The aim of the paper is to study the role and features of proofs in mathematics. Formal and informal proofs are distinguished. It is stressed that the main roles played by proofs in mathematical research are verification and explanation. The problem of the methods acceptable in informal proofs, in particular of the usage of computers, is considered with regard to the proof of the Four-Color Theorem. The features of in-formal and formal proofs are compared and contrasted. It is stressed that (...) the concept of an informal proof is not precisely defined, it is simply practised and any attempts to define it fail. It is — so to speak — a practical notion, psychological, sociological and cultural in character. The second one is precisely defined in terms of logical con-cepts. Hence it is a logical concept which is rather theoretical than practical in char-acter. The first one is — in part at least — semantical in nature, the second is entirely syntactical. A proof-theoretical thesis, similar to the Turing-Church Thesis in the re-cursion theory, is formulated. It says that both concepts of a proof in mathematics are equivalent. Arguments for and against it are formulated. (shrink)
W pracy rozważa się problem pojęcia prawdy w matematyce. Punktem wyjścia jest definicja prawdziwości Tarskiego. Dyskutuje się tło filozoficzne tej definicji, jej znaczenie dla języka matematyki i dla filozofii, stosunek do różnych definicji prawdy. Rozważa się też związek dowodliwości i prawdziwości w matematyce. Korzystając z wyników logiki matematycznej wykazuje się, że warunki z definicji Tarskiego nie zapewniają jedyności interpretacji predykatu prawdy. Pokazuje się też, że pojęcia semantyczne, takie jak spełnianie i prawdziwość nie są pojęciami finitystycznymi i wymagają użycia pojęcia nieskończoności.
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.
In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated.
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)
The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel?s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...) that are also considered. (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.
Zygmunt Ratajczyk was a deep and subtle mathematician who, with mastery, used sophisticated and technically complex methods, in particular combinatorial and proof-theoretic ones. Walking always along his own paths and being immune from actual trends and fashions he hesitated to publish his results, looking endlessly for their improvement.
Already after sending the first two parts of this paper (, ) 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.