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1

Matematyka - nauka o fikcjach?

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2009

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issue 45

3-26

PL

According to mathematical realism, mathematics describes an abstract realm of mathematical entities, and mathematical theorems are true in the classical sense of this term. In particular, mathematical realism is claimed to be the best theoretical explanation of the applicability of mathematics in science. According to Quine's indispensability argument, applicability is the best argument available in favor of mathematical realism. However, Quine's point of view has been questioned several times by the adherents of antirealism. According to Field, it is possible to show, that - in principle - mathematics is dispensable, and that so called synthetic versions of empirical theories are available. In his 'Science Without Numbers' Field follows the 'geometric strategy' - his aim is to reconstruct standard mathematical techniques in a suitable language, acceptable from the point of view of the nominalist. In the first part of the article, the author briefly presents Field's strategy. The second part is devoted to Balaguer's fictionalism, according to which mathematics is indispensable in science, but nevertheless can be considered to be a merely useful fiction.

2

Dowód matematyczny – argumentacja czy derywacja? – część II

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2011

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issue 49

81-97

PL

In the first part of the paper, Azzouni’s derivation–indicator view was presented. In the second part it is analyzed in a detailed way. It is shown, that many problems arise, which cannot be explained in a satisfactory way in Azzouni’s theory, in particular the problem of the explanatory role of proof, of its epistemic role; the relationship between first–order and second–order versions of proofs is also not clear. It is concluded, that Azzouni’s theory does not provide a satisfactory account of mathematical proof, but inspires an interesting discussion. In the article, some of the mentioned problems are discussed.

3

Dowód matematyczny – argumentacja czy derywacja? – część I

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2011

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issue 49

63-80

PL

The article is devoted to the problem of status of mathematical proofs, in particular it tries to capture the relationship between the real, „semantic” notion of mathematical proof, and its formal (algorithmic) counterpart. In the first part, Azzouni’s derivation–indicator view is presented in a detailed way. According to the DI view, there is a formal derivation underlying every real proof.

4

Empiricist Inspirations in Philosophy of Mathematics. Part I

100%

Wójtowicz K.

Filozofia Nauki

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2015

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vol. 23

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issue 3

57-75

PL

The article is devoted to the problem of mathematical empiricism: what could the phrase “to be a mathematical empiricist” possibly mean? There are many interpretations of this stance in philosophy of mathematics. The essay discusses relevant views of Mill, Berkeley, and Carnap. Some detailed questions concerning e.g. the applicability of mathematics, the status of mathematical proof, the status of mathematical axioms, are examined from the point of view of these philosophies.

5

Analysis of the Modal Antirealism

100%

Wójtowicz K.

Filozofia Nauki

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2000

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vol. 8

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issue 2

45-79

PL

In the recent years we can observe a sort of renaissance of the philosophy of mathematics. More and more papers and books are published. A few years ago a new journal (Philosophia Mathematica) devoted exclusively to the philosophy of mathematics started appearing. In the contemporary discussions - especially in the context of the question of the applicability of mathematics to the description of the physical world - the issue of the existence and the ontological status of mathematical objects plays a particular role. Many new conceptions have appeared - both realistic and antirealistic. In the paper one of the antirealistic conceptions formulated in the recent years is presented - namely a conception by C. Chihara [1990]. The author rejects both Gödel's and Quine's arguments for the realistic standpoint. First, these standpoints will be briefly summarised, since they play a significant role in the contemporary philosophy of mathematics - and are particularly important in the context of the discussion upon Chihara's conception. Next, Chihara's arguments will be analysed in detail. As a result it will turn out that these arguments are not conclusive. Chihara's system relies on certain unjustified assumptions. Moreover, the philosophical difficulties that it encounters are not sufficiently discussed.

6

Antyrealistyczna ucieczka w sferę możliwości

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2008

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issue 42

15-27

PL

The article is devoted to a popular presentation of two important styles of thinking concerning the problem of existence of mathematical objects: Chihara's linguistic constructivism, and Hellman's modal structuralism. According to Chihara, mathematical statements should be interpreted as referring to certain linguistic construction; according to Hellman, mathematics is the science of possible structures. The motivations and main ideas are examined (without going into technical details), and the similarities and differences between these two viewpoints are highlighted.

7

The Status of the Continuum Hypothesis in the Light of Woodin’s Argumentation

100%

Wójtowicz K.

Filozofia Nauki

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2011

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vol. 19

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issue 4

67-82

PL

In the article, Woodin’s program (for setting up axioms, which decide the continuum hypothesis) is presented, and some philosophical aspects of it are discussed. In particular, the general problem of justifying axioms of set theory is discussed in the context of the relation between set theory and mainstream mathematics.

8

Ontological Reduction in Mathematics. Part I

100%

Wójtowicz K.

Filozofia Nauki

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2008

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vol. 16

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issue 3-4

105-118

PL

The article is the first part of a series of papers devoted to the problem of ontological reductions in mathematics – in particular, of choosing the basic category of mathematical entities. The received view is that such a category is provided by set theory, which serves as the ontological framework for the whole of mathematics (as all mathematical entities can be represented as sets). However, from the point of view of "naive mathematical realism" we should rather think of the mathematical universe as populated by a variety of diverse mathematical objects, and the set-theoretic reduction seems to be rather unnatural. In the first (introductory) part I discuss the general problem of providing an ontological foundation for mathematics.

9

Legal aspects of the United Kingdom’s withdrawal from the European Union

100%

Wójtowicz K.

Constitutional Review

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2017

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issue 2

PL

Following the electorate’s will expressed in the June 23, 2016 referendum, the European Union (Notification of Withdrawal) Act of 2017 was passed, and authorized ministers to notify under Article 50 TEU. This notification was given on March 29, 2017. The withdrawal of the United Kingdom from the European Union will result in significant changes in the system of the sources of law applied in the domestic legal order. Simply repealing European Communities Act of 1972 would leave large holes in the legal system making it incomplete. In order to avoid it the European Union (Withdrawal) Bill, known as the Great Repeal Bill, is proposed. The main purpose of this framework legislation is to convert directly-applicable EU laws into UK laws and to provide a power to use delegated legislation, when necessary, to rectify problems occurring as a consequence of the withdrawal.

10

Ontological Reductions in Mathematics. Part III: On Reconstruction of Some Parts of Mathematics

100%

Wójtowicz K.

Filozofia Nauki

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2011

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vol. 19

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issue 3

49-62

PL

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.

11

Abstract Model Theory – The Semantic Account of Logic

100%

Wójtowicz K.

Filozofia Nauki

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2015

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vol. 23

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issue 2

69-82

PL

The article presents main motivations underlying investigations into abstract model theory. It introduces basic technical notions and describes some philosophical problems associated with the theory.

12

Ontological Reductions in Mathematics. Part II: Argumentational Strategies for Realism

100%

Wójtowicz K.

Filozofia Nauki

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2011

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vol. 19

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issue 2

29-40

PL

This is the second part of the study concerning the problem of ontological reductions in mathematics. In this part, some major strategies of argumentation in favor of mathematical realism are presented. The versions to be considered are: Gödel’s realism, Quine’s quasi-empiricism and Balaguer’s Full-Blooded Platonism. Some introductory remarks considering the problem of ontological reductions in the context of these three stances are also presented.

13

Is Mathematics Indispensable in Science?

100%

Wójtowicz K.

Filozofia Nauki

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1994

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vol. 2

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issue 3-4

141-160

PL

This article consists of the two parts: the first on presents Hartry Field's nominalistic theory of science contained in his „Science Without Numbers”. The second part points to certain difficulties, which the realization of Field's program is faced with. The problem of an exact translation of nominalistic theories to mathematical theories, the connection between the incompleteness of the nominalistic theory and the conservativeness of its mathematical extension and an example of a theorem about finite sets, which needs some strong assumptions about infinite sets in its proof are presented.

14

Structuralizm versus Object Realism – A Genuine Argument?

100%

Wójtowicz K.

Filozofia Nauki

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2012

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vol. 20

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issue 2

115-128

PL

In the article, I discuss the differences between mathematical structuralism and object realism. I argue that they are partly only a matter of formulation and that some basic theses of both standpoints can be translated into the opponent’s language. I also indicate some problems of both these standpoints. From the point of view of mathematical structuralism, object realism has to face the problem of reference of mathematical terms. From the point of view of mathematical realism, mathematical structuralism relies on quite strong metaphysical assumptions. I also claim that structuralism also violates our intuitive understanding of some mathematical

15

Podstawowe założenia strukturalizmu w filozofii matematyki

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

|

2009

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issue 44

40-60

PL

The notion of a structure is one of fundamental notions in mathematics: we speak of geometrical, topological, probabilistic, differential etc. structures. This notion is also important in the philosophical discussion concerning ontology for mathematics. In the last decades, the stance of mathematical structuralism attracts more and more attention. In this article the author discusses the motivations which lie behind mathematical structuralism and briefly present Shapiro's 'ante rem' structuralism.

16

Oblicza matematycznego quasi-empiryzmu

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2009

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issue 44

61-83

PL

The received view concerning mathematics is the one, that mathematics is a priori, and that mathematical knowledge develops via 'intelektuelle Anschauung' rather than by analyzing empirical data. Mathematical proofs seems to be immune to empirical refutation, and in particular the development of mathematics does not in any way resemble the development of e.g. physics. On the other hand, it is quite clear, that mathematics play a fundamental role in science, and it is often considered to be rather just a useful tool, which provides a language and a conceptual system allowing to express statements concerning empirical world. Such views stress the dependence of mathematics upon physics. In the article, the author presents two quite different aspects of this problem: the ontological and the methodological aspects. According to Quine, our argumentation in favor of mathematical realism should be based on the analysis of ontological commitment of empirical theories. There is no other compelling argument for mathematical realism. According to Lakatos, mathematical knowledge develops in a way similar to empirical science: it is fallible, and the proper model to describe it is the model of proofs and refutations. In the article the author describes and contrast these two points of view.

17

Rozumienie dowodu matematycznego a zagadnienie wyjaśnienia w matematyce

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2015

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issue 58

89-114

EN

In the article, I present two possible points of view concerning mathematical proofs: (a) the formal view (according to which the formalized versions of mathematical proofs reveal their “essence”); (b) the semantic view (according to which mathematical proofs are sequences of intellectual acts, and a form of intuitive “grasp” is crucial). The problem of formalizability of mathematical proofs is discussed, as well as the problem of explanation in mathematics – in particular the problem of explanatory versus non-explanatory character of mathematical proofs. I argue, that this problem can be analyzed in a fruitful way only from the semantic point of view.

18

Kilka uwag o (meta)filozofii matematyki

100%

Wójtowicz K.

Zagadnienia Filozoficzne w Nauce

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2007

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issue 40

12-29

PL

The present essay deals with the problem of how to choose the correct method of doing philosophy of mathematics taking into account the importance of technical mathematical results for philosophical analysis. After a short historical introduction presenting the formation of the present mathematical paradigm, it is pointed out that the current mathematical praxis has, in principle, no connection with philosophical investigations. Two radically different approaches to philosophy of mathematics are outlined. Basing on selected examples it is argued that the correct method of doing philosophy of mathematics should take into account both technical results obtained by mathematicians (which often throw a new light on old philosophical questions) and the autonomy of philosophical method.

19

Empiricist Inspirations in Philosophy of Mathematics. Part II

100%

Wójtowicz K.

Filozofia Nauki

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2015

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vol. 23

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issue 4

71-87

PL

The article is devoted to the problem of mathematical empiricism: what could the phrase "to be a mathematical empiricist" mean? There are many interpretations of this position in philosophy of mathematics. The essay discusses relevant views of Quine and Putnam. Some detailed questions concerning e.g. the applicability of mathematics, the status of mathematical proof, the status of mathematical axioms are examined from the point of view of these philosophies.

20

On the Problem of Set-Theoretic Realism

100%

Wójtowicz K.

Filozofia Nauki

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1995

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vol. 3

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issue 4

113-130

PL

The paper is devoted to the problem of the existence of mathematical objects. The ideas of Godel and the Quine-Putnam indispensability argument are discussed. A „qualitative” version of this argument, in which the results of reverse mathematics are used, is presented.

Refine search results

Matematyka - nauka o fikcjach?

Dowód matematyczny – argumentacja czy derywacja? – część II

Dowód matematyczny – argumentacja czy derywacja? – część I

Empiricist Inspirations in Philosophy of Mathematics. Part I

Analysis of the Modal Antirealism

Antyrealistyczna ucieczka w sferę możliwości

The Status of the Continuum Hypothesis in the Light of Woodin’s Argumentation

Ontological Reduction in Mathematics. Part I

Legal aspects of the United Kingdom’s withdrawal from the European Union

Ontological Reductions in Mathematics. Part III: On Reconstruction of Some Parts of Mathematics

Abstract Model Theory – The Semantic Account of Logic

Ontological Reductions in Mathematics. Part II: Argumentational Strategies for Realism

Is Mathematics Indispensable in Science?

Structuralizm versus Object Realism – A Genuine Argument?

Podstawowe założenia strukturalizmu w filozofii matematyki

Oblicza matematycznego quasi-empiryzmu

Rozumienie dowodu matematycznego a zagadnienie wyjaśnienia w matematyce

Kilka uwag o (meta)filozofii matematyki

Empiricist Inspirations in Philosophy of Mathematics. Part II

On the Problem of Set-Theoretic Realism