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Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late eighteenth century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.
相似文献Elements of military life can create challenges for all family members, including military-connected adolescents, and can have detrimental consequences for their adjustment. Although research with samples of military-connected adolescents has examined the influences of military stressors for adolescent adjustment (e.g., depressive symptoms, anxiety), less research has identified possible mechanisms responsible for these effects, particularly the role of specific familial factors. Drawing from social ecological theory and attachment theory, we examined the associations between military stressors (e.g., parental rank, combat deployments, permanent change of station moves) and self-reported adolescent adjustment (e.g., depressive symptoms, self-efficacy) along with examining adolescents’ perceptions of parent-adolescent relationship quality with both the active duty and civilian parent as a linking mechanism. Using a path analysis, data from 265 Army families were examined to identify the direct and indirect associations between military stressors and adolescent adjustment through parent-adolescent relationship quality. Most military stressors were not significantly related to relationship quality of either parent or indicators of adolescent adjustment. However, parent-adolescent relationship quality with each parent (active duty and civilian parent) was uniquely related to adolescents’ adjustment. Discussion is provided regarding how military stressors and familial factors are conceptualized within the context of military families and implications for future research, family therapy, and policies are suggested.
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