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I intend to write a paper that will address among other issues the informal communication between mathematicians. My point of origin is the view that every proof can be represented by a sequence of gestalt switches, each described by, sometimes idiosyncratic, metaphor. I further believe that mathematicians develop gestalt and metaphors when processing proofs and that the Gestalt & metaphors they generate constitute the arsenal used in their research. I conjecture that because their understanding of the subject matter rests on combinations of gestalt and metaphors, these are reflected in their informal discussions.

Thanks. Hope I made myself clear.

It is enough if you just say that you have witnessed occasion when in a discussion of research mathematics a mathematician was using metaphors, gestures and the like.

Arik
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    Yeah: Compactness! ^^ – C-star-W-star May 09 '14 at 08:31
  • For example: a bump function is a compactly supported smooth function into the reals ...compactness is only important when building distributions however in general compactness is counterproductive as when proving localness where exactly the opposite is needed – C-star-W-star May 09 '14 at 09:01
  • Or another one: Mathematicians talk about nowhere dense sets. Then one might guess there is somewhere dense and everywhere dense sets but when working out these things bearing in mind Baire's category theorem one notices that this gave actually the wrong intuition as this is not the crucial point there. – C-star-W-star May 09 '14 at 09:11
  • Linear independence is also an example as this is not the crucial point in building bases – C-star-W-star May 09 '14 at 09:13
  • Measurable functions being the analogue of continuous functions cool yes but when establishing integrals of Banach space valued functions one encounters the problem that measurability is not a good idea since the situation is measure space to topological space ...Walter Rudin even goes so far to define measurability not completely analogues to continuity but in this more fitted sense – C-star-W-star May 09 '14 at 09:16
  • @Freeze_S I have no idea what you mean by "compactness is only important when..." No, that's just false. "One might guess there is somewhere dense and everywhere dense sets" Yes, and there are, so the guess is accurate. Your cryptic comment about Baire category seems wrong as well. Anyway, your comments do not seem to address the question at all. "Linear independence is also an example". An example of what? Of gestalt and metaphors when developing proofs? – Andrés E. Caicedo May 10 '14 at 14:18
  • Oh im sorry I meant compactness can be counterproductive. I doubt in no way that compactness is one of the strongest tools in mathematics over all branches since it allows to give sense to issues of infinite expressions by translating them to finite ones. ...Terry Tao even devotes a whole book on compactness - very nice btw ;) – C-star-W-star May 10 '14 at 15:35
  • As to denseness. Guess: what is somewhere dense? Take the closure and then the interior points are those where a set is dense according to the definition used in the framework of baires category theorem. But taking the complement as it is considered as complementary in baires category theorem is, well, not an everywhere dense set but sth absolutely unrealted to the notion of somewhere dense... So thats why metaphor here is really misleading! – C-star-W-star May 10 '14 at 15:40

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