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Unfolding finitist arithmetic
, 2010
"... The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significan ..."
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The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significance was previously carried out for a system of nonfinitist arithmetic, NFA; it was shown that U(NFA) is prooftheoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the socalled Bar Rule. It is shown that U(FA) and U(FA + BR) are prooftheoretically equivalent, respectively, to Primitive Recursive Arithmetic, PRA, and to Peano Arithmetic, PA.
Info‐Computational Philosophy Of Nature: An Informational Universe With Computational Dynamics
, 2011
"... Starting with the Søren Brier’s Cybersemiotic critique of the existing practice of Wissenshaft, this article develops the argument for an alternative naturalization of knowledge production. It presents the framework of natural info‐computationalism, ICON, as a new Natural Philosophy based on concep ..."
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Starting with the Søren Brier’s Cybersemiotic critique of the existing practice of Wissenshaft, this article develops the argument for an alternative naturalization of knowledge production. It presents the framework of natural info‐computationalism, ICON, as a new Natural Philosophy based on concepts of information (structure) and computation (process). In this approach, which is a synthesis of informationalism (the view that nature is informational) and computationalism (the view that nature computes its own time development), computation is in general not a substrate‐independent disembodied symbol manipulation. Based on the informational character of nature, where matter and informational structure are equivalent, information processing in general is embodied and in general substrate specific. The Turing Machine model of abstract discrete sequential symbol manipulation is a subset of the Natural computing model. With this generalized idea of Natural computing and Informational Structural Realism, Info‐computationalism (ICON), adopting scientific third‐person account, covers the entire list of requirements for naturalist knowledge production framework from Brier (2010) except for qualia as experienced in a first‐person mode.
The Impact of the Incompleteness Theorems On Mathematics
, 2006
"... In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits an ..."
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In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about what he calls The Paradox of Irrelevance: “There are many math problems that have achieved the cachet of tremendous significance, e.g., Fermat, fourcolor, Kepler’s packing, Gödel, etc. Of Fermat, I have read: ‘the most famous math problem of all time’. Of Gödel, I have read: ‘the most mathematically significant achievement of the 20th century’. … Yet, these problems have engaged the attention of relatively few research mathematicians—even in pure math. ” What accounts for this disconnect between fame and relevance? Before going into the question for Gödel’s theorems, it should be distinguished in one respect from the other examples mentioned, which in any case form quite a mixed bag. Namely, each of the Fermat, fourcolor, and Kepler’s packing problems posed a standout challenge following extended efforts to settle them; meeting the challenge in each case required new ideas or approaches and intense work, obviously of different degrees. By contrast, Gödel’s theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things. SetSolomon Feferman is professor of mathematics and philosophy, emeritus, at Stanford University. His email address is
Emergence: an algorithmic formulation
, 2005
"... When the microequations of a dynamical system generate complex macrobehaviour, there can be an explanatory gap between the smallscale and largescale descriptions of the same system. The microdynamics may be simple, but its relationship to the macrobehaviour may seem impenetrable. This phenomenon, ..."
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When the microequations of a dynamical system generate complex macrobehaviour, there can be an explanatory gap between the smallscale and largescale descriptions of the same system. The microdynamics may be simple, but its relationship to the macrobehaviour may seem impenetrable. This phenomenon, known as emergence, poses problems for the nature of scientific understanding. How do we reconcile two radically different modes of description? Emergence is formulated using the powerful tools of algorithmic information and computational theory. This provides the ground for an extension and generalisation of the phenomenon. Mathematics itself is analysed as an emergent system, linking formalist notions of mathematics as a string manipulation game with the more abstract ideas and proofs that occupy mathematicians. A philosophical problem that has plagued emergence is whether the whole can be more than the sum of its parts. This possibility, known as strong emergence, manifests when emergent macrostructures introduce brand new causal dynamics into a system. A new perspective on this
Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits
"... The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicia ..."
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The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the farreaching mathematical implications thereof. 2 I.
Gödel and the Metamathematical Tradition
, 2007
"... The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although Gödel’s work in logic fits squarely in that tradition, one often finds him curiously at odds with the asso ..."
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The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although Gödel’s work in logic fits squarely in that tradition, one often finds him curiously at odds with the associated methodological orientation. This essay explores that tension and what lies behind it. 1
Introducción a la computación cuántica
, 2007
"... La mecánica cuántica es la rama de la física que describe el comportamiento de la naturaleza a escalas muy pequeñas (por ejemplo, el comportamiento de los átomos). La teoría de la computación se encarga de estudiar si un problema es susceptible de ser resuelto utilizando una computadoa, así como la ..."
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La mecánica cuántica es la rama de la física que describe el comportamiento de la naturaleza a escalas muy pequeñas (por ejemplo, el comportamiento de los átomos). La teoría de la computación se encarga de estudiar si un problema es susceptible de ser resuelto utilizando una computadoa, así como la cantidad de recursos (tiempo, energía) que se debe invertir en caso de existir solución. En consecuencia, la computación cuántica hace uso de la mecánica cuántica con el objetivo de incrementar nuestra capacidad computacional para el procesamiento de información y solución de problemas. Por otra parte, la teoría de la información cuántica estudia los métodos, capacidades y límites que las leyes de la física imponen en la transmisión y recuperación de información. El estudio formal de la computación cuántica comenzó con las preguntas que Richard Feynman planteó sobre dos temas: 1) la posibilidad de simular sistemas cuánticos, y 2) las leyes de la física que caracterizan al proceso de calcular [92, 93]. A partir de ese trabajo, la computación cuántica ha avanzado a paso firme; por ejemplo, se ha definido formalmente la estructura de una computadora cuántica [3], se han encontrado resultados espectaculares como el algoritmo de Shor [4] (capaz de factorizar un número entero muy largo en tiempo razonable utilizando una computadora cuántica [4, 7]) y el algoritmo de Grover [5] (este algoritmo encuentra elementos en conjuntos desordenados de forma más eficiente que cualquier algoritmo posible ejecutado en computadoras convencionales [5, 7]), y se ha diseñado una teoría y práctica de la criptografía usando las propiedades de la física cuántica [6]. En el futuro mediato, la computación cuántica tendrá gran impacto en la industria de la computación y el desarrollo de protocolos de criptografía y seguridad computacional [12–14]. La computación y la información cuánticas representan un reto teórico y experimental por la cantidad y complejidad de problemas a resolver. A pesar de dichos retos, los avances realizados hasta ahora permiten ya pensar en aplicaciones de esta disciplina en áreas del conocimiento tales como la
Published In Gödel and the metamathematical tradition∗
, 2007
"... The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although Gödel’s work in logic fits squarely in that tradition, one often finds him curiously at odds with the a ..."
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The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although Gödel’s work in logic fits squarely in that tradition, one often finds him curiously at odds with the associated methodological orientation. This essay explores that tension and what lies behind it. 1