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

320 jeremy avigadIn that respect, logic does not tell us the whole story:Logic teaches us that on such and such a road we are sure of not meeting anobstacle; it does not tell us which is the road that leads to the desired end. (Ibid.,pp. 129–130)Philosophers of science commonly distinguish between the ‘logic of justification’and the ‘logic of discovery’. Factors that guide the process of discoveryalso fall under the general category of ‘understanding’, and, indeed, understandingand discovery are often linked. For example, understanding a proofmay involve, in part, seeing how the proof could have been discovered; or, atleast, seeing how the train of inferences could have been anticipated.It seems to me, then, as I repeat an argument I have learned, that I could havediscovered it. This is often only an illusion; but even then, even if I am not cleverenough to create for myself, I rediscover it myself as I repeat it. (Ibid., PartI,Chapter III, p. 50)While knowing the relevant definitions may be enough to determine that aproof is correct, understanding is needed to find the definitions that make itpossible to discover a proof. Poincaré characterized the process of discovery,in turn, as follows:Discovery consists precisely in not constructing useless combinations, but in constructingthose that are useful, which are an infinitely small minority. Discoveryis discernment, selection. (Ibid., p.51)These musings provide us with some helpful metaphors. Mathematicspresents us with a complex network of roads; understanding helps us navigatethem, and find the way to our destination. Mathematics presents us witha combinatorial explosion of options; understanding helps us sift throughthem, and pick out the ones that are worth pursuing. Without understanding,we are lost in confusion, wandering blindly, unable to cope. When we domathematics, we are like Melville’s sailors, swimming in a vast expanse. Just asthe sailors cling to sides of their ship, we rely on our understanding to guideus and support us.12.2 Understanding and abilityLet us see if we can work these metaphors into something more definite. Onething to notice is that there seems to be some sort of reciprocal relationshipbetween mathematics and understanding. That is, we speak of understanding