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1

All the mathematics in the world: logical validity and classical set theory

100%

McCarty D. C.

Zagadnienia Filozoficzne w Nauce

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2017

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issue 63

5-29

EN

A recognizable topological model construction shows that any consistent principles of classical set theory, including the validity of the law of the excluded third, together with a standard class theory, do not suffice to demonstrate the general validity of the law of the excluded third. This result calls into question the classical mathematician's ability to offer solid justifications for the logical principles he or she favors.

2

Filozofia i logika intuicjonizmu

94%

Fila M.

Semina Scientiarum

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2015

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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.

3

Kryzys podstaw matematyki przełomu XIX i XX wieku

94%

Fila M.

Semina Scientiarum

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2014

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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.

4

Special Issue on point-free geometry and topology. An introduction

84%

Coppola C., Gerla G.

Logic and Logical Philosophy

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2013

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vol. 22

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issue 2

139-143

EN

In the first section we briefly describe the methodological assumptions of point-free geometry and topology. We also outline the history of geometrical theories based on the notion of a region. The second section is devoted to a concise presentation of the content of the LLP special issue on point-free theories of space.

5

What are the limits of mathematical explanation? Interview with Charles McCarty by Piotr Urbańczyk

67%

McCarty D. C., Urbańczyk P.

Zagadnienia Filozoficzne w Nauce

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2016

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issue 60

119-137

EN

An interview with Charles McCarty by Piotr Urbańczyk concerning mathematical explanation.

6

Wokół intuicjonizmu metaetycznego Mariana Przełęckiego

67%

Jedynak A.

Edukacja Filozoficzna

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2019

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issue 67

63-81

PL

Choć Marian Przełęcki uważał poznanie moralne za nienaukowe, poglądy na nie modelował według swoich wcześniejszych poglądów na poznanie w naukach empirycznych. Nauka wychodzi od zdań jednostkowych, opartych na poznaniu zmysłowym i doświadczeniu, a następnie je indukcyjnie uogólnia, dochodząc do zasad. Poznanie moralne zaś wychodzi od jednostkowych ocen, których źródłem jest intuicja czy zmysł moralny, po czym uogólnia je indukcyjnie do postaci zasad etycznych. Swoje stanowisko metaetyczne nazwał zatem indukcjonistycznym intuicjonizmem. Niniejszy tekst przedstawia trzy refleksje, do jakich motywuje przemyślenie tego stanowiska: 1. Granica między intuicjonizmem a naturalizmem jest nieostra. Co więcej, wbrew rozpowszechnionemu przekonaniu i przekonaniu samego Przełęckiego, każde z tych stanowisk musi zawierać jakąś przymieszkę drugiego. 2. Istnieją pewne interesujące analogie (wykraczające poza projekt Przełęckiego) między złudzeniami zmysłowymi a złudzeniami intuicji moralnej. 3. A także analogie, dotyczące relacji, w jakie rozum wchodzi z jednej strony z poznaniem zmysłowym, a z drugiej – z intuicyjnym poznaniem moralnym.

EN

Although Marian Przełęcki considered moral cognition as non-scientific, he shaped his views on its sources and structure according to his earlier views on cognition in natural sciences. The starting point in the latter are individual sentences based on experience and sensual cognition, which are then generalized as empirical principles or laws. On the other hand, the starting point in moral cognition are individual assessments rooted in intuition or moral sense, next generalized inductively as moral principles. Therefore, Przełęcki named his metaethical views an inductive intuitionism. The paper presents three reflections evoked by those views: 1. In metaethics, the border between intuitionism and naturalism is vague. Moreover, in spite of the common view and Przełęcki’s conviction, each of those two concepts must include an admixture of the other. 2. There are some interesting analogies (going beyond Przełecki’s project) between sensual illusions and moral-sense illusions. 3. As well as there exist analogies dealing with the relations between the reason and, on the one hand, empirical cognition, and on the other hand, intuitive moral cognition.

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