Online research in philosophy

The aim of this paper is to present and discuss the philosophical views concerning mathematics of the founders of the so called Warsaw Mathematical School, i.e., Wacław Sierpiński, Zygmunt Janiszewski and Stefan Mazurkiewicz. Their interest in the philosophy of mathematics and their philosophical papers will be considered. We shall try to answer the question whether their philosophical views influenced their proper mathematical investigations. Their views towards set theory and its rôle in mathematics will be emphasized.

Review of Douglas Patterson. Alfred Tarski: Philosophy of Language and Logic.

The aim of the paper is to present the main trends and tendencies in the philosophy of mathematics in the 20th century. To make the analysis more clear we distinguish three periods in the development of the philosophy of mathematics in this century: (1) the first thirty years when three classical doctrines: logicism, intuitionism and formalism were formulated, (2) the period from 1931 till the end of the fifties - period of stagnation, and (3) from the beginning of the sixties (...) till today when new tendencies putting stress on the knowing subject and the research practice of mathematicians arose. (shrink)

The author shows in his article how the awareness of the difference between truth and provability in mathematics has developed. He points out the role played in this process by Gödel's results concerning incompleteness of formalised theories and also indicates the attempts at overcoming these limitations by giving up the finitistic condition and by allowing infinitary methods in the notion of mathematical proof. The philosophical assumptions that one accepts are important for the problem under discussion. For strict formalists and intuitionists (...) the problem of distinguishing between truth and proof does not exist at all. For them a mathematical statement is true if it is provable, where proofs are considered to be our own constructions - syntactic or mental. The situation is entirely different for the proponents of platonism (realism) in the philosophy of mathematics. It can be said that it is just the platonist approach to mathematics that made it possible for Gödel to both pose the problem and to understand and show the difference between provability and truth. (shrink)

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.

Already after sending the first two parts of this paper ([5], [6]) to the editor, two new results on the subject have appeared — namely the results of G. Wilmers and Z. Ratajczyk. So for the sake of completeness let us review them here.