Bernard Bolzano (1781-1848, Prague) was a remarkable thinker and reformer far ahead of his time in many areas, including philosophy, theology, ethics, politics, logic, and mathematics.Aimed at historians and philosophers of both mathematics and logic, and research students in those fields, this volume contains English translations, in most cases for the first time, of many of Bolzano's most significant mathematical writings.These are the primary sources for many of his celebrated insights and anticipations, including: clear topological definitions of various geometric extensions; an effective statement and use of the Cauchy convergence criterion before it appears in Cauchy's work; proofs of the binomial theorem and the intermediate value theorem that are more general and rigorous than previous ones; an impressive theory of measurable numbers (a version of real numbers), a theory of functions including theconstruction of a continuous, non-differentiable function (around 1830); and his tantalising conceptual struggles over the possible relationships between infinite collections.Bolzano identified an objective and semantic connection between truths, his so-called 'ground-consequence' relation that imposed a structure on mathematical theories and reflected careful conceptual analysis. This was part of his highly original philosophy of mathematics that appears to be inseparable from his extraordinarily fruitful practical development of mathematics in ways that remain far from being properly understood, and may still be of relevance today.
Bernard Bolzano (1781-1848, Prague) was a remarkable thinker and reformer far ahead of his time in many areas, including philosophy, theology, ethics, politics, logic, and mathematics.Aimed at historians and philosophers of both mathematics and logic, and research students in those fields, this volume contains English translations, in most cases for the first time, of many of Bolzano's most significant mathematical writings.These are the primary sources for many of his celebrated insights and anticipations, including: clear topological definitions of various geometric extensions; an effective statement and use of the Cauchy convergence criterion before it appears in Cauchy's work; proofs of the binomial theorem and the intermediate value theorem that are more general and rigorous than previous ones; an impressive theory of measurable numbers (a version of real numbers), a theory of functions including theconstruction of a continuous, non-differentiable function (around 1830); and his tantalising conceptual struggles over the possible relationships between infinite collections.Bolzano identified an objective and semantic connection between truths, his so-called 'ground-consequence' relation that imposed a structure on mathematical theories and reflected careful conceptual analysis. This was part of his highly original philosophy of mathematics that appears to be inseparable from his extraordinarily fruitful practical development of mathematics in ways that remain far from being properly understood, and may still be of relevance today.