VI.28 Bernard Bolzano

b. Prague, 1781; d. Prague, 1848
Catholic priest and Professor of Theology, Prague (1805-19)

Bolzano was concerned with problems connected with finding the “correct,” or the most appropriate, proofs and definitions in analysis and related areas. In 1817 he proved an early version of the intermediate value theorem for continuous functions—he was among the first to have a rigorous conception of a continuous function—and in the course of doing so proved the following important lemma. If a property M does not apply to all values of a variable x but does apply for all values smaller than a certain u, then there is always a quantity U, which is the greatest of those of which it can be asserted that all smaller x possess the property M. The value u in this formulation is a lower bound for the (nonempty) set of numbers with the property not-M. Bolzano’s lemma is therefore equivalent to what nowadays might be called the “greatest lower bound” axiom (or, more commonly and equivalently, the “least upper bound” axiom). It is also equivalent to the Bolzano-Weierstrass theorem (that every bounded infinite set in , or more generally ⁿ, has an accumulation point). It is likely that WEIERSTRASS [VI.44] independently rediscovered the Bolzano-Weierstrass theorem, but it is also likely that he knew, and was influenced by, Bolzano’s proof technique of iterated bisection (used by Bolzano in 1817).

In the early 1830s it was widely believed that a continuous function must be differentiable except at some isolated points. But at that time Bolzano constructed a counterexample (although he did not publish it), and proved that it was such—more than thirty years before the well-known counterexample due to Weierstrass.

Bolzano had a surprising variety of insights and successful proof techniques that were well ahead of their time: notably in analysis, topology, dimension theory, and set theory.