{"page_number":173,"title":"Page 173","overview":"This page, part of \"The Britannica Guide to Analysis and Calculus,\" provides biographical and intellectual summaries of two significant mathematicians: Bernard Bolzano and Luitzen Egbertus Jan Brouwer. It details their key published works, philosophical stances, and contributions to the foundations of mathematics.","text_summary":"The page is divided into two main sections, each dedicated to a prominent mathematician:\n\n1.  **Bernard Bolzano**: The first section lists several of Bolzano's influential published works, providing both their original German titles and English translations, along with their publication years. These works cover a range of topics including \"The Binomial Theorem\" (1816), \"Pure Analytic Proof\" (1817), \"Functions Model\" (1834), \"Scientific Model\" (1834, in 4 volumes), \"An Attempt at a New Presentation of Logic\" (1837, in 4 volumes), and \"Paradoxes of Infinity\" (1851).\n\n2.  **Luitzen Egbertus Jan Brouwer**: This section provides a more detailed account of Brouwer's life and work. Born in 1881 and deceased in 1966, Brouwer is identified as a Dutch mathematician and the founder of mathematical intuitionism. The text explains intuitionism as a doctrine that views mathematics as mental constructions derived from self-evident laws, emphasizing the study of fundamental properties of geometric surfaces and configurations. Brouwer studied mathematics at the University of Amsterdam from 1897 to 1904 and showed a strong interest in philosophical matters, which was evident in his doctoral thesis, \"Leven, Kunst, en Mystiek\" (1905), translated as \"Life, Art, and Mysticism.\" His subsequent thesis, \"Over de grondslagen der wiskunde\" (1907), or \"On the Foundations of Mathematics,\" saw him challenge the logical foundations of mathematics, particularly the efforts of David Hilbert and Bertrand Russell. Brouwer was a key figure in shaping the intuitionist school, notably rejecting the principle of the excluded middle (or excluded third) in mathematical proofs. He argued that this principle, which states every mathematical statement is either true or false, does not apply to infinite sets.","content_markdown":"# Page 173\n\n### Page Overview\nThis page, part of \"The Britannica Guide to Analysis and Calculus,\" provides biographical and intellectual summaries of two significant mathematicians: Bernard Bolzano and Luitzen Egbertus Jan Brouwer. It details their key published works, philosophical stances, and contributions to the foundations of mathematics.\n\n### Text Content Summary\nThe page is divided into two main sections, each dedicated to a prominent mathematician:\n\n1.  **Bernard Bolzano**: The first section lists several of Bolzano's influential published works, providing both their original German titles and English translations, along with their publication years. These works cover a range of topics including \"The Binomial Theorem\" (1816), \"Pure Analytic Proof\" (1817), \"Functions Model\" (1834), \"Scientific Model\" (1834, in 4 volumes), \"An Attempt at a New Presentation of Logic\" (1837, in 4 volumes), and \"Paradoxes of Infinity\" (1851).\n\n2.  **Luitzen Egbertus Jan Brouwer**: This section provides a more detailed account of Brouwer's life and work. Born in 1881 and deceased in 1966, Brouwer is identified as a Dutch mathematician and the founder of mathematical intuitionism. The text explains intuitionism as a doctrine that views mathematics as mental constructions derived from self-evident laws, emphasizing the study of fundamental properties of geometric surfaces and configurations. Brouwer studied mathematics at the University of Amsterdam from 1897 to 1904 and showed a strong interest in philosophical matters, which was evident in his doctoral thesis, \"Leven, Kunst, en Mystiek\" (1905), translated as \"Life, Art, and Mysticism.\" His subsequent thesis, \"Over de grondslagen der wiskunde\" (1907), or \"On the Foundations of Mathematics,\" saw him challenge the logical foundations of mathematics, particularly the efforts of David Hilbert and Bertrand Russell. Brouwer was a key figure in shaping the intuitionist school, notably rejecting the principle of the excluded middle (or excluded third) in mathematical proofs. He argued that this principle, which states every mathematical statement is either true or false, does not apply to infinite sets.\n\n### Visual Elements (Diagrams, Figures, Graphs, Portraits, Illustrations)\n*No visual elements on this page.*","has_visuals":0,"visual_count":0,"visuals":[]}