lakatos is talking about how people often deal with criticisms of their proofs by barricading their definitions against pathological cases

e.g.

"here is my lemma that you can divide a polyhedron's surfaces into a network of triangles and remove the triangles one at a time without changing the value of the sum of edges, faces, and vertices"

"well, what if the polyhedron was two pyramids connected at their narrow points?"

"oh, that's not actually a polyhedron, a polyhedron is [refines the definition to exclude this weird case]"

the interesting thing is, this isn't really how math is supposed to work! you're supposed to work from the definitions themselves

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i think the larger point that lakatos is driving at in this book is that the ways we use logic, which make logical sense, tend to escape our formalizations of logical processes

@dankwraith yeah that's a great point actually; that's fundamentally a rhetorical, not a logical move, and while appearing to be more precise it is actually, just, not.

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