Beth an Bernays on Intuitionism

What is common to Beth’s and Bernays’ reflections about intuitionism concerns its philosophical aspects, in particular its basic notion : mental evidence as something irrefutable and fixed forever. At the biginning Bernays, as a collaborator of Hilbert’s, considered intuitionism too extremistic since it assigned a foundational role exclusively to (mental) evidence, neglecting the role of abstraction. Later, when he approached Gonseth’s epistemology, he even stated that evidence as something fixed forever cannot exist. What is evident can vary when the “horizon of experience” varies : evidences are acquired. On his side Beth at first believed that intuitionism was the only reliable foundational school and used this fact to defend Kant’s epistemology. Later he started to doubt the evidence of natural numbers and more and more came to believe that in general evidence is not reliable. In 1950 he enlarged on his reflection about evidence by including it among the postulates of the Aristotelian theory of science, deducibility and reality being the other postulates. All of these are unreliable. As Kant shared the Aristotelian theory of science, Beth concluded to the necessity of abandoning Kant’s philosophical system. Furthermore, as Kant’s and Brouwer’s thought had led him to underestimate logic, Beth felt the need to re-evaluate logic and to devote himself to it (and obtained many interesting metatheorical results). Beth and Bernays had direct exchanges of ideas about the notion of evidence. In 1943, when Bernays still believed in a limited philosophical role for evidence, Beth wrote him that, although some evidences in mathematics may exist, it is very difficult to express them in an unexceptionable way, and that language itself contributes to make concepts evident. Later, in 1958, when Bernays did not believe in evidence at all, Beth, starting from a philosophical analysis of evidences, shared with Bernays the idea of acquired evidences. Finally, Beth stressed that in the literature “intuition” is often confused with “evidence”. He recognized the presence in mathematics of a “creative intuition”, of a “global intuition” and of an “intuition of the infinite”. The existence of intuition was also supported by the Löwenheim-Skolem paradox and had as a consequence that reality is not a unique block but is built by various spheres (logic being one of them). The relationship between the spheres was described by Beth with reference to Bernays’ notion of complementarity.