The hundred pages of that remarkable essay ring changes on a single geometrical theme: Euler’s law that the faces and vertices of a polyhedron together outnumber the edges by two. After explaining the classical proof, Lakatos produces an exception: a hollow solid whose surfaces are a cube within a cube. Its faces and vertices outnumber its edges by four. Then he examines the classical proof to see how it falls foul of such examples, and what stipulations would be suitable for excluding them. Having thus narrowed the scope of Euler’s law, he produces a further exception: a solid consisting of two tetrahedra with only an edge or vertex in common. A further tightening of the law is thus indicated, and still the exceptions are not at an end. A polyheron with a square tunnel through it occasions a further restriction; a cube with a penthouse on top occasions yet a further restriction; and so the dialectic of revision and exception goes its oscillating way.
The geometry is fascinating, but the purspose is philosophical. Lakatos is opposing the formalists’ conception of mathematical proofs, which represents them as effectively testable and, once tested, incontrovertible. He is opposing the notion, so central to logical positivism, that mathematics and natural science are methodologically unlike.
Taken from W. V. Quine review of Imre Lakato’s Proofs and Refutations, 1977.