10,392 research outputs found
Information Physics: The New Frontier
At this point in time, two major areas of physics, statistical mechanics and
quantum mechanics, rest on the foundations of probability and entropy. The last
century saw several significant fundamental advances in our understanding of
the process of inference, which make it clear that these are inferential
theories. That is, rather than being a description of the behavior of the
universe, these theories describe how observers can make optimal predictions
about the universe. In such a picture, information plays a critical role. What
is more is that little clues, such as the fact that black holes have entropy,
continue to suggest that information is fundamental to physics in general.
In the last decade, our fundamental understanding of probability theory has
led to a Bayesian revolution. In addition, we have come to recognize that the
foundations go far deeper and that Cox's approach of generalizing a Boolean
algebra to a probability calculus is the first specific example of the more
fundamental idea of assigning valuations to partially-ordered sets. By
considering this as a natural way to introduce quantification to the more
fundamental notion of ordering, one obtains an entirely new way of deriving
physical laws. I will introduce this new way of thinking by demonstrating how
one can quantify partially-ordered sets and, in the process, derive physical
laws. The implication is that physical law does not reflect the order in the
universe, instead it is derived from the order imposed by our description of
the universe. Information physics, which is based on understanding the ways in
which we both quantify and process information about the world around us, is a
fundamentally new approach to science.Comment: 17 pages, 6 figures. Knuth K.H. 2010. Information physics: The new
frontier. J.-F. Bercher, P. Bessi\`ere, and A. Mohammad-Djafari (eds.)
Bayesian Inference and Maximum Entropy Methods in Science and Engineering
(MaxEnt 2010), Chamonix, France, July 201
A Potential Foundation for Emergent Space-Time
We present a novel derivation of both the Minkowski metric and Lorentz
transformations from the consistent quantification of a causally ordered set of
events with respect to an embedded observer. Unlike past derivations, which
have relied on assumptions such as the existence of a 4-dimensional manifold,
symmetries of space-time, or the constant speed of light, we demonstrate that
these now familiar mathematics can be derived as the unique means to
consistently quantify a network of events. This suggests that space-time need
not be physical, but instead the mathematics of space and time emerges as the
unique way in which an observer can consistently quantify events and their
relationships to one another. The result is a potential foundation for emergent
space-time.Comment: The paper was originally titled "The Physics of Events: A Potential
Foundation for Emergent Space-Time". We changed the title (and abstract) to
be more direct when the paper was accepted for publication at the Journal of
Mathematical Physics. 24 pages, 15 figure
Semantics out of context: nominal absolute denotations for first-order logic and computation
Call a semantics for a language with variables absolute when variables map to
fixed entities in the denotation. That is, a semantics is absolute when the
denotation of a variable a is a copy of itself in the denotation. We give a
trio of lattice-based, sets-based, and algebraic absolute semantics to
first-order logic. Possibly open predicates are directly interpreted as lattice
elements / sets / algebra elements, subject to suitable interpretations of the
connectives and quantifiers. In particular, universal quantification "forall
a.phi" is interpreted using a new notion of "fresh-finite" limit and using a
novel dual to substitution.
The interest of this semantics is partly in the non-trivial and beautiful
technical details, which also offer certain advantages over existing
semantics---but also the fact that such semantics exist at all suggests a new
way of looking at variables and the foundations of logic and computation, which
may be well-suited to the demands of modern computer science
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
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