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  1. 2 Semina Scientiarum

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  1. 2 Fila M.

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  2. 1 2014

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Filozofia i logika intuicjonizmu

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Fila M.
Semina Scientiarum
|
2015
|
vol. 14
EN
At the end of the 19th century in the fundamentals of mathematics appeared a crisis. It was caused by the paradoxes found in Cantor’s set theory. One of the ideas a resolving the crisis was intuitionism – one of the constructivist trends in the philosophy of mathematics. Its creator was Brouwer, the main representative was Heyting. In this paper described will be attempt to construct a suitable logic for philosophical intuitionism theses. In second paragraph Heyting system will be present – its axioms and matrices truth-. Later Gödel theorem about the inadequacy of finite dimensional matrices for this system will be explained. At the end this paper an infinite sequence of matrices adequate for Heyting axioms proposed by Jaśkowski will be described.
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Kryzys podstaw matematyki przełomu XIX i XX wieku

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Fila M.
Semina Scientiarum
|
2014
|
vol. 13
EN
By the end of the 19th century, mathematics had become very intensively developed. Mathematical logic became an independent discipline, and in the 1880s Cantor published his work on set theory. All this led to questions about the consistency of mathematical theories and decidability theorems. Therefore, for the second time in the history of mathematics, there emerged a crisis of the basis of mathematics.There were a few ideas for overcoming the crisis. In this paper, there will be described three trends in the philosophy of mathematics in the late 19th and early 20th centuries: logicism (Frege), intuitionism (Brouwer) and formalism (Hilbert). These three trends were described from the philosophical point of view and in the context of the crisis. Moreover, for each of them there will be present the most important methodological assumptions, and I will briefly describe attempts to achieve them. This will describe the problem in such a way that allows for the grasping of important differences and similarities between logicism, intuitionism and formalism and better understand their causes.
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