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2 Przegląd Filozoficzny. Nowa Seria

2 2008

2 Wojtowicz K.

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Why No Possible World is Mathematical

100%

Wojtowicz K.

2008

tom 17

nr 2(66)

59-77

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.

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

75%

Wojtowicz K.

2008

tom 17

nr 4(68)

281-293

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