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Philosophy

*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.