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

266 jamie tappendenfunctions ...) The upshot of the general investigation is a collection of generaltheories regarded by mathematicians (and hobbyists) of a range of differentbackgrounds and subspecialties as explaining the astonishing connection betweenarbitrary odd primes. The judgement that the Legendre symbol carves at ajoint interacts with a delicate range of mathematical facts and judgements:verified conjectures, the practice of seeking explanations and understanding,efforts to resolve more general versions of known truths (and the evaluationsof ‘proper’ generalizations that support this practice), judgements of similarityand difference, judgements about what facts would be antecedently expected(quadratic reciprocity not among them), and more.The history of quadratic reciprocity also illustrates the importance ofinductive reasoning in mathematics. Euler conjectured the law decadesbefore Gauss proved it, on the basis of enumerative induction from cases.²²This issue will be revisited in the research article, so I’ll just notethe key point.²³ In many cases, the natural/artificial distinction is linkedto projectibility: Natural properties (‘green’) support correct predictions andartificial ones (‘grue’) don’t. It is common, as in the influential work ofSydney Shoemaker (1980a, b), to connect this with a thesis about causality.Simplifying Shoemaker’s picture drastically: natural properties are those thatenter into causal relations, and it is because of this that only natural propertiessupport induction properly. Euler and quadratic reciprocity reveal alimit to this analysis: induction, as a pattern of reasoning, does not dependfor its correctness on physical causation. The properties supporting Euler’scorrect inductive reasoning have the same claim to naturalness deriving fromprojectibility that ‘green’ has. This is consistent with the observation thatmathematical properties don’t participate in causation as Shoemaker understandsit. Even though there is much more about inductive reasoning inmathematics that we need to understand better, and we should have dueregard for the differences between mathematical and empirical judgements,we shouldn’t underestimate the affinities.Delicate issues of identity and difference of content also arise, as in Dedekind’sproof of quadratic reciprocity in (Dedekind, 1877/1996). Dedekind describeshimself as presenting ‘essentially the same as the celebrated sixth proof ofGauss’ (from Gauss, 1817). The derivation recasts the treatment of cyclotomicextensions in section 356 of (Gauss, 1801). Dedekind is plausibly describedas presenting the same argument in a conceptual form avoiding most of the²² Edwards (1983), Cox (1989, pp. 9–20), and Weil (1984) are excellent treatments of the inductivereasoning that led Euler to his conjecture. There is a particularly beginner-friendly discussion at theonline Euler archive .²³ For more on plausible reasoning in mathematics see Jeremy Avigad’s first article.