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Mathematics
Bolzano made several original contributions to mathematics.
His overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics (Boyer 1959, pp. 268?269).
To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works 論文1 (1810), 論文2(1816) and 論文3 (1817).
These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years later when they came to the attention of Karl Weierstrass (O'Connor & Robertson 2006).
To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε-δ definition of a mathematical limit.
Bolzano, like several others of his day, was skeptical of the possibility of Gottfried Leibniz's infinitesimals, that had been the earliest putative foundation for differential calculus.
Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals,
must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity.
Bolzano also gave the first purely analytic proof of the fundamental theorem of algebra, which had originally been proven by Gauss from geometrical considerations.
He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano's theorem).
Today he is mostly remembered for the Bolzano?Weierstrass theorem,
which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered (Boyer & Merzbach 1991, p. 561).