Wyniki wyszukiwania - Biblioteka Nauki

1

“Mathesis Universalis na nasze czasy: Wkład Fregego, Cantora i Gödla

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Marciszewski W.

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nr 4(190)

455-471

EN

The phrase "Mathesis Universalis" (MU) denotes the project of unifying the whole of knowledge with the help of mathematical methods; under this title the project has appeared at the eve of modernity, having been somehow anticipated by antiquity and middle ages. Its main proponents were Descartes and Leibniz. Leibniz's approach is radically formalistic, being thereby tractable for a machine, while Descartes's is decidedly antiformalistic. The article focuses on Leibniz's project as one being continued in modern science. Its crucial idea that MU should operate through an universal symbolism and a logical calculus is being nowadays realized with respect to the whole of mathematics. The fact that this gets accomplished is owed to those devices, to wit an ingenious notation and a logical calculus, which have been created firstly by Gottlob Frege (1879). An essential contribution is due to Georg Cantor as the author of the theory dealing with powers of sets. This theory introduces the distinction of countably infinite sets and those having the power of continuum. This makes it possible to prove the two propositions fundamental for contemporary MU, one due to Gödel, stating the undecidability of natural arithmetic, the other stating the undecidability of logic. The latter has been demonstrated by Alan Turing. He proved that the programs able to decide about the values of a definite function (i.e. to compute the values) form a countable set, while the set of problems to be decided possesses the power of continuum. Hence there are problems which are not decidable in a machine-like manner what implies, in turn, undecidability of logic. An analogous result of Gödel is accompanied by his very important statement that the limitations of mechanical decidability are just relative to the current state of a formalized system, and can get overcome by creative constructing new devices: primitive notions, axioms and methods of research. Thus the science enjoys a perspective of incessant progress. Such a dynamic vision is a feature to essentially distinguish the modern MU version from the classical one having been static.

2

On advancing frontiers of science. A pragmatist approach

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Marciszewski W.

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tom 47

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nr 4

51-71

EN

The pragmatist approach, as stated in this essay, takes into account two features of knowledge, both having an enormous potential of growth: the scope of science, whose frontiers can be infinitely advanced, while firmness of its propositions grows with consolidating once attained frontiers. An opposite view may be called limitativist as it conservatively sticks to some a priori limiting principles which do not allow progressing in certain directions. Some of them influence science from outside, like ideological constraints, others are found inside science itself. The latter can be exemplified by principles like these: (1) there can be no action at a distance; (2) there are no necessary truths; (3) there are no abstract objects. The first might have happened to limit physics with rejecting the theory of gravitation. The second entails that arithmetical propositions are either devoid of (clasical) truth or are not necessary; this would limit its role to being a mere calculating machine, without giving any insights into reality. The third, for instance, limits logic to the first-order level (since in the second-order variables range over abstract sets). The history of ideas shows that such limiting principles, had they been obeyed, would have hindered some great achievements of science. This is why we should not acknowledge them as necessarily true, that is, winning in confrontation with any view contrary to them. Such principles should compete with other ones on equal terms in obtaining as high degree of epistemic necessity as they may prove worth of. To the core of the pragmatist approach there belongs treating epistemic necessity as a gradable attribute of propositions. In accordance with ordinary usage, "necessary" is a gradable adjective, having a comparative form. The degree of epistemic necessity of a scientific statement depends on how much it is needed for the rest of the field of knowledge (Quine's metaphor). The greater damage for knowledge would be caused by getting rid of the point in question, the greater is its epistemic necessity. At the top of such a hierarchy are laws of logic and arithmetic. Among physical laws at a very high level there is the law of gravitation, owing both to its universality, that is, a colossal scope of possible applications (advancement of frontiers), and its having been empirically confirmed with innumerable cases (consolidation of frontiers). Such a success has proved possible owing to the bold transgression of the limiting principle 1 (see above), and this has resulted in so high a degree of unavoidability.

PL

Jak to wyrażamy w prezentowanym tekście, w ramach ujęcia pragmatycznego bierze się pod uwagę dwie charakterystyczne cechy wiedzy, przy czym obydwie mają ogromną możliwość wzrostu: zasięg nauki, której granice mogą się przesuwać w nieskończoność i stanowczość jej sądów, która wzrasta razem z umocnieniem granic osiągniętych. Przeciwnym podejściem, które można nazwać limitywistycznym, jest takie, w którym konserwatywnie a priori formułuje się limitacyjne (lepiej: limitujące)zasady nie pozwalające na postęp w niektórych kierunkach. Niektóre z takich zasad wpływają na naukę z zewnątrz, np. wymogi ontologiczne, inne możemy znaleźć w samej nauce. Przykładami tych ostatnich mogą być takie zasady jak: (1) nie ma ruchu bez odległości,(2) nie istnieją prawdy konieczne, (3) nie ma obiektów abstrakcyjnych. Pierwsza z wymienionych zasad być może została sformułowania w celu ograniczenia fizyki, w której odrzuca się teorię grawitacji. Druga z nich pociąga za sobą koncepcję, zgodnie z którą twierdzenia arytmetyczne są bądź pozbawione prawdy w sensie klasycznym, bądź nie są konieczne. Trzecia z wymienionych zasad ogranicza logikę do logiki pierwszego rzędu (ponieważ w rzędzie drugim zmienne przebiegają abstrakcyjne zbiory). Historia idei pokazuje nam, że tego rodzaju zasady limitacyjne (limitujące),którym były owe idee podporządkowane, utrudniały niektóre ważne osiągnięcia naukowe(lepiej: stały na przeszkodzie w realizacji niektórych przedsięwzięć naukowych). Z tego też powodu nie powinniśmy ich uznawać jako koniecznie prawdziwe tj. jako zwyciężające w konfrontacji z każdym przeciwnym względem nich poglądem. Tego rodzaju zasady powinny na równi rywalizować z innymi co do przyznania im tak wysokiego stopnia epistemicznej konieczności, jak pozwala na to ich uzasadnienie. Do istoty podejścia pragmatycznego należy traktowanie epistemicznej konieczności jako stopniowalnego atrybutu sądów. W zgodzie z potocznym użyciem „konieczny”jest przymiotnikiem stopniowalnym, wobec tego że ma formę względną (porównawczą). Stopień epistemicznej konieczności twierdzenia naukowego zależy od tego, jak bardzo jest ono niezbędne w ramach danego zakresu wiedzy (metafora Quine’a). Tym większą szkodą dla wiedzy byłoby porzucenie takiego punktu widzenia, im wyższa jest owa epistemiczna konieczność. Na szczycie tego rodzaju hierarchii są prawa logiki i arytmetyki. Do fizycznych praw na wysokim poziomie (epistemicznej konieczności)zaliczylibyśmy prawo grawitacji, w związku zarówno z jego uniwersalnością, tj. kolosalnym zakresem możliwych aplikacji (przesuwanie granic) jak i faktem, iż jest ono empirycznie potwierdzone przez niezliczone przypadki (zcalenie granic).

3

A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

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Marciszewski W.

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tom 153

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nr 2

125-40

EN

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions $c_p(X)$ onto $c_p(X)$ × ℝ$. In particular, $c_p(X)$ is not linearly homeomorphic to $c_p(X)$ × ℝ$. One of these examples is compact. This answers some questions of Arkhangel'skiĭ.

4

The Computational and Pragmatic Approach to the Dynamics of Science

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Marciszewski W.

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nr 1

31-67

EN

Science means here mathematics and those empirical disciplines which avail themselves of mathematical models. The pragmatic approach is 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 computational approach 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 computertheoretic representation of intuition or invention. Computational approach in another sense means considering problem-solving processes in terms of logical gates, supposed to be a physical basis for solving problems with a reasoning. Pragmatic rationalism about science, seen at the background of classical rationalism (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 version 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’s exponential and Thomas Kuhn’s paradigm model (the most familiar instances). This essay suggests considering Turing’s idea of oracle as a complementary model to explain most adequately, in terms of exceptional inventiveness, the dynamics of mathematics and mathematizable empirical sciences.

5

On sequential convergence in weakly compact subsets of Banach spaces

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Marciszewski W.

Studia Mathematica

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1994-1995

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tom 112

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nr 2

189-194

EN

We construct an example of a Banach space E such that every weakly compact subset of E is bisequential and E contains a weakly compact subset which cannot be embedded in a Hilbert space equipped with the weak topology. This answers a question of Nyikos.

6

Kazimierz Ajdukiewicz and the Polish Debate on Universals

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Marciszewski W.

Filozofia Nauki

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1999

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tom 7

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nr 3-4

5-13

PL

When discussing Kazimierz Ajdukiewicz's role in philosophy, it is worthwhile recalling his participation in scholarly controversies. It was characteristic of his open mind that his taking part in debates was motivated by a vivid interest in various ways of thinking. Ajdukiewicz's intellectual power consisted, so to speak, in his ability of not to understand. This ability has brought him success in some important debates, concerning i.a. the classical logical concept of contradiction and the debate on universals raised in modern Poland with the nominalistic program of Stanislaw Lesniewski and Tadeusz Kotarbiński. In this latter debate Ajdukiewicz shows that when one says that individuals exist, the word „exist" refers to something different that in the statement that universals exist. In other words, the functor „is" has a different category in the definition of an individual from that appearing in the definition of a universal; hence there must be two different senses of the word „exist".

7

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

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Marciszewski W.

Studies in Logic, Grammar and Rhetoric

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2015

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tom 40

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nr 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.

8

Some remarks on universality properties of $ℓ_∞/c₀$

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Krupski M. , Marciszewski W.

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nr 2

187-195

EN

We prove that if 𝔠 is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into $ℓ_∞/c₀$. We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $ℓ_∞/c₀$, but fails to embed isometrically. As far as we know it is the first example of this kind.

9

Failure of the Factor Theorem for Borel pre-Hilbert spaces

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Dobrowolski T. , Marciszewski W.

Fundamenta Mathematicae

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2002

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tom 175

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nr 1

53-68

EN

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σδσ}$-subset of X and contains a retract R so that $R × E^{ω}$ is not homeomorphic to $E^{ω}$. This shows that Toruńczyk's Factor Theorem fails in the Borel case.

10

A contribution to the topological classification of the spaces Ср(X)

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Cauty R. , Dobrowolski T. , Marciszewski W.

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tom 142

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nr 3

269-301

EN

We prove that for each countably infinite, regular space X such that $C_p(X)$ is a $Z_σ$-space, the topology of $C_p(X)$ is determined by the class $F_0(C_p(X))$ of spaces embeddable onto closed subsets of $C_p(X)$. We show that $C_p(X)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set $Ω_α$ for the multiplicative Borel class $M_α$ if $F_0(C_p(X)) = M_α$. For each ordinal α ≥ 2, we provide an example $X_α$ such that $C_p(X_α)$ is homeomorphic to $Ω_α$.

11

Classification of function spaces with the pointwise topology determined by a countable dense set

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Dobrowolski T. , Marciszewski W.

Fundamenta Mathematicae

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1995

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tom 148

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nr 1

35-62

12

Konferencja „Moc obliczeniowa dla badań społecznych. Idee von Neumanna”

38%

Marciszewski W.

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nr 1

13

The ethics of counterexample and the endeavour for great syntheses in social sciences

38%

Marciszewski W.

Nauka

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2010

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nr 3

14

A function space C(K) not weakly homeomorphic to C(K)×C(K)

38%

Marciszewski W.

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tom 88

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nr 2

129-137