Bolzano's Analytic Proof of the Intermediate Value Theorem Bernhard Bolzano (17811848)
After the rapid advances in calculus and its applications in the 18th Century by the Bernoullis, Euler, Lagrange, Lacroix and others, it became apparent through contradictions and paradoxes that there were gaps in the foundations. For example, functions that could not be expressed as power series, series which did not converge, Maclaurin series which converged but not to the function from which they were derived, series of continuous functions whose limit was not continuous, functions which were continuous on a domain but not differentiable anywhere, branching of solutions to differential equations, definition of derivative for functions with complex domains and so on. It was clear that the most important lacks were firstly, a logically coherent definition of limit, and secondly, criteria for convergence of arbitrarily defined series. Behind these of course was a need for an axiomatic definition of real numbers with which to work. One of the first to attack this problem was Bernhard Bolzano, a Czech priest of Italian descent who was Professor in the Philosophy of Religion at the University in Prague, then part of the Austrian empire. The Chair had been expressly established by the Emperor Franz I to counter the spread of enlightenment in Europe following the French Revolution, but he chose the wrong incumbent. Bolzano spread his own enlightened ideas in his lectures. He was eventually dismissed and even arrested on suspicion of heresy. His philosophical training attracted him to questions about the foundations of mathematics, including infinity and its paradoxes, and the properties of real numbers. | |
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