The main aim of the paper was to draw attention to Wacław Sierpiński as not only a great mathematician but also a philosopher. We undertook the attempt of reconstruction of Sierpiński’s philosophy. To aim this goal we mainly based ourselves on Sierpiński’s habilitation lecture entitled _The concept of correspondence in mathematics_. The complementation of Sierpiński’s philosophical views were conclusions from his mathematical achievements, his scheme of research on The Axiom of Choice, and his attitude to this axiom.

Praca poświęcona jest prezentacji i krytycznej ocenie poglądów filozoficznych Jana Franciszka Drewnowskiego na matematykę i logikę. Podstawą rozważań są cztery źródła. Podstawowym źródłem jest Drewnowskiego "Zarys programu filozoficznego", dalej dwa artykuły, a mianowicie "Stosowanie logiki symbolicznej w filozofii" oraz "Uwagi o stosowaniu logiki symbolicznej", w końcu fragmenty dziennika. Pokazuje się w niej, jak Drewnowski rozumiał matematykę i teorie matematyczne, jak widział logikę i jej rolę w nauce oraz na ile był zaznajomiony ze współczesnymi osiągnięciami w zakresie logiki matematycznej i podstaw (...) matematyki oraz z ich konsekwencjami dla filozofii matematyki. (shrink)

Artykuł dotyczy zagadnienia, w jakim sensie można stosować kategorię wyjaśnienia do interpretacji filozofii matematyki Kurta Gödla. Gödel – jako realista matematyczny – twierdzi bowiem, że w wypadku matematyki mamy do czynienia z niezależnymi od nas faktami. Jednym z owych faktów jest właśnie rozwiązywalność wszystkich dobrze postawionych problemów matematycznych – i ten fakt domaga się wyjaśnienia. Kluczem do zrozumienia stanowiska Gödla jest identyfikacja założeń, na których się opiera: metafizyczny realizm: istnieje uniwersum matematyczne, ma ono charakter obiektywny, niezależny od nas; (...) optymizm epistemologiczny: jesteśmy wyposażeni w wystarczająco dobre środki poznawcze, aby uzyskać wgląd w owo uniwersum. Pojęcie rozwiązania problemu matematycznego Gödel rozumie znacznie szerzej niż jako podanie matematycznego dowodu – chodzi raczej o znalezienie wiarogodnych aksjomatów, prowadzących do rozwiązania. Stawiany w artykule problem analizuję na przykładzie hipotezy kontinuum. (shrink)

The origins of mathematics, a close connection and interpenetration of its parts, and uniform procedures of dealing with the mathematical matter - all of them speak in favour of the integrality of mathematics. It seems that a strong argument for such a view is a fundamental object of contemporary mathematics; namely a real line, which contains real numbers (so arithmetics as well) and constitutes a basis of geometry, mathematical analysis and all derivative branches. From the basic-structures perspective it is clear (...) that the real line is an exceptionally complex structure, for it contains the ordered structure (generated by the less-than relation), the algebraic structure (generated by addition and multiplication), the gemetrical structure (generated by translations and reflections) and the topological structure (generated by open intervals). This example explains, at least to some extent, the integrality and also the vivacity of mathematics. On the other hand, the integrality has not been confirmed by comprising the whole mathematics in one axiomatised, deductive theory. Moreover, the increasing „volume” of mathematics, unwillingness of mathematicians to cross the specialisation-barriers, emphasis on utility (models) and difficulties with axiomatisations of some parts of mathematics cast some doubts on the integrality in question. (shrink)

Na pytanie o rozumienie pojecia przyrody udzielano w historii filozofii różnych odpowiedzi. W książce Mutschlera możemy znaleźć obszerny przegląd współczesnego rozumienia tego pojecia. Zasadniczy przełom w poglądach na przyrodę dokonał się na początku XVII wieku, kiedy to przy jej badaniu zaczęto się odwoływać do powtarzalnych eksperymentów, do metod ilościowych oraz do matematyki. Przykładem takiego postępowania jest metoda zaproponowana przez Galileusza, dla którego „księga przyrody" jest napisana w języku matematyki. Przed Galileuszem nikt nie wpadł na pomysł, zęby przyrody traktować (...) jako księgę napisaną w języku matematyki. Zwrotu „księga przyrody" lub „księga natury" używano już w średniowieczu i przeciwstawiano „księdze objawienia". Tego rodzaju przeciwstawienie odpowiadało równocześnie podziałowi na to, co „naturalne" i na to, co „ponadnaturalne". (shrink)

In the contemporary philosophy of mathematics there is an ongoing discussion concerning the issue of the justification of mathematical axioms and independent sentences. The works of P. Maddy in which the author focuses on the set theory are extensive studies devoted to that problem. In one of the monographs she deals with the problem of the justification of set theory's sentences in a new way, assuming a different metaphilosophical standpoint. In the paper (which is the first part of the two-part (...) whole) Maddy's conception is faithfully presented. The author makes no comments of his own, except for the first - introductory - chapter and footnotes, in which some additional explanations and comments can be found. (shrink)

Na pytanie o rozumienie pojecia przyrody udzielano w historii filozofii różnych odpowiedzi. W książce Mutschlera możemy znaleźć obszerny przegląd współczesnego rozumienia tego pojecia. Zasadniczy przełom w poglądach na przyrodę dokonał się na początku XVII wieku, kiedy to przy jej badaniu zaczęto się odwoływać do powtarzalnych eksperymentów, do metod ilościowych oraz do matematyki. Przykładem takiego postępowania jest metoda zaproponowana przez Galileusza, dla którego „księga przyrody" jest napisana w języku matematyki. Przed Galileuszem nikt nie wpadł na pomysł, zęby przyrody traktować (...) jako księgę napisaną w języku matematyki. Zwrotu „księga przyrody" lub „księga natury" używano już w średniowieczu i przeciwstawiano „księdze objawienia". Tego rodzaju przeciwstawienie odpowiadało równocześnie podziałowi na to, co „naturalne" i na to, co „ponadnaturalne". (shrink)

The article deals with philosophy of mathematics called quasi-empiricism, with special attention paid to concepts of changes in mathematics as a result of social activity during its development. Three outlooks on historical changes in the development of mathematics are presented: Wilder's description of mathematical practise in a cultural framework, Garbiner's analysis of „scientific revolution” in mathematics in the end of 18th century and Kitcher's criticism of invariability and cummulativity of mathematical knowledge and his application of Kuhn's philosophy of science to (...) mathematics. (shrink)

The article starts with various definitions of music and its components (taken es-pecially from medieval sources). Then the Statuit introit is presented as the main example useful for mathematical analysis in the paper. We discuss the distinc-tion between a musical work and its performance and give a review of possible representations/ notations of music with their short mathematical characteristics. The next point is devoted to the concept of modality, Gregorian scales and selected theories linking musical experience with mathematics. We use (...) the Statuit introit as the illustration for the construction of its statistical analysis (for notes, intervals and progresses). The results of this analysis serve as a reference point to indicate the connection between the process of musical composition and the apparatus of mathematical linguistics. Based on the above considerations we suggest that the pattern analysis could be useful as a tool unifying some aspects of music and mathematics and explaining the perception of musical works. (shrink)

It has been widely believed that David Hilbert, a precursor of formalism, consid-ered mathematics an ordinary “game of signs”. In this paper, I try to show that Hilbert worked out and applied in his research a cohesive, holistic and organic, as well as universal picture of mathematics (as a supranational discipline) and a vision of its development. This scientific area was supposed to constitute a complex construction, and was perceived by him in an optimistic and humanistic way (it enabled solution (...) to any research problem and fulfilled diverse human needs). External experience (e.g. in the case of geometric pictures) and abstraction of its internal structures (a priori state, mathematical analysis of infinity) are particularly important for this exact science. Mathematics creates a bridge between theory and practice, between thinking and ob-servation. (shrink)

Autorka artykułu odniosła się do założeń rozprawy doktorskiej Mariana Massoniusa, a także uwzględniła założenia jego kilku pomniejszych prac dotyczących sposobu ujęcia Kantowskiego systemu filozofii krytycznej. Poddała analizie problem sądów analitycznych i syntetycznych oraz sądów syntetycznych a priori. Podjęła zagadnienie możliwości istnienia czystej matematyki, problem czasu i przestrzeni, opisała także sposoby ujęcia Kantowskich antynomii oraz teorii poznania rozwijanej w kontekście idealizmu oraz realizmu czasu i przestrzeni. Uwagę skupiła również na problemie umiarkowanego agnostycyzmu w filozofii Massoniusa oraz ujęciu filozofii w perspektywie (...) nauki. Podjęła również wysiłek udzielenia odpowiedzi na fundamentalne pytanie: dlaczego Marian Massonius, podobnie jak w 1865 roku wczesny neokantysta Otto Liebmann, rzucał wyzwanie powrotu do Kanta (Zurück zu Kant!) oraz głosił postulat koniecznego wypracowania krytycznej formuły apriorycznych form umysłu. (shrink)

The paper is an attempt to present the cognitive functions of mathematics in relation to empirical sciences. Firstly - mathematics is a 'generator' of mathematical categories used in natural sciences. In this sense mathematics is a science about the tools of cognition which it creates or perfects. Secondly - mathematics plays the role of 'prism' through which we view the world because some phenomena e.g. from the micro- and macro-world can only be seen through the prism of mathematical structures and (...) notions. And thirdly - mathematics is also a 'selector' of cognitive content. It eliminates from the cognitive field these phenomena which presently can't be grasped by the existing mathematical structures. It is worth noticing that the functions of mathematics: 'selector' and 'prism' are complementary in some sense; although they are opposites ('prism' lets you 'see' something, and 'selector' restricts this 'seeing'), they don't exclude each other. (shrink)

In the first part of the article, Giuseppe Veronese’s concept of arithmetical constructivism is reconstructed from his dispersed remarks. It is pointed out that although for Veronese time is a necessary condition for the construction of natural numbers by an individual subject and the subject cognizes time in an a priori way, it is not a (proto-)intuition of the subject. This is a fundamental difference between the concept proposed by Veronese and the constructivism of Kant and Brouwer. Veronese’s justification of (...) the subject’s ability to cognize infinity, represented in his text by infinite sequences, is analyzed in the second part of the paper. The only condition for the cognition of the infinite sequence is to have a rule according to which its successive terms “follow one another.” The price of the ability to cognize the infinitive “whole” (sequence) may be the uncognizability of certain “parts” (terms of sequence). Infinite sequences, cognizable according to Veronese, are allowed as objects of mathematical research, although they do not meet the condition of constructability. (shrink)

This is an attempt to defend Field's nominalistic program from the criticism raised by K. Wójtowicz in his article. The author argues for the following theses: (a) that Wójtowicz uses the notion of „mathematical theory” broader than Field does it; (b) that he misinterprets the conception of the „abstract counterparts” of nominalistic statements; (c) and that his general evaluation of Field's program is based on too high methodological standards which he applies to the possible nominalistic versions of empirical theories. The (...) second part of this paper contains an attempt to generalize the results of Field's analysis. The following fact is proved: every open sentence expressible in the language of an empirical theory and being empirically contentful is implicitly definable by the set of certain qualitative predicates. In the case of first-order language this result can be strengthened via Beth's definability theorem, to the theorem stating that every open sentence fulfilling conditions formulated above is definable explicitly with the help of certain nominalistic formula. The philosophical significance of this result is that each mathematized empirical theory for which representation theorem is true, can be translated into purely qualitative version. (shrink)

The article is an attempt at collecting and systematising views on the role and place of computers in mathematics, in particular the views on the consequences of using computers in proving mathematical theorems. The following issues are considered in the article: the problem connected with the concept of mathematical proof and its features; the attempts at answering the question whether computer proofs are genuine mathematical proofs; the problems with methods of checking the correctness of classical and computer-assisted proofs; and finally (...) the problem of the reliability of proofs and a related problem of the validity of mathematical conjectures. The main issue under discussion is the problem of philosophical consequences of regarding computer-assisted proofs as rightful in mathematics and especially its possible influence on the opinion that mathematical knowledge is a priori. (shrink)

The aim of this paper is to analyze structuralism as an alternative view to platonism in the philosophy of mathematics. We also try to find out if ontological and epistemological problems of platonism can be avoided by admitting the principles of structuralism. Structuralism claims that mathematical objects are merely positions in structures and have no identity or in general any important features outside these structures. Such view allows to avoid problems of the nature of numbers and other mathematical objects. But (...) the chief motivation of structuralism is in fact epistemological. It offers a nice way of explaining how one can get the knowledge of mathematical objects by the process of abstraction. On the other hand, there are many problems connected with mathematical structuralism. Some of them are analyzed in the paper. We come to the conclusion that mathematical structuralism, up to now, can't express the whole truth about mathematics. (shrink)

Jacek Urbaniec: Platonism in the philosophy of mathematics does not necessarily imply that there are links between the temporal physical world and the eternal world of mathematics..

In the paper there are presented two proposals of the interpretations of the applications of mathematics in the natural sciences - realistic and instrumentalistic. The realistic conception, in accordance with the successes of science, maintains that there exists some kind of correspondence between the mathematical structures and the internal structure of the world. It is expressed in the thesis of the mathematicality of nature. The instrumentalistic approach separates the cognitive content of the scientific theory from the mathematical means of expression (...) of the content. In this approach the mathematical categories do not represent any aspect of the modelled reality. In the final part of the paper the author undertakes such an attempt of approaching the applications of mathematics which, while combining the advantages of the above discussed conceptions, would be devoid of their disadvantages. (shrink)

This is the third part of the study concerning the problem of ontological reductions in mathematics. In this part, the problem of reconstruction of (parts of) mathematics in theories weaker than full ZFC is discussed. The tools from reverse mathematics are used, and the results are discussed from the point of view of various versions of realism (Gödel’s realism, Quine’s quasi-empiricism and Balaguer’s Full-Blooded Platonism). Some problems concerning the possibility of discussing these problem outside the conceptual system of set theory (...) are also addressed. (shrink)

The article summarizes the book Filozofia Jezuitów na ziemiach dawnej Rzeczypospolitej w XIX wieku [The Philosophy of the Jesuit in the Terriroties of the Former Commonwealth: Poland, Lithuania, Belarus and Ukraine in the 19th Century], by Roman Darowski.