The purpose of this lecture, and of the lectures that will follow it, is a controversial one. It is, plainly and simply, to reopen some questions in the foundations of probability, and thus also a much touchier and much neglected subject, the foundations of statistics. There seems to have been a conspiracy of silence on such foundational questions among mathematicians (not so among philosophers), who would like to believe that the subject has been definitively closed by Kolmogorov, and many of whom react unfavorably to any suggestion of revision of Kolmogorov's elegant setup.