Proving quadratic reciprocity: explanation,disagreement, transparency and depth

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.

     

Altruistic Eudaimonism and the Self-Absorption Objection

The Journal of Value Inquiry -     

Nonsuicidal Self-injury,Suicide Planning,and Suicide Attempts Among High-risk Adolescents Prior to Psychiatric Hospitalization

Research on Child and Adolescent Psychopathology - The purpose of this study was to understand the trajectories of nonsuicidal self-injury (NSSI) and suicide plans (SP) in the 90 days...     

Review of Julie Chajes,Recycled Lives: A History of Reincarnation in Blavatsky’s Theosophy,Oxford, Oxford University Press, 2019, xii + 215 p., Hardcover,ISBN 978-0-19-090913-0, £64

Studies in East European Thought - This is a review of Julie Chajes, Recycled Lives: A History of Reincarnation in Blavatsky’s Theosophy, Oxford University Press, Oxford, 2019. The book,...     

Military Stressors,Parent-Adolescent Relationship Quality,and Adolescent Adjustment

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.

     

Grandfamilies and the Opioid Epidemic: A Systemic Perspective and Future Priorities

Clinical Child and Family Psychology Review - As a result of the devastating impact of the opioid epidemic, increased numbers of children are being raised by their grandparents in what are known as...     

Réflexions critiques sur les notions d’abolition et d’altération de la faculté à discerner

Inasmuch as the constitution of the mens rea of any offence remains suspended to its integrity, the capacity for discernment represents a condition of accountability whose definition responds to issues that are as much technical – with respect to psychiatric and psychological penal expertise – as they are theoretical – with respect to our conception of criminal responsibility and the psychological prerequisites that underlie it. Now, the fact is that since its introduction in 1992 with article 122–1 of the Penal Code, this notion of discernment is characterized by a certain conceptual inconsistency, giving rise to a damaging plurality of meanings making the verb discern the condensed version of a heterogeneous network of distinct psychological functions. The polysemic term of discernment thus gives rise to a fundamental indetermination as to the nature of the faculty that must be examined in order to decide on its abolition or its preservation at the moment of the act. This is not without favouring the appearance and persistence of arbitrary interpretations, and consequently discordant expert conclusions. Even more problematic, however, is the idea of an alteration of discernment, given the irreducible discrepancy between the continuous nature of an alteration and the discontinuous nature of the conclusion to which the agent is subject, in that the legal demand requires a decision – in a discretization of what is continuous – between the presence and absence of something that is lacking. We will in fact show in what way this concept of alteration as it applies to discernment tends to be either meaningless, or redundant – and therefore useless – in relation to that of abolition. So much so that the only conceivable solution to maintain a certain gradation of degrees of non-accountability seems to consist in having to specify the different types of abolition of discernment potentially observed, and then to propose a hierarchization according to their greater or lesser compatibility with the conservation of a criminal responsibility. .