Mathematics
Bolzano's mathematical teachings were not quite understood by his contemporaries, and most of his deep insights into the foundations of mathematical analysis long remained unrecognized. A famous theorem in the early stages of a modern presentation of the calculus is known as the Bolzano-Weierstrass theorem, but another masterful anticipation (by more than forty years) of Karl Theodor Wilhelm Weierstrass's discovery that there exist functions that are everywhere continuous but nowhere differentiable remained buried in manuscripts until the 1920s. But perhaps more important than Bolzano's actual discoveries of new theorems was the meticulousness with which he endeavored to lay new foundations for the Grössenlehre, the science of quantity—which was how Bolzano, using a very broad interpretation of "quantity," designated mathematics. In particular, his insistence that no appeal to any intuition of space and time should be acknowledged for this purpose and that only "purely analytical" methods were to be recognized put him in opposition to the then current Kantian ways of thinking and back into the Leibnizian tradition.
Bolzano's most famous posthumously published work is Paradoxien des Unendlichen (F. Prihonsky, ed., Leipzig, 1851; translated by D. A. Steele as The Paradoxes of the Infinite, London, 1950), in which he anticipated certain basic ideas of set theory, developed only a generation later by Georg Cantor, who fully acknowledged his indebtedness to Bolzano in this respect.
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