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1

THE COMPUTER EXPERIMENT - THE SOURCE OF MATHEMATICAL KNOWLEDGE

100%
Wojtowicz K.
Kwartalnik Filozoficzny
|
2008
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vol. 36
|
issue 1
87-104
EN
In the article, the status of the computer proof of the four-color theorem is discussed. After a short survey of contemporary literature, the general status of the mathematical proof (as an idealized logical procedure versus a part of mathematical practice) is discussed. The solution of the problem of the computer proof, which amounts to accepting the four-color theorem as a new axiom justified by the computer experiment, is considered (and rejected). Finally, the author argues that the best explanation is offered within the theoretical framework of quasi-empiricism.
2

Why No Possible World is Mathematical

100%
Wojtowicz K.
|
2008
|
vol. 17
|
issue 2(66)
59-77
EN
The notion of possible mathematical world is discussed. The problem is analyzed from the point of view of two classical realistic stances in the philosophy of mathematics (Quine's realism and Gödel's Platonism), and from the point of view of Balaguer's full-blooded Platonism. Balauger's stance seems to be compatible with the use of the notion of a possible mathematical world (a universe), but as a matter of fact it is not. If one adopts such a notion, several profound philosophical problems arise, concerning e.g. the criterion of identity, the problem of the 'borders of mathematicity', the problem of singling out the actual world from a possible one. In conclusion the author claims that the notion of possible world is not clear enough to be used in ontological discussions concerning mathematics.
3

Philosophy of Mathematics

100%
Wojtowicz K.
Przegląd Filozoficzny. Nowa Seria
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2006
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vol. 15
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issue 4(60)
155-168
EN
In this article, J.St.Mill's philosophy of mathematics is presented and discussed. First, Mill's views concerning geometry (which - in his opinion - is a general science concerning physical space) is presented. Knowledge of geometrical principles is obtained via inductive generalizations of our observations of physical phenomena. A short discussion follows whose conclusion is that Mill's explanation of the nature of geometrical knowledge is not satisfactory. Nevertheless, his ideas are of certain importance in the philosophy of mathematics, as Mill can be considered a predecessor of the quasi-empiricist philosophy of mathematics of Putnam and Quine.
4

W.V.O. Quine's Philosophy of Mathematics Confronted with Henkin and Boolos Quantifiers

100%
Wojtowicz K.
|
2008
|
vol. 17
|
issue 4(68)
281-293
EN
The article contains a brief presentation of Quine's arguments in favor of metamathematical realism and raises the problem of the dependence of ontological commitments of a theory on 'ideological' assumptions concerning the range of logical concepts. The problem is discussed by reference to Henkin and Boolos quantifiers. If these quantifiers are chosen and introduced as semantically primitive terms, fully understandable without recourse to set theoretical semantics and paraphrases presented as logical sentences with function variables and set variables, the resulting theory makes different ontological commitments from those that are made by Quine. The difference is illustrated by discussing a theory proposed by Hellman
5
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THE OVERSIGHT FUNCTION OF THE SEJM IN RELATION TO EUROPEAN INTEGRATION

100%
Wojtowicz K.
Przegląd Sejmowy
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2008
|
issue 3(86)
EN
The oversight function of the Sejm in matters of European integration should be understood widely as it covers both the stage of delegation to the European Union the powers typical of the sovereign authority of the State, as well as the exercise of those powers within a particular separate institutional structure. As concerns the first aspect, the Sejm (sometimes jointly with the Senate) takes part in the procedure of granting consent for ratification of the accession treaty itself, or ratification of possible modifications introduced to treaties establishing the European Union. Thereby, the Sejm, as part of Polish parliament participates in the oversight of the intensity and pace of integration, as well as its territorial extent. Moreover, the exercise of oversight on the way in which the Union uses the powers conferred on her, should take into account the specific nature of that organization, manifested in the exercise of the lawmaking function (in general) by an international organ - the Council of the European Union. For that reason, the influence of national parliaments on decisions taken within the Union is mainly determined by the degree to parliaments are able to oversee the activities of government within the framework of appropriate institutions. The Sejm exercise oversight over the government in a form specified in the Cooperation Act. The Sejm (its European Affairs Committee) may, in particular, be provided with information and documents, including legislative proposals of the EU and to express adequate opinions which, however, do not obligate the government to take any position within the EU organ. Additionally, the Sejm (its European Affairs Committee) gives its opinion about the candidates nominated by the government for certain posts in the EU. In the event that the Lisbon Treaty enters into force, the scope of oversight function of the Sejm (parliament) will be considerably extended, particularly in relation to the observance of the subsidiarity by the Union. The exercise by the Sejm (parliament) of the oversight function, within the above-said meaning, contributes to strengthening of the democratic basis of integration and legitimizes the process of making decisions in the European Union.
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