1,511 research outputs found
Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality
In this paper the claim that Zeno's paradoxes have been solved is contested.
Although no one has ever touched Zeno without refuting him (Whitehead), it will
be our aim to show that, whatever it was that was refuted, it was certainly not
Zeno. The paper is organised in two parts. In the first part we will
demonstrate that upon direct analysis of the Greek sources, an underlying
structure common to both the Paradoxes of Plurality and the Paradoxes of Motion
can be exposed. This structure bears on a correct - Zenonian - interpretation
of the concept of division through and through. The key feature, generally
overlooked but essential to a correct understanding of all his arguments, is
that they do not presuppose time. Division takes place simultaneously. This
holds true for both PP and PM. In the second part a mathematical representation
will be set up that catches this common structure, hence the essence of all
Zeno's arguments, however without refuting them. Its central tenet is an
aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some
number theoretic and geometric implications will be shortly discussed.
Furthermore, it will be shown how the Received View on the motion-arguments can
easely be derived by the introduction of time as a (non-Zenonian) premiss, thus
causing their collapse into arguments which can be approached and refuted by
Aristotle's limit-like concept of the potentially infinite, which remained -
though in different disguises - at the core of the refutational strategies that
have been in use up to the present. Finally, an interesting link to Newtonian
mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure
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
Review of 'The Outer Limits of Reason' by Noson Yanofsky 403p (2013) (review revised 2019)
I give a detailed review of 'The Outer Limits of Reason' by Noson Yanofsky from a unified perspective of Wittgenstein and evolutionary psychology. I indicate that the difficulty with such issues as paradox in language and math, incompleteness, undecidability, computability, the brain and the universe as computers etc., all arise from the failure to look carefully at our use of language in the appropriate context and hence the failure to separate issues of scientific fact from issues of how language works. I discuss Wittgenstein's views on incompleteness, paraconsistency and undecidability and the work of Wolpert on the limits to computation. To sum it up: The Universe According to Brooklyn---Good Science, Not So Good Philosophy.
Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book âThe Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searleâ 2nd ed (2019). Those interested in more of my writings may see âTalking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019
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"
Unknowable Truths: The Incompleteness Theorems and the Rise of Modernism
This thesis evaluates the function of the current history of mathematics methodologies and explores ways in which historiographical methodologies could be successfully implemented in the field. Traditional approaches to the history of mathematics often lack either an accurate portrayal of the social and cultural influences of the time, or they lack an effective usage of mathematics discussed. This paper applies a holistic methodology in a case study of Kurt Gödelâs influential work in logic during the Interwar period and the parallel rise of intellectual modernism. In doing so, the proofs for Gödelâs Completeness and Incompleteness theorems will be discussed as well as Gödelâs philosophical interests and influences of the time. To explore the intersection of these worlds, practices are borrowed from the fields of intellectual history and history of science and technology to analyze better the effects of society and culture on the mind of mathematicians like Gödel and their work
Topological set theories and hyperuniverses
We give a new set theoretic system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK and the theory of hyperuniverses. On the other hand, it retains most of the expressiveness of these theories and has the same consistency strength as ZF. We single out the additional axiom of the universal set as the one that increases the consistency strength to that of GPK and explore several other axioms and interrelations between those theories.
Hyperuniverses are a natural class of models for theories with a universal set. The Aleph_0- and Aleph_1-dimensional Cantor cubes are examples of hyperuniverses with additivity Aleph_0, because they are homeomorphic to their hyperspace. We prove that in the realm of spaces with uncountable additivity, none of the generalized Cantor cubes has that property.
Finally, we give two complementary constructions of hyperuniverses which generalize many of the constructions found in the literature and produce initial and terminal hyperuniverses
- âŠ