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.

     

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...     

An X Windows tool for synthesizing face images from eigenvectors

An X Windows software tool for the construction of faces with a weighted combination of eigenvectors is described. The eigenvectors were extracted from an autoassociative matrix that comprised 100 face images. The program input consists of eigenvectors and sets of weights that describe individual faces and combines these to create face images. The tool creates a panel of buttons that permits the display of individual eigenvectors and the display of an average face as well. Facilities for on-line changes to the intensity of individual eigenvectors can be used to change the appearance of a face. Previously, O’Toole, Abdi, Deffenbacher, and Bartlett (1991) have shown that the intensity of certain individual eigenvectors contains reliable information for determining the sex and race of the face.     

Perceived motion in structure from motion: Pointing responses to the axis of rotation

We investigated the ability to match finger orientation to the direction of the axis of rotation in structure-from-motion displays. Preliminary experiments verified that subjects could accurately use the index finger to report direction. The remainder of the experiments studied the perception of the axis of rotation from full rotations of a group of discrete points, the profiles of a rotating ellipsoid, and two views of a group of discrete points. Subjects’ responses were analyzed by decomposing the pointing responses into their slant and tilt components. Overall, the results indicated that subjects were sensitive to both slant and tilt. However, when the axis of rotation was near the viewing direction, subjects had difficulty reporting tilt with profiles and two views and showed a large bias in their slant judgments with two views and full rotations. These results are not entirely consistent with theoretical predictions. The results, particularly for two views, suggest that additional constraints are used by humans in the recovery of structure from motion.     

Positive and negative mood following imaging-guided core needle breast biopsy and receipt of biopsy results

Positive and negative mood are independent psychological responses to stressful events. Negative mood negatively impacts well-being and co-occurring positive mood leads to improved adjustment. Women undergoing core needle breast biopsies (CNB) experience distress during CNB and awaiting results; however, influences of mood are not well known. This longitudinal study examines psychosocial and biopsy- and spirituality-related factors associated with mood in patients day of CNB and one week after receiving results. Ninety women undergoing CNB completed questionnaires on psychosocial factors (chronic stress, social support), biopsy experiences (pain, radiologist communication), and spirituality (peace, meaning, faith) day of CNB. Measures of positive and negative mood were completed day of CNB and one week after receiving results (benign n = 50; abnormal n = 25). Multiple linear regression analyses were conducted. Greater positive mood correlated with greater peace (β = .25, p = .02) day of CNB. Lower negative mood correlated with greater peace (β = ?.29, p = .004) and there was a trend for a relationship with less pain during CNB (β = .19, p = .07). For patients with benign results, day of CNB positive mood predicted positive mood post-results (β = .31, p = .03) and only chronic stress predicted negative mood (β = .33, p = .03). For women with abnormal results, greater meaning day of CNB predicted lower negative mood post-results (β = ?.45, p = .03). Meaning and peace may be important for women undergoing CNB and receiving abnormal results.     

The Spiritual and Theological Challenges of Stillbirth for Bereaved Parents

Journal of Religion and Health - Stillbirth is recognized as one of the most challenging experiences of bereavement raising significant spiritual and theological questions. Semi-structured...