The classical example of a successful research programme is Newton's gravitational theory: possibly the most successful research programme ever.

No experimental result can ever kill a theory: any theory can be saved from counterinstances either by some auxiliary hypothesis or by a suitable reinterpretation of its terms.

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

In degenerating programmes, however, theories are fabricated only in order to accommodate known facts

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.

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