearning systems become more and more popular in daily education,
available applications lack opportunities to structure, annotate and manage
their contents in a high-level fashion. General efforts to improve these
deficits are taken by initiatives to define rich meta data sets and a
semanticWeb layer. In the present paper we introduce Hylos, an online learning
system. Hylos is based on a cellular eLearning Object (ELO) information model
encapsulating meta data conforming to the LOM standard. Content management is
provisioned on this semantic meta data level and allows for variable,
dynamically adaptable access structures. Context aware multifunctional links
permit a systematic navigation depending on the learners and didactic needs,
thereby exploring the capabilities of the semantic web. Hylos is built upon the
more general Multimedia Information Repository (MIR) and the MIR adaptive
context linking environment (MIRaCLE), its linking extension. MIR is an open
system supporting the standards XML, Corba and JNDI. Hylos benefits from
manageable information structures, sophisticated access logic and high-level
authoring tools like the ELO editor responsible for the semi-manual creation of
meta data and WYSIWYG like content editing.Comment: 11 pages, 7 figure
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