After Cantor first demonstrated that the real numbers (continuum) were uncountable, the hypothesis arose that the set of the real numbers was "the smallest" uncountable set. In 1900 David Hilbert made settling the continuum hypothesis the first problem on his now famous list of problems for this century. The author will discuss some of the historical, philosophical, and mathematical  developments connected to this problem proceeding from issues of definition of the real numbers and proofs of uncountability to issues of consistency and models and proofs of the independence of this hypothesis and possibly some comments on its current status. (Received September 14, 2001)

       Outline of possible Discussion (depending on time allowed).
       I. "Greek" Views - Control early Second Millenium.

       II. Pre-Calculus and Early Calculus Views

       III. The Age of Development and Conventions
       Euler:1707-1783
       Bolzano 1781- 1848 (1817 paper)
       Cauchy:1789-1857
       Dirichlet: 1805-1859

       IV. The Age of Critical Awareness and Foundations
       Weierstrass: 1815- 1897
       Dedekind: 1831-1916
       Frege: 1848-1925
       Peano: 1858-1932

       V. Cantor 1845-1918: Investigation of discontinuities with Fourier series and Set Theory Beginnings.

       VI. The XXth Century: An Age of Exploration and Discovery.
Hilbert: (1862-1943)
Brouwer: (1881-1966)
Godel: (1906-1978)
Cohen: (1934- )
Dana Scott/ Solovay

       VII.? The XXIst Century: The Hypothesis is False?

Now onto a slightly more detailed ouline.