# Lecture 5: Bernard Bolzano

Bernard Bolzano (1781-1848) was not only a fine Logician, he also spear headed
real analysis. He is usually credited for freeing calculus from the concept of infinitesimal
but his work on calculus only was recognized after his death. He anticipated for example
the concept of Cauchy sequence years before Cauchy and anticipated the concept of cardinality
before Cantor. He can be seen as a quite modern ethical philolospher seeing not so much doctrine
or Kant's imperative as fundamental but a more pragmatic utilitarinism. He also is an early
researcher in education and pedagogy like investigating the methodology of textbooks or to
investigate the theory of science.
There are three important theorems of him. They are all related:
Theorem 1: Extremal Value Theorem: A continuous function on an interval [a,b] has a maximum.
Theorem 2: Intermediate Value theorem: A continuous function on [a,b] with f(a) f(b) ≤ 0 has a root in (a,b).
Theorem 3: Bolzano-Weierstrass: an infinite set of numbers in [a,b] has an accumulation point.