Blind commitment to a theory is not an intellectual virtue: it is an intellectual crime.

The proving power of the intellect or the senses was questioned by the skeptics more than two thousand years ago; but they were browbeaten into confusion by the glory of Newtonian physics.

Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.

The history of mathematics, lacking the guidance of philosophy, [is] blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, is empty.

That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.

Research programmes, besides their negative heuristic, are also characterized by their positive heuristic.

There is no falsification before the emergence of a better theory.