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Uniwersalność systemów funkcyjnych a całkowitość dziedzin funkcji – granice konfliktu i wzajemnego wpływu

100%
Mycka J.
Zagadnienia Filozoficzne w Nauce
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2017
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issue 63
31-58
PL
The article presents several examples of different mathematical structures and interprets their properties related to the existence of universal functions. In this context, relations between the problem of totality of elements and possible forms of universal functions are analyzed. Furthermore, some global and local aspects of the mentioned functional systems are distinguished and compared. In addition, the paper attempts to link universality and totality with the dynamic and static properties of mathematical objects and to consider the problem of limitations in the construction of structures combining harmoniously the availability of information at the local and global level.
2
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Gödel’s Philosophical Challenge (to Turing)

75%
Sieg W.
Studia Semiotyczne
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2020
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vol. 34
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issue 1
EN
The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind.
3
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O różnych sposobach rozumienia analogowości w informatyce

71%
Stacewicz P.
Semina Scientiarum
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2017
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vol. 16
EN
Two different types of analog computations are discussed in the paper: 1)analog-continuous computations (performed physically upon continuous signals),2) analog-analogical computations (performed naturally by means of socalled natural analogons of mathematical operations). They are analyzed withregard to such questions like: a) are continuous computations physically implementable?b) what is the actual computational power of different analogtechniques? c) can natural (empirical) computations be such reliable as digital?d) is it possible to develop universal analog computers (assuming that theyshould be functionally similar to universal Turing machine)? Presented analysesare rather methodological than formal.
4

On Accelerations in Science Driven by Daring Ideas: Good Messages from Fallibilistic Rationalism

51%
Marciszewski W.
Studies in Logic, Grammar and Rhetoric
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2015
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vol. 40
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issue 1
19-41
EN
The first good message is to the effect that people possess reason as a source of intellectual insights, not available to the senses, as e.g. axioms of arithmetic. The awareness of this fact is called rationalism. Another good message is that reason can daringly quest for and gain new plausible insights. Those, if suitably checked and confirmed, can entail a revision of former results, also in mathematics, and - due to the greater efficiency of new ideas - accelerate science’s progress. The awareness that no insight is secured against revision, is called fallibilism. This modern fallibilistic rationalism (Peirce, Popper, Gödel, etc. oppose the fundamentalism of the classical version (Plato, Descartes etc.), i.e. the belief in the attainability of inviolable truths of reason which would forever constitute the foundations of knowledge. Fallibilistic rationalism is based on the idea that any problem-solving consists in processing information. Its results vary with respect to informativeness and its reverse - certainty. It is up to science to look for highly informative solutions, in spite of their uncertainty, and then to make them more certain through testing against suitable evidence. To account for such cognitive processes, one resorts to the conceptual apparatus of logic, informatics, and cognitive science.
5
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The Computational and Pragmatic Approach to the Dynamics of Science

51%
Marciszewski W.
Filozofia i Nauka
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2020
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vol. 8
6
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The Computational and Pragmatic Approach to the Dynamics of Science

51%
Marciszewski W.
Filozofia i nauka. Studia filozoficzne i interdyscyplinarne
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2020
|
vol. 8/1
31-67
EN
Sciencemeans here mathematics and those empirical disciplines which avail themselves of mathematical models. The pragmaticapproachis conceived in Karl R. Popper’s The Logic of Scientific Discovery(p.276) sense: a logical appraisal of the success of a theory amounts to the appraisal of its corroboration. This kind of appraisal is exemplified in section 6 by a case study—on how Isaac Newton justified his theory of gravitation. The computationalapproach in problem-solving processes consists in considering them in terms of computability: either as being performed according to a model of computation in a narrower sense, e.g., the Turing machine, or in a wider perspective—of machines associated with a non-mechanical device called “oracle”by Alan Turing (1939). Oracle can be interpreted as computer-theoretic representation of intuitionor invention. Computational approach in an-other sense means considering problem-solving processes in terms of logical gates, supposed to be a physical basis for solving problems with a reasoning.Pragmatic rationalismabout science, seen at the background of classical ration-alism (Descartes, Gottfried Leibniz etc.), claims that any scientific idea, either in empirical theories or in mathematics, should be checked through applications to problem-solving processes. Both the versions claim the existence of abstract objects, available to intellectual intuition. The difference concerns the dynamics of science: (i) the classical rationalism regards science as a stationary system that does not need improvements after having reached an optimal state, while (ii) the pragmatical ver-sion conceives science as evolving dynamically due to fertile interactions between creative intuitions, or inventions, with mechanical procedures.The dynamics of science is featured with various models, like Derek J.de Solla Price’sexponential and Thomas Kuhn’s paradigm model (the most familiar instanc-es). This essay suggests considering Turing’s idea of oracle as a complementary model to explain most adequately, in terms of exceptional inventiveness, the dynam-ics of mathematics and mathematizable empirical sciences.
7
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Diagonal Anti-Mechanist Arguments

51%
Kashtan D.
Studia Semiotyczne
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2020
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vol. 34
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issue 1
EN
Gödel’s first incompleteness theorem is sometimes said to refute mechanism about the mind. §1 contains a discussion of mechanism. We look into its origins, motivations and commitments, both in general and with regard to the human mind, and ask about the place of modern computers and modern cognitive science within the general mechanistic paradigm. In §2 we give a sharp formulation of a mechanistic thesis about the mind in terms of the mathematical notion of computability. We present the argument from Gödel’s theorem against mechanism in terms of this formulation and raise two objections, one of which is known but is here given a more precise formulation, and the other is new and based on the discussion in §1.
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