Reason and Argument
Volume 1 / 1989 to Volume 4 / 1991
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Chapter
Gödel Incompleteness and the Empirical Sciences
We show how widespread are metamathematical phenomena in mathematics and in the sciences which rely on mathematics. We will consider specific examples of undecidable sentences in mathematics, physics and econo...
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Article
Non-reflexive Logical Foundation for Quantum Mechanics
On the one hand, non-reflexive logics are logics in which the principle of identity does not hold in general. On the other hand, quantum mechanics has difficulties regarding the interpretation of ‘particles’ a...
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Article
Definability and Invariance
In his thesis Para uma Teoria Geral dos Homomorfismos (1944), the Portuguese mathematician José Sebastião e Silva constructed an abstract or generalized Galois theory, that is intimately linked to F. Klein’s Erla...
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Chapter
On Some Recent Undecidability and Incompleteness Results in the Axiomatized Sciences
We summarize in an intuitive vein a few recent results by the authors on the incompleteness of elementary real analysis and its consequences to the axiomatized sciences, from chaos theory to the dynamics of po...
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Chapter and Conference Paper
H-computation
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Chapter
Structures, Suppes Predicates, and Boolean-Valued Models in Physics
An old and important question concerning physical theories has to do with their axiomatization [47]. The sixth problem in Hilbert’s celebrated list of mathematical problems deals with its desirable (or ideal) ...
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Article
Undecidable hopf bifurcation with undecidable fixed point
We exhibit a polynomial dynamical system where one cannot decide whether a Hopf bifurcation occurs. Therefore one cannot decide whether there will be parameter values such that a stable fixed point becomes an ...
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Article
Two questions on the geometry of gauge fields
We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for.....
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Article
Dynamical system where proving chaos is equivalent to proving Fermat's conjecture
We prove that we can explicitly construct the expression for a low-dimensional Hamiltonian system where proving the existence of a Smale horseshoe is equivalent to proving that Fermat's Conjecture is true. We ...
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Chapter and Conference Paper
On Arnol'd's Hilbert symposium problems
We prove that stability is undecidable for dynamical systems whose right-hand side is explicitly written in the language of elementary analysis.
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Article
Classical physics and Penrose's thesis
We expose and discussPenrose's thesis: “Nature produces harnessable noncomputable processes, but none at the classical level.” We then suggest a partial counterexample to it, based on aGedanken experiment about a...
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Article
Undecidability and incompleteness in classical mechanics
We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary real-valued functions and problems. We then derive a general undecidability and incompleteness result fo...
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Book Series
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Book
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Article
A Suppes predicate for general relativity and set-theoretically generic spacetimes
We summarize ideas from Zermelo-Fraenkel set theory up to an axiomatic treatment for general relativity based on a Suppes predicate. We then examine the meaning of set-theoretic genericity for manifolds that u...
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Article
Book reviews
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Article
A new formulation of discussive logic
S. Jaśkowski introduced the discussive prepositional calculus D 2as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. ...
