Polish mathematicians and mathematics in World War I: Part II. Russian Empire

In the second part of our article we continue presentation of individual fates of Polish mathematicians (in a broad sense) and the formation of modern Polish mathematical community against the background of the events of World War I. In particular we focus on the situations of Polish mathematicians in the Russian Empire (including those affiliatedwith the University of Warsaw, reactivated by Germans, and the Warsaw Polytechnics, founded already by Russians) and other countries.In the second part of our article we continue presentation of individual fates of Polish mathematicians (in a broad sense) and the formation of modern Polish mathematical community against the background of the events of World War I. In particular we focus on the situations of Polish mathematicians in the Russian Empire (including those affiliatedwith the University of Warsaw, reactivated by Germans, and the Warsaw Polytechnics, founded already by Russians) and other countries. Polscy matematycy i polska matematyka w czasach I wojny światowej. Część II. Cesarstwo Rosyjskie Abstrakt W drugiej części artykułu kontynuujemy przedstawianie indywidualnych losów matematyków polskich (w szerokim sensie) oraz kształtowanie się nowoczesnego polskiego środowiska matematycznego na tle wydarzeń pierwszej wojny światowej. W szczególności skupiamy się na sytuacji matematyków polskich w Cesarstwie Rosyjskim (także tych związanych z reaktywowanym przez Niemców Uniwersytetem Warszawskim i utworzoną jeszcze przez Rosjan Politechniką Warszawską) i innych krajach

The Problem of Intuition in Mathematics in the Thoughts and Creativity of Selected Polish Mathematicians in the Context of the Nineteenth-Century Breakthrough in Mathematics

In the article, I examine the presence and importance of intuitive cognition in mathematics. I show the occurrence of mathematical intuition in four contexts: discovery, understanding, justification, and acceptance or rejection. I will deal with examples from the history of mathematics, when new mathematical theories were being created (the end of the nineteenth and the beginning of the twentieth century will be particularly important, including the period of establishing the Polish mathematical school). I will also refer to the research (mainly) of Polish philosophers and mathematicians in this field. The goal of the article is also an attempt to understand the breakthrough that took place in mathematics at the turn of the nineteenth century. The analysis also shows, by highlighting the specifics of intuition and mathematical creativity, the difficulties that arise when acquiring new concepts and mathematical arguments. Research goes in the direction of deepening research on the very phenomenon of intuition in cognition, by pointing to the universal nature of mathematical intuition.In the article, I examine the presence and importance of intuitive cognition in mathematics. I show the occurrence of mathematical intuition in four contexts: discovery, understanding, justification, and acceptance or rejection. I will deal with examples from the history of mathematics, when new mathematical theories were being created (the end of the nineteenth and the beginning of the twentieth century will be particularly important, including the period of establishing the Polish mathematical school). I will also refer to the research (mainly) of Polish philosophers and mathematicians in this field. The goal of the article is also an attempt to understand the breakthrough that took place in mathematics at the turn of the nineteenth century. The analysis also shows, by highlighting the specifics of intuition and mathematical creativity, the difficulties that arise when acquiring new concepts and mathematical arguments. Research goes in the direction of deepening research on the very phenomenon of intuition in cognition, by pointing to the universal nature of mathematical intuition

Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

Foundations of Mathematics and Mathematical Practice. The Case of Polish Mathematical School

The foundations of mathematics cover mathematical as well as philosophical problems. At the turn of the 20th century logicism, formalism and intuitionism, main foundational schools were developed. A natural problem arose, namely of how much the foundations of mathematics influence the real practice of mathematicians. Although mathematics was and still is declared to be independent of philosophy, various foundational controversies concerned some mathematical axioms, e.g. the axiom of choice, or methods of proof (particularly, non-constructive ones) and sometimes qualified them as admissible (or not) in mathematical practice, relatively to their philosophical (and foundational) content. Polish Mathematical School was established in the years 1915–1920. Its research program was outlined by Zygmunt Janiszewski (the Janiszewski program) and suggested that Polish mathematicians should concentrate on special branches of studies, including set theory, topology and mathematical logic. In this way, the foundations of mathematics became a legitimate part of mathematics. In particular, the foundational investigations should be conducted independently of philosophical assumptions and apply all mathematically accepted methods, finitary or not, and the same concerns other branches of mathematics. This scientific ideology contributed essentially to the remarkable development of logic, set theory and topology in Poland

Philosophy of exact sciences (logic and mathematics) in Poland in 1918–1939

This paper describes the philosophy of logic and mathematics in Poland in the years 1918‒1939. The special attention is attributed to the views developed in the Polish Mathematical School and the Warsaw School of Logic. The paper indicates various differences between mathematical circles in Warszawa, Lvov and Kraków

Philosophical reflection on mathematics in Poland in the interwar period

AbstractIn the paper the views and tendencies in the philosophical reflection on mathematics in Poland between the wars are analyzed. Views of most outstanding representatives of Lvov–Warsaw Philosophical School and of Polish Mathematical School are presented. Their influence on logical and mathematical researches is considered