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The Computational and Pragmatic Approach to the Dynamics of Science

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Marciszewski W.
Filozofia i nauka. Studia filozoficzne i interdyscyplinarne
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2020
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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.
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COMPUTATIONAL DYNAMICS OF COMPLEX SYSTEMS. A NEW WAY OF DOING SCIENCE

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Marciszewski W.
Studies in Logic, Grammar and Rhetoric
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2007
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issue 11(24)
107-123
EN
The essay comments on Klaus Mainzer's book 'Thinking in Complexity. The Computational Dynamics of Matter, Mind and Mankind' (Springer 2004). The way of perceiving the world called 'thinking in complexity' proves revolutionary in science and philosophy as it radically changes our world perspective. This is Mainzer's point which is reported in some details. Besides such reports, the essay poses some related questions among which are the following. The revolutionary initiative is due to mathematical logic at this point from which theoretical computer science has emerged, that is, the discoveries concerning computability as due to Gödel, Church, Turing, Post etc. This software complexity is entangled with the complexity of hardware, the latter meaning dynamic systems changing in time as studied by physics, technology, biology etc. The essay distinguishes two uses of 'complexity'. (1) The most general one regarding any system, i.e. any set having elements interrelated with one another, which are either (A) abstract and static, as are the domains of mathematical theories etc. or (B) dynamic, that is, changing in time as are bodies, ecosystems, minds, societies, etc. (2) A more specific notion refers only to those dynamic systems which behave in a way we call nonlinear. There is a historical parallel between hardware complexity and the ancient Atomists' notion of complexity on the one hand, and between sofware complexity and the Stoic idea of 'rational seeds' as programming reality. Against the background of the old atomic notion, there appears the striking novelty of the modern definition of complexity in terms of feedbacks and non-linear behaviour. Special attention is paid to the problem of insights which produce new algorithms to reduce complexity of some computational procedures, esp. mathematical proofs. This is perfectly exemplified with Gödel's statement to the effect that: passing to the logic of the next higher order has the effect, not only of making formally (i.e. algorithmically) provable certain propositions that were not formally provable before, but also of making it possible to shorten, by an extraordinary amount, infinitely many of the formal proofs already available. Such a shortening means reducing complexity.
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ON THE POWER AND GLORY OF DEDUCTIVISM

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Marciszewski W.
Studies in Logic, Grammar and Rhetoric
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2009
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issue 16(29)
353-355
EN
This comment is meant as a couple of glosses on the margin of Dale Jacquette'e contribution (this issue) 'Deductivism in Formal and Informal Logic'.
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Intuicja matematyczna w ujęciu nowoczesnego racjonalizmu

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Marciszewski W.
Edukacja Filozoficzna
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2019
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issue 68
169-194
PL
Nowoczesny racjonalizm, w skrócie: neoracjonalizm, jest prądem, w którym mieszczą się m.in. Frege, Russell, Church, Bernays, Gödel (najwyraziściej), Quine, Putnam, Kreisel, Chaitin. Przypisuje on istnienie abstraktom, a umysłowi zdolność ich poznawania w sposób intuicyjny. W przypadku obiektów matematycznych, jak uzyskiwane w wyniku abstrakcji zbiory, liczby, algorytmy etc., mówimy o intuicji matematycznej; na niej koncentruje się artykuł. Nazwa „nowoczesny” uwydatnia różnicę w stosunku do racjonalizmu klasycznego z XVII w. Polega to na poniechaniu tezy o doskonałej wiarogodności intuicji matematycznej. Neoracjonalizm opowiada się w kwestii intuicji za fallibilizmem oraz stopniowaniem wiarogodności: tym wyższy jej stopień, im mocniej jest ugruntowana we wrodzonym wyposażeniu biologicznym (co oznacza natywizm w stylu Chomsky’ego) i w doświadczeniu zmysłowym. Ze względu na fallibilizm, pewne zbliżenie do empiryzmu i odniesienie do biologii, mylące jest nazywanie tego prądu „platonizmem”, stąd propozycja nazwy „neoracjonalizm”.
EN
Modern rationalism, abbr. neorationalism, is a philosophical orientation to include Frege, Russell, Church, Bernays, Gödel (most distinctly), Quine, Putnam, Kreisel, Chaitin, etc. It claims the existence of abstract entities as classes, numbers, algorithms etc., and mind’s ability to intuitively learn about them. When meaning mathematical entities, we speak of mathematical intuition, being in focus of this paper. The adjective “modern” highlights the difference in relation to the classical rationalism of the 17-th century. The modern one denies the mathematical intuition to possess a perfect reliability, and sees it as a gradable faculty which does not enjoy an assured infallibility. The degree of reliability depends on how close is intuition to an inborn biological equipment (what means nativism in Chomsky’s style), and to sensory experiences. What is called neorationalism in this paper happens to be called mathematical platonism by other authors. However, on account of fallibilism, a certain tilt toward empiricism, and a significant reference to biology, “Platonism” (as lacking these traits) proves to be less fitting term than is “neorationalism”.
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Does Science Progress Towards Ever Higher Solvability Through Feedbacks Between Insights and Routines?

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Marciszewski W.
Studia Semiotyczne
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2018
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vol. 32
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issue 2
153-185
EN
The affirmative answer to the title question is justified in two ways: logical and empirical. (1) The logical justification is due to Gödel’s discovery (1931) that in any axiomatic formalized theory, having at least the expressive power of PA (Peano Arithmetic), at any stage of development there must appear unsolvable problems. However, some of them become solvable in a further development of the theory in question, owing to subsequent investigations. These lead to new concepts, expressed with additional axioms or rules. Owing to the so-amplified axiomatic basis, new routine procedures like algorithms, can be reached. Those, in turn, help to gain new insights which lead to still more powerful axioms, and consequently again to ampler algorithmic resources. Thus scientific progress proceeds to an ever higher scope of solvability. (2) The existence of such feedback cycles – in a formal way rendered with Turing’s systems of logic based on ordinal (1939) – gets empirically supported by the history of mathematics and other exact sciences. An instructive instance of such a process is found in the history of the number zero. Without that insight due to some ancient Hindu mathematicians there could not arise such an axiomatic theory as PA. It defines the algorithms of arithmetical operations, which in turn help intuitions; those, in turn, give rise to algorithmic routines, not available in any of the previous phases of the process in question. While the logical substantiation of the point of this essay is a well-established result of logico-semantic inquiries, its empirical claim, based on historical evidences, remains open for discussion. Hence the author’s intention to address philosophers and historians of science, and to encourage their critical responses.
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