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

The great scientific achievements are research programmes which can be evaluated in terms of progressive and degenerative problemshifts; and scientific revolutions consist of one research programme superceding (overtaking in progress) another. This methodology offers a new rational reconstruction of science.

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.

Intellectual honesty consists in stating the precise conditions under which one will give up one's belief.

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

Philosophy of science without history of science is empty; history of science without philosophy of science is blind.

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.