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

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"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

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@dankwraith well go on, what's that mean in the broader context of his argument?

@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.

@prophet_goddess this is sort of lakatos's point, that we start with a definition that seems simple and reasonable but they can become complicated as the logical ideas reflect on them

@dankwraith yeah i mean there's no reason why a concept like "polyhedron" should refer to one thing over another other than that we all agree what "polyhedron" means. so it makes just as much sense to change a definition as it does to change an argument from a definition

@prophet_goddess yep! but ideal formalized axiomatic systems have no accounting for this, you're supposed to start from a definition and build up theorems from there. from a metamathematical perspective you are just not really supposed to do this, but the development of mathematics has lots of examples where this sort of thing happened

@dankwraith i am confused as to why someone felt the need to write a book about this because i consider it blindingly obvious

@prophet_goddess @dankwraith do mathematicians forget that formalized definitions start from informal rules governing the use of a term

@catalina @dankwraith love too pretend language has any necessary relationship to reality. i also love it when the map is the territory

@prophet_goddess this book was written in the 70s, everyone was still stoned on logical positivism

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