Polish mathematicians proved invaluable during the Second World War. In 1932 three young mathematicians, Marian Rejewski, Jerzy Ró¿ycki, and Henryk Zygalski, started out on a joint project with the Cryptology Office of the Polish intelligence service and broke the code of the German Enigma encoding machine; however, the Germans continued to make improvements to the machine. In July 1939 the Poles shared their information on the Enigma machine with the Allies. After the outbreak of the War, all three mathematicians managed to escape with two copies of the machine to Rumania, and then to France. Thanks to their work, an Englishman named Alan Mathison Turing constructed a very fast and precise machine known as the Colossus to decode the Enigma's messages.

Did you know that... The Polish mathematician Wac³aw Sierpiñski was a leading topologist who served as Vice-President of the International Mathematical Union.

A group of Polish scholars made outstanding contributions in the field of logic. At the turn of the 19th and 20th centuries they founded the Lwów and Warsaw School of Logic, led by Kazimierz Twardowski. It was the third such school in the world, after the Vienna Circle and the school of Franz Brentano. Kazimierz Twardowski introduced a fundamental differentiation between the content and the subject of presentations. Alfred Tarski developed a concept of truth that was adopted by the Vienna Circle. Kazimierz Ajdukiewicz demonstrated the existence of certain categories of concepts which are mutually untranslatable and was the creator of what is known as the "Categorical Grammar"; this was later applied in the development of machines to translate texts from one language to another. Tadeusz Kotarbiñski's philosophy negated the existence of mental subjects and events, i.e., there is no rain, there are only raindrops that are falling, which is to say that in reality, only things exist. He described his own epistemological approach as "radical realism". During the interwar period Warsaw was one of the "logic capitals" of the world, as the outstanding German logician Heinrich Scholz once said.

Did you know that... Franciszek Leja developed the concept of the continuous group, one of the fundamental ideas in modern mathematics.

Stefan Banach is one of the best-known Polish mathematicians. He was one of the creators of functional analysis, and the author of Théorie des opérations linéaires (1932), the first work to elaborate the general theory of linear spaces. Jerzy Neyman-Sp³awa, a pioneer in modern statistics, worked with Ronald Fisher and Egon Pearson. His work dealing with confidence intervals, hypothesis testing, and survey sampling revolutionised the field. Stanis³aw Ulam (1909-1984), who devised the Monte Carlo method, was also interested in similar problems. This method uses statistical sampling to solve mathematical problems. He is nevertheless better known for his work for the American project to create the hydrogen bomb. He is also known for his work with J. C. Everett on the Orion plan for nuclear-powered space vehicles. Zenon Mróz is also well-known abroad, particularly for his work describing the processes of  material deformation; major companies, such as for example General Motors, use this information to calculate the metal fatigue in automobile components. His work has also been used in the construction of nuclear reactors, drilling platforms and earthquake-resistant buildings. Professor Zenon Mróz is a consultant for the American company Mechanical Testing Systems.

"There is probably no country which has contributed, relative to the size of its population, so much to mathematical logic and set theory as Poland." Abraham Fraenkel and Yehoshuah Bar-Hillel, Foundations of Set Theory (1958)   "Thanks to Jan £ukasiewicz, over the last ten years Poland has become the main country for the academic study of logic, and Warsaw has become its main centre." Professor Heinrich Scholz, Historia logiki (1931)