In 1955, Paul Lorenzen Clears the Sky in Foundations of Mathematics for Hermann Weyl

Source abstract

Abstract In 1955, Paul Lorenzen is devoting all his research to the foundations of mathematics. After having provided a proof of consistency for arithmetic in 1944, he inquires still further and arrives at the conviction that analysis can also be given a predicative foundation. Wilhelm Ackermann, as well as Paul Bernays, have pointed out to him in 1947 that his views are very close to those proposed by Hermann Weyl in Das Kontinuum (1918): Sets are not postulated to exist beforehand; they are being generated in an ongoing process of comprehension. This seems to be the reason for Lorenzen to get into contact with Weyl, who develops a genuine interest in Lorenzen’s operative mathematics and welcomes with great enthusiasm his Einführung in die operative Logik und Mathematik (1955), whose aim is to grasp the objects of analysis by means of inductive definitions. 167This mathematical kinship is brutally interrupted by Weyl’s death in 1955; a planned visit by Lorenzen at the Institute for Advanced Study in Princeton takes place only in 1957‒1958. In the following years, Lorenzen abandons language levels and simplifies his presentation of analysis by distinguishing only between “definite” and “indefinite” quantifiers: The former govern domains for which a proof of consistency is available and secures the use of the law of excluded middle; the latter govern those for which there isn’t, for example, the real numbers. Lorenzen states in his foreword to Differential und Integral (1965) that he is faithful to Weyl’s approach of Das Kontinuum in this simplification.