Penrose can't argue for his hypothesis
No scientist can either establish or refute the hypothesis that mathematical insight is non-algorithmic / noncomputable (see detailed text).
- An empirically decidable property must be "a A2 property in the Borel hierarchy". Computability and noncomputability lack that property. Therefore, the hypothesis that mathematical intuition is non-computable is empirically undecidable.
- Penrose's ability to recognise mathematical truths by insight is consistent both with the hypothesis that insight is algorithmic and the hypothesis that it isn't. So, Penrose's intuitions fail to establish his hypothesis one way or other.
Clark Glymour and Kevin Kelly (1990).