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

32 marcus giaquintothe internal angles of a triangle sum to two right angles provides an example.Norman (2006) makes a strong case that following this proof requires visualthinking, and that the visual thinking is not replaceable by non-visual thinking.Returning to our visual route to the formula for triangular numbers, we cancheck that the method works for all positive integers after the first, usingvisual reasoning to assure ourselves that it works for 2 and that if the methodworks for k it works for k + 1. Together with this reflective thinking, thevisual thinking sketched earlier constitutes following a proof of the formulafor the nth triangular number for all integers n > 1, at least if the relaxed viewof thinking through a proof is correct. In some cases, then, the danger ofunwarranted generalization in visual reasoning can be overcome.1.3 DiscoveringThough philosophical discussion of visual thinking in mathematics has concentratedon its role in proof, visual thinking more often shows its worth indiscovery than in proof. By ‘discovering’ a truth I will mean coming to believeit by one’s own lights (as opposed to reading it or being told) in way that isreliable and involves no violation of epistemic rationality (given one’s epistemicstate). Priority is not the point: discovering something, in this sense, entailsseeing it for oneself rather than seeing it before anyone else. The differencebetween merely discovering a truth and proving it is a matter of transparency:for proving or following a proof the subject must be aware of both the wayin which the conclusion is reached and the soundness of that way; this is notrequired for discovery.The oldest and best known discussion of visual discovery is to be found inPlato’s Meno (82b–86b). Using the diagram of Fig. 1.3, it appears quite easy toFig. 1.3.