64 research outputs found
Zeno meets modern science
``No one has ever touched Zeno without refuting him''. We will not refute
Zeno in this paper. Instead we review some unexpected encounters of Zeno with
modern science. The paper begins with a brief biography of Zeno of Elea
followed by his famous paradoxes of motion. Reflections on continuity of space
and time lead us to Banach and Tarski and to their celebrated paradox, which is
in fact not a paradox at all but a strict mathematical theorem, although very
counterintuitive. Quantum mechanics brings another flavour in Zeno paradoxes.
Quantum Zeno and anti-Zeno effects are really paradoxical but now experimental
facts. Then we discuss supertasks and bifurcated supertasks. The concept of
localization leads us to Newton and Wigner and to interesting phenomenon of
quantum revivals. At last we note that the paradoxical idea of timeless
universe, defended by Zeno and Parmenides at ancient times, is still alive in
quantum gravity. The list of references that follows is necessarily incomplete
but we hope it will assist interested reader to fill in details.Comment: 40 pages, LaTeX, 10 figure
Some Thoughts on Hypercomputation
Hypercomputation is a relatively new branch of computer science that emerged
from the idea that the Church--Turing Thesis, which is supposed to describe
what is computable and what is noncomputable, cannot possible be true. Because
of its apparent validity, the Church--Turing Thesis has been used to
investigate the possible limits of intelligence of any imaginable life form,
and, consequently, the limits of information processing, since living beings
are, among others, information processors. However, in the light of
hypercomputation, which seems to be feasibly in our universe, one cannot impose
arbitrary limits to what intelligence can achieve unless there are specific
physical laws that prohibit the realization of something. In addition,
hypercomputation allows us to ponder about aspects of communication between
intelligent beings that have not been considered befor
The dome: An unexpectedly simple failure of determinism
Newton's equations of motion tell us that a mass at rest at the apex of a dome with the shape specified here can spontaneously move. It has been suggested that this indeterminism should be discounted since it draws on an incomplete rendering of Newtonian physics, or it is "unphysical," or it employs illicit idealizations. I analyze and reject each of these reasons. Copyright 2008 by the Philosophy of Science Association. All rights reserved
How to Build an Infinite Lottery Machine
An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough
Temporal breakdown and Borel resummation in the complex Langevin method
We reexamine the Parisi-Klauder conjecture for complex e^{i\theta/2} \phi^4
measures with a Wick rotation angle 0 <= \theta/2 < \pi/2 interpolating between
Euclidean and Lorentzian signature. Our main result is that the asymptotics for
short stochastic times t encapsulates information also about the equilibrium
aspects. The moments evaluated with the complex measure and with the real
measure defined by the stochastic Langevin equation have the same t -> 0
asymptotic expansion which is shown to be Borel summable. The Borel transform
correctly reproduces the time dependent moments of the complex measure for all
t, including their t -> infinity equilibrium values. On the other hand the
results of a direct numerical simulation of the Langevin moments are found to
disagree from the `correct' result for t larger than a finite t_c. The
breakdown time t_c increases powerlike for decreasing strength of the noise's
imaginary part but cannot be excluded to be finite for purely real noise. To
ascertain the discrepancy we also compute the real equilibrium distribution for
complex noise explicitly and verify that its moments differ from those obtained
with the complex measure.Comment: title changed, results on parameter dependence of t_c added,
exposition improved. 39 pages, 7 figure
How to Build an Infinite Lottery Machine
An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough
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