On Development of the Awareness of the Difference between Truth and Provability in Mathematics

Abstract

The author shows in his article how the awareness of the difference between truth and provability in mathematics has developed. He points out the role played in this process by Gödel's results concerning incompleteness of formalised theories and also indicates the attempts at overcoming these limitations by giving up the finitistic condition and by allowing infinitary methods in the notion of mathematical proof. The philosophical assumptions that one accepts are important for the problem under discussion. For strict formalists and intuitionists the problem of distinguishing between truth and proof does not exist at all. For them a mathematical statement is true if it is provable, where proofs are considered to be our own constructions - syntactic or mental. The situation is entirely different for the proponents of platonism (realism) in the philosophy of mathematics. It can be said that it is just the platonist approach to mathematics that made it possible for Gödel to both pose the problem and to understand and show the difference between provability and truth.