Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos (1977) is a delightful book that I clearly need to read again (and again; many tnx to GdM for recommending it some years ago). Lakatos explores, through parable and dialog, the sources of truth in mathematics: why definitions necessarily evolve over time, which forces drive the search for theorems, and where good hypotheses come from. It's all done via a meditation on Euler's Formula, a deceptively simple relationship between the number of edges, vertices (corners), and faces (sides) for a solid polyhedron: Vertices - Edges + Faces = 2. (Try it for various examples: a cube has 8 vertices, 12 edges, and 6 faces, and 8-12+6=2; a classic pyramid has 5 vertices, 8 edges, and 5 faces, and 5-8+5=2; etc.)

But that formula breaks down sometimes — when shapes have holes in them, for instance, or if a polyhedron is made of pieces stuck together in certain funny ways. Precisely why the theorem breaks down, and how to repair and extend it, are key themes of Lakatos's book. And the same principles of knowledge and creativity apply to areas outside of mathematics.

"Seek simplicity and distrust it," as Alfred North Whitehead famously said. And as mothers everywhere have admonished, "Clean your plate!" — a metaphor for one of Lakatos's precepts, that one must always strive to get as much as possible from the inputs to a theorem. Waste not, want not ...

(see also GreatIdeas (3 May 1999), ComplexityFromSimplicity (5 Aug 1999), ScienceAndPseudoscience (6 Oct 2001), NoFinalAnswers (11 Mar 2002), HighPrecision (16 Aug 2002), ... )

TopicScience - TopicPhilosophy - TopicLiterature - 2004-06-24

(correlates: IdeaGardening, BeUnprepared, AirFlow, ...)