December 2012
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197 Reads
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1 Citation
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Publications (97)
December 2012
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127 Reads
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32 Citations
January 2012
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63 Reads
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3 Citations
Lecture Notes in Computer Science
This note gives some information about the magical number Ω and why it is of interest. Our purpose is to explain the significance of recent work by Calude and Dinneen attempting to compute Ω. Furthermore, we propose measuring human intellectual progress (not scientific progress) via the number of bits of Ω that can be determined at any given moment in time using the current mathematical theories.
June 2011
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785 Reads
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11 Citations
Manuscrito
We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel (1871-1956).
February 2011
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70 Reads
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2 Citations
February 2011
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159 Reads
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14 Citations
In this essay we present an information-theoretic perspective on epistemology using software models. We shall use the notion of algorithmic information to discuss what is a physical law, to determine the limits of the axiomatic method, and to analyze Darwin’s theory of evolution. Weyl, Leibniz, complexity and the principle of sufficient reason The best way to understand the deep concept of conceptual complexity and algorithmic information, which is our basic tool, is to see how it evolved, to know its long history. Let’s start with Hermann Weyl and the great philosopher/mathematician G. W. Leibniz. That everything that is true is true for a reason is rationalist Leibniz’s famous principle of sufficient reason. The bits of Ω seem to refute this fundamental principle and also the idea that everything can be proved starting from self-evident facts. What is a scientific theory?
January 2011
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66 Reads
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12 Citations
This essay is intended as a contribution to The Once and Future Turing edited by S. Barry Cooper and Andrew Hodges, which will be published by Cambridge University Press as part of the Turing Centenary celebration.
January 2011
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69 Reads
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4 Citations
December 2010
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97 Reads
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4 Citations
Leibniz in 1686 in his Discours de mtaphysique points out that if an arbitrarily complex theory is permitted then the notion of theory” becomes vacuous because there is always a theory. This idea is developed in the modern theory of algorithmic information, which deals with the size of computer programs and provides a new view of Gdel’s work on incompleteness and Turing’s work on uncomputability.This will be a first-person account of some doubts and speculations about the nature of mathematics that I have entertained for the past three decades.
January 2010
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100 Reads
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13 Citations
Notices of the American Mathematical Society
Turing's famous 1936 paper \On computable numbers, with an application to the Entschei- dungsproblem" denes a computable real num- ber and uses Cantor's diagonal argument to ex- hibit an uncomputable real. Roughly speaking, a computable real is one that one can calculate digit by digit, that there is an algorithm for approximating as closely as one may wish. All the reals one normally en- counters in analysis are computable, like , p 2 and e. But they are much scarcer than the un- computable reals because, as Turing points out, the computable reals are countable, whilst the uncomputable reals have the power of the con- tinuum. Furthermore, any countable set of re- als has measure zero, so the computable reals have measure zero. In other words, if one picks a real at random in the unit interval with uni- form probability distribution, the probability of obtaining an uncomputable real is unity. One may obtain a computable real, but that is in- nitely improbable.
Citations (67)
... In the theory of computability, Turing [17] introduced a fundamental result on the unsolvability of the Halting Problem. Chaitin [5] defines the Halting probability and shows that it is an uncomputable irrational number Ω that if were known, then based on the first n bits of its binary expansion, it would be possible to answer the halting problem for all programs of length n or less. ...
- Citing Article
- Full-text available
June 2011
Manuscrito
... As such their emphasis is not on general computations but rather on showing feasibility of specific computations in the laboratory. Articles [8, 9, 13, 31, 48] directly address Turing completeness, but the algorithmic or programming aspects are not easy to see. How our approach is different: Contrary to several existing models , our atomic notion (the " blob " ) carries a fixed amount of data and has a fixed number of possible interaction points with other blobs. ...
- Citing Chapter
January 2011
... We have as Boltzmann entropy, the equalization of particles giving an evenness. Information entropy is regarded as the ability to predict (Shannon, 1948;Chaitin, 2001). It is uniformity that allows the experimental scientific method (Whewell, 1847), but is not uniformity randomness? ...
- Citing Chapter
January 2001
... However, in mathematics, randomness is closely tied to undecidability-the existence of propositions that cannot be decided within a given axiomatic framework [13]. ...
- Citing Chapter
January 2002
... Still others, -like A.V. Aho [2], or Searle (again) [18] require that there must be some computational model supporting the computation. Fredkin [13] has put it like this: The thing about a computational process is that we normally think of it as a bunch of bits that are evolving over time plus an engine -the computer. Deutsch [10] holds an even tougher view: there must also be a physical realization of a computational model. ...
- Citing Chapter
December 2012
... Mathematics relies mainly on numerical values, and researchers initially embraced natural numbers, followed by positive integers, negative integers, rational and irrational numbers, and, eventually, the concept of imaginary numbers. Refer to Figure 2, depicting a number line and the categorizations of numbers [6]. Integers are categorized into positive, negative, and zero. ...
- Citing Article
July 1999
Nature
... The cumulative evolution model is defined in [36,37] as a sequence of sole (resourceunbounded) Turing machines that evolve over time due to the transformations effected by randomly generated algorithmic mutations. Thus, in accordance with evolutionary biology, not only are these Turing machines subjected to randomly generated mutations and natural selection, but they may also inherit information from their predecessors. ...
- Citing Article
December 2012
... Another analogue is from recursive analysis: Specker sequences of computable numbers converge to an uncomputable limit [35][36][37][38]. One example is Chaitin's constant, the halting probability of prefix-free program codes on a universal computer [39,40], whose rate of convergence is tied to the halting time, and therefore, 'grows faster' than any computable function. ...
- Citing Article
January 2010
Notices of the American Mathematical Society
... We do not assert the metrological viability of sociocognitive measurement without recognizing that local realizations and interpretations of even physical units of measurement may vary across communities of research and practice in divergent ways (Galison, 1997;Tal, 2014;Woolley & Fuchs, 2011); that irreducible randomness, incompleteness, and inconsistencies permeate elementary number theory, arithmetic, and Newtonian mechanics (Chaitin, 2003); and that longstanding calls for clearly distinguishing levels of complexity (Rousseau, 1985;Star & Ruhleder, 1996, p. 118) typically go unheeded. ...
- Citing Article
January 1996
JOURNAL OF UNIVERSAL COMPUTER SCIENCE
... Chaitin is a well known mathematician who proved that in mathematics exists an infinite number of statements which can not be proved nor disproved from the existing set of axioms. He discussed this in an article from the Scientific American magazine in 2006 and before that in standard scientific publications [65]. Similarly before him, Gödel presented his incompleteness theorem. ...
- Citing Article
November 1993
Applied Mathematics and Computation