On the constructivity of proofs. A debate among Behmann, Bernays, Gödel, and Kaufmann

Summary

Introduction The question ofwhether proofs by contradiction can be eliminated in favor of direct proofs has occupied the minds of many philosophers and mathematicians for centuries. Aristotle, Arnauld, Kant, Bolzano, and many others up to our day have tackled this topic. In the present paper I present the details of a debate centered around the relationship between constructivity and indirect proofs which took place in 1930 and involved some of the most able logicians and philosophers of mathematics of the century, namely Heinrich Behmann, Paul Bernays, Rudolf Carnap, Kurt Gödel, and Felix Kaufmann. The debate never made its way into print but it is still accessible in its entirety in documents preserved in the Kaufmann archive in Konstanz, the Bernays archive at the ETH in Zurich, and the Behmann archive in Erlangen. Related materials are also found in the Carnap archive in Pittsburgh. The Kaufmann archive in Konstanz contains, among other things, a large folder entitled Zur Frage der Konstruktivität von Beweisen which is of great interest to the historian of logic and the foundations of mathematics. Felix Kaufmann was an associate of the Vienna circle and he was in constant contact with, among others, Carnap, Gödel, Hahn, Waismann, Hempel, and Behmann. Most of the material we will discuss is found in this folder but supplementary materials from the other archives are used to fill in the picture when needed.

A conjecture by Kaufmann On October 24, 1930 Heinrich Behmann sent a letter to Paul Bernays, assistant editor of Mathematische Annalen, submitting a manuscript for publication entitled Zur Frage der Konstruktivität von Beweisen. He explained to Bernays the circumstances from which the paper originated:

[The paper] concerns the solution of a problem Herr Kaufmann suggested to me duringmy stay inVienna in September. Although I do not share the constructivist point of view I still deem such investigations important so that the practical import of the constructivist principles will not be overestimated and consequently great parts of mathematics be, unjustifiably, put into doubt.

He also added “Herr Carnap has already checked the proof and he agrees with it in every respect.”