Must all mathematicians be platonists?: a case on Penrose's use of Gödel

Abstract

Roger Penrose has received broad attention for his arguments with the conclusion “the human mind is nonalgorithmic”. From this position he concludes that a Quantum Gravitational Theory must be nonalgorithmic. In his writings Penrose discusses Gödel's famous incompleteness sentence and his epistemological position he calls Mathematical Platonism. In our article we reconstruct the implicit logical structure of Penrose's argumentation with the following results: First, we show that his conclusion “the human mind is nonalgorithmic” can be obtained if both Gödel's sentence and Mathematical Platonism are taken as premises. Second, we show that Penrose originally derives his conclusion solely from Gödel's sentence, using a certain interpretation that differs from Gödel's own. Third, it is shown that in both cases the practice of mathematics would prescribe a certain epistemology, namely Mathematical Platonism. We argue that Penrose did not recognize the constitutional, indispensable function of Mathematical Platonism for his considerations and the ensuing consequences for the epistemological state of his whole argument.