The Mathematical Works of Bernard BolzanoOUP Oxford, 9 déc. 2004 - 732 pages 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 the construction 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. |
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Expressions et termes fréquents
absolute value actual number already angle arbitrary assumption axiom become as small become smaller belongs Bernard Bolzano binomial binomial series binomial theorem Bolzano certainly concept consequence considered continuous function Corollary decrease indefinitely definition denominator denote derivative determined difference distance dx dy equal equation example exponent expression F(x+Ax finite follows function Fx geometry given quantity greater greatest holds infinite multitude infinite number infinitely large infinitely small judgements kind latter law of continuity limits magnitude mathematics measurable number measuring fraction merely negative increase number of terms pair points positive number positive or negative possible Proof proper fraction proposition proved quotient rational relationship remains represents respect side similar small as desired spatial object straight line surface taken Taylor's theorem Theorem theory things translation triangle truth values of x variable zero
Fréquemment cités
Page xxiv - While content and language form a certain unity in the original, like a fruit and its skin, the language of the translation envelops its content like a royal robe with ample folds
Page 7 - by proposition in itself I mean any assertion that something is or is not the case, regardless whether or not somebody has put it into words, and regardless even whether or not it has been thought.