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

mathematical concepts and definitions 259arise in mathematical practice as somehow ‘merely pragmatic’ questions of‘mathematical convenience’?⁴ Here too we can’t evaluate the importance ofmathematical practice for the metaphysical questions unless we get a bettersense of just what theoretical choices are involved. To make progress we needillustrations with enough meat on them to make clear how rich and intricatejudgements of naturalness can be in practice. The next two sections sketch twoexamples.9.3 Fruitfulness and stipulative definition:the Legendre symbolSpivak’s remark suggests that one of the criteria identifying ‘properly defined’terms is that they are fruitful, in that they support ‘trivial’ results with ‘significantconsequences’. It is an important part of the picture that the consequencesare ‘significant’. (Any theorem will have infinitely many consequences, fromtrivial inferences like A ⊢ A&A.) So what makes a consequence ‘significant’?I won’t consider everything here, but one will be especially relevant in thesequel: a consequence is held in practice to be significant if it contributesto addressing salient ‘why?’ questions. Evaluations of the explanatoriness ofarguments (theories, principles, etc.) and evaluations of the fruitfulness ofdefinitions (theories, principles, etc.) interact in ways that make them hard tosurgically separate. I’m not suggesting that considerations about explanationexhaust the considerations relevant to assessing whether or not a consequenceis significant or a concept fruitful because it has significant consequences. I’mjust making the mild observation that explanation is easier to nail down andbetter explored than other contributors to assessments of significance, so itis helpful as a benchmark. As a contrast, it is also common for proofs andprinciples to be preferred because they are viewed as more natural.⁵ However,⁴ This is a pivotal argumentative support in Sider (1996) to cite just one example. Discussion of thearguments would go beyond this survey, so I’ll leave it for the companion article.⁵ For example, many number theorists count the cyclotomic proof as particularly natural. (Frölichand Taylor, 1991, p.204) opine that this proof is ‘most natural’ (or rather: ‘ ‘‘most natural’’ ’). Similarly,in the expository account of Artin’s life by Lenstra and Stevenhagen (2000, p.48) we read: ‘Artin’sreciprocity law over Q generalizes the quadratic reciprocity law and it may be thought that itsmysteries lie deeper. Quite the opposite is true: the added generality is the first step on the wayto a natural proof. It depends on the study of cyclotomic extensions.’ (p. 48). Gauss, on the otherhand, though one of his proofs exploits cyclotomy, preferred a more direct argument using whatis now called ‘Gauss’s lemma’. Of other proofs he wrote: ‘Although these proofs leave nothing tobe desired as regards rigor, they are derived from sources much too remote... I do not hesitate tosay that until now a natural proof has not been produced.’ (Gauss, 1808, p.1). Gauss might have