Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

24 marcus giaquintowhere that excludes what is involved in the constructive phase, such as gettingthe main ideas for a proof, but covers thinking through the steps in a proof,either for the first time or following a given proof, in such a way that thesoundness of the argument is apparent to the thinker.1.2 ProvingWe should distinguish between a proof and a presentation of a proof. A proofcan be presented in different ways. How different can distinct presentationsbe and yet be presentations of the same proof? There is no context-invariantanswer to this, and even within a context there may be some indeterminacy.Usually mathematicians are happy to regard two presentations as presentingthe same proof if the central idea is the same in both cases. But if one’s mainconcern is with what is involved in thinking through a proof, its central ideais not enough to individuate it: the overall structure, the sequence of stepsand perhaps other factors affecting the cognitive processes involved will berelevant. Even so, not every cognitive difference in the processes of followinga proof will entail distinctness of proofs: in some cases, presumably, the samebits of information in the same order can be given in ink and in Braille.Once individuation of proofs has been settled, we can distinguish betweenreplaceable thinking and superfluous thinking. In the process of thinkingthrough a proof, a given part of the thinking is replaceable if thinking of someother kind could stand in place of the given part in a process that would countas thinking through the same proof. A given part of the thinking is superfluousif its excision without replacement would be a process of thinking through thesame proof. Let us agree that there can be superfluous diagrammatic thinkingin thinking through a proof, thinking which serves merely to facilitate orreinforce understanding of the text. This leaves several possibilities.(a) All thinking that involves a diagram in thinking through a proof issuperfluous.(b) Not all thinking that involves a diagram in thinking through a proofis superfluous; but if not superfluous it will be replaceable by nondiagrammaticthinking.(c) Some thinking that involves a diagram in thinking through a proof isneither superfluous nor replaceable by non-diagrammatic thinking.The negative view stated earlier that diagrams can have no role in proofentails claim (a). The idea behind (a) is that, because visual reasoning is