my forty years on his shoulders - Department of Mathematics

38However, many mathematical logicians, particularly those in set theory,take a quite different view. This includes Kurt Gödel. They take the view that thecontinuum hypothesis is a well defined mathematical assertion with a definitetruth value. The problem is to determine just what this truth value is.The idea here is that there is a definite system of objects that existsindependently of human minds, and that human minds can no more manipulatethe truth value of statements of set theory than they can manipulate the truthvalue of statements about electrons and stars and galaxies.This is the so called Platonist point of view that is argued so forcefully andexplicitly in (Gödel 1947,1964).The late P.J. Cohen led a panel discussion at the Gödel Centenary calledOn Unknowability, where he conducted a poll roughly along these lines. Thequestion he asked was, roughly, “does the continuum hypothesis have a definiteanswer”, or “does the continuum hypothesis have a definite truth value”.The response from the audience appeared quite divided on the issue.Of the panelists, the ones who have expressed very clear views on thistopic were most notably Cohen and Woodin. Cohen took a formalist viewpoint,whereas Woodin takes a Platonist one. See their respective contributions to thisvolume.My own view is that we simply do not know enough in the foundations ofmathematics to decide the truth or appropriateness of the formalist versus thePlatonist viewpoint - or, for that matter, what mixture of the two is true orappropriate.But then it is reasonable to place the burden on me to explain what kind ofadditional knowledge could be relevant for this issue.