Kant, Gödel, and the problem of synthetic a priori judgementsAbstract
Debates over Kant’s famous postulate about the existence of synthetic a priori judgements in mathematics, formulated in the Critique of Pure Reason, have been raging for over two centuries. On the one hand, it was fiercely criticised by neo-positivists in the early 20th century. On the other hand, Kant’s ideas on constructive nature of mathematics served as a philosophical framework for LEJ Brouwer’s programme of intuitionism in the foundations of mathematics. Of interest are the ideas of the great logician and mathematician Kurt Gödel about the analytical nature of mathematics, put forward in a number of his works on philosophy of mathematics. Although he never mentions synthetic a priori judgements, the course of his reasoning about analytical judgements is close to that employed by Kant. As early as the mid-20th century, Gödel’s incompleteness theorems and the works of Church and Turing constituted arguments in favour of the existence of synthetic a priori judgements. The American logician Irving Copi was the first to use Gödel’s first incompleteness theory to that end. While his small work went almost unnoticed, such ideas were expressed by at least two other mathematicians. In modern mathematics, particularly, Martin-Löf type theory, the existence of synthetic a priori truths, is considered justified. Although it is based on different grounds than those mentioned above, it is nevertheless compatible with Gödel’s results. Analysing proofs of existence of synthetic a priori judgements helps demonstrate that a solution to this problem is determined by the implicitly or explicitly accepted image of logic, whose key parameter is the object of logic or, in other worlds, the ideas about the nature of the logical and, therefore, the ideas about the boundaries of logic and mathematics.